1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

1001

1002

1003

1004

1005

1006

1007

1008

1009

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

1084

1085

1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

1101

1102

1103

1104

1105

1106

1107

1108

1109

1110

1111

1112

1113

1114

1115

1116

1117

1118

1119

1120

1121

1122

1123

1124

1125

1126

1127

1128

1129

1130

1131

1132

1133

1134

1135

1136

1137

1138

1139

1140

1141

1142

1143

1144

1145

1146

1147

1148

1149

1150

1151

1152

1153

1154

1155

1156

1157

1158

1159

1160

1161

1162

1163

1164

1165

1166

1167

1168

1169

1170

1171

1172

1173

1174

1175

1176

1177

1178

1179

1180

1181

1182

1183

1184

1185

1186

1187

1188

1189

1190

1191

1192

1193

1194

1195

1196

1197

1198

1199

1200

1201

1202

1203

1204

1205

1206

1207

1208

1209

1210

1211

1212

1213

1214

1215

1216

1217

1218

1219

1220

1221

1222

1223

1224

1225

1226

1227

1228

1229

1230

1231

1232

1233

1234

1235

1236

1237

1238

1239

1240

1241

1242

1243

1244

1245

1246

1247

1248

1249

1250

1251

1252

1253

1254

1255

1256

1257

1258

1259

1260

1261

1262

1263

1264

1265

1266

1267

1268

1269

1270

1271

1272

1273

1274

1275

1276

1277

1278

1279

1280

1281

1282

1283

1284

1285

1286

1287

1288

1289

1290

1291

1292

1293

1294

1295

1296

1297

1298

1299

1300

1301

1302

1303

1304

1305

1306

1307

1308

1309

1310

1311

1312

1313

1314

1315

1316

1317

1318

1319

1320

1321

1322

1323

1324

1325

1326

1327

1328

1329

1330

1331

1332

1333

1334

1335

1336

1337

1338

1339

1340

1341

1342

1343

1344

1345

1346

1347

1348

1349

1350

1351

1352

1353

1354

1355

1356

1357

1358

1359

1360

1361

1362

1363

1364

1365

1366

1367

1368

1369

1370

1371

1372

1373

1374

1375

1376

1377

1378

1379

1380

1381

1382

1383

1384

1385

1386

1387

1388

1389

1390

1391

1392

1393

1394

1395

1396

1397

1398

1399

1400

1401

1402

1403

1404

1405

1406

1407

1408

1409

1410

1411

1412

1413

1414

1415

1416

1417

1418

1419

1420

1421

1422

1423

1424

1425

1426

1427

1428

1429

1430

1431

1432

1433

1434

1435

1436

1437

1438

1439

1440

1441

1442

1443

1444

1445

1446

1447

1448

1449

1450

1451

1452

1453

1454

1455

1456

1457

1458

1459

1460

1461

1462

1463

1464

1465

1466

1467

1468

1469

1470

1471

1472

1473

1474

1475

1476

1477

1478

1479

1480

1481

1482

1483

1484

1485

1486

1487

1488

1489

1490

1491

1492

1493

1494

1495

1496

1497

1498

1499

1500

1501

1502

1503

1504

1505

1506

1507

1508

1509

1510

1511

1512

1513

1514

1515

1516

1517

1518

1519

1520

1521

1522

1523

1524

1525

1526

1527

1528

1529

1530

1531

1532

1533

1534

1535

1536

1537

1538

1539

1540

1541

1542

1543

1544

1545

1546

1547

1548

1549

1550

1551

1552

1553

1554

1555

1556

1557

1558

1559

1560

1561

1562

1563

1564

1565

1566

1567

1568

1569

1570

1571

1572

1573

1574

1575

1576

1577

1578

1579

1580

1581

1582

1583

1584

1585

1586

1587

1588

1589

1590

1591

1592

1593

1594

1595

1596

1597

1598

1599

1600

1601

1602

1603

1604

1605

1606

1607

1608

1609

1610

1611

1612

1613

1614

1615

1616

1617

1618

1619

1620

1621

1622

1623

1624

1625

1626

1627

1628

1629

1630

1631

1632

1633

1634

1635

1636

1637

1638

1639

1640

1641

1642

1643

1644

1645

1646

1647

1648

1649

1650

1651

1652

1653

1654

1655

1656

1657

1658

1659

1660

1661

1662

1663

1664

1665

1666

1667

1668

1669

1670

1671

1672

1673

1674

1675

1676

1677

1678

1679

1680

1681

1682

1683

1684

1685

1686

1687

1688

1689

1690

1691

1692

1693

1694

1695

1696

1697

1698

1699

1700

1701

1702

1703

1704

1705

1706

1707

1708

1709

1710

1711

1712

1713

1714

1715

1716

1717

1718

1719

1720

1721

1722

1723

1724

1725

1726

1727

1728

1729

1730

1731

1732

1733

1734

1735

1736

1737

1738

1739

1740

1741

1742

1743

1744

1745

1746

1747

1748

1749

1750

1751

1752

1753

1754

1755

1756

1757

1758

1759

1760

1761

1762

1763

1764

1765

1766

1767

1768

1769

1770

1771

1772

1773

1774

1775

1776

1777

1778

1779

1780

1781

1782

1783

1784

1785

1786

1787

1788

1789

1790

1791

1792

1793

1794

1795

1796

1797

1798

1799

1800

1801

1802

1803

1804

1805

1806

1807

1808

1809

1810

1811

1812

1813

1814

1815

1816

1817

1818

1819

1820

1821

1822

1823

1824

1825

1826

1827

1828

1829

1830

1831

1832

1833

1834

1835

1836

1837

1838

1839

1840

1841

1842

1843

1844

1845

1846

1847

1848

1849

1850

1851

1852

1853

1854

1855

1856

1857

1858

1859

1860

1861

1862

1863

1864

1865

1866

1867

1868

1869

1870

1871

1872

1873

1874

1875

1876

1877

1878

1879

1880

1881

1882

1883

1884

1885

1886

1887

1888

1889

1890

1891

1892

1893

1894

1895

1896

1897

1898

1899

1900

1901

1902

1903

1904

1905

1906

1907

1908

1909

1910

1911

1912

1913

1914

1915

1916

1917

1918

1919

1920

1921

1922

1923

1924

1925

1926

1927

1928

1929

1930

1931

1932

1933

1934

1935

1936

1937

1938

1939

1940

1941

1942

1943

1944

1945

1946

1947

1948

1949

1950

1951

1952

1953

1954

1955

1956

1957

1958

1959

1960

1961

1962

1963

1964

1965

1966

1967

1968

1969

1970

1971

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

2021

2022

2023

2024

2025

2026

2027

2028

2029

2030

2031

2032

2033

2034

2035

2036

2037

2038

2039

2040

2041

2042

2043

2044

2045

2046

2047

2048

2049

2050

2051

2052

2053

2054

2055

2056

2057

2058

2059

2060

2061

2062

2063

2064

2065

2066

2067

2068

2069

2070

2071

2072

2073

2074

2075

2076

2077

2078

2079

2080

2081

2082

2083

2084

2085

2086

2087

2088

2089

2090

2091

2092

2093

2094

2095

2096

2097

2098

2099

2100

2101

2102

2103

2104

2105

2106

2107

2108

2109

2110

2111

2112

2113

2114

2115

2116

2117

2118

2119

2120

2121

2122

2123

2124

2125

2126

2127

2128

2129

2130

2131

2132

2133

2134

2135

2136

2137

2138

2139

2140

2141

2142

2143

2144

2145

2146

2147

2148

2149

2150

2151

2152

2153

2154

2155

2156

2157

2158

2159

2160

2161

2162

2163

2164

2165

2166

2167

2168

2169

2170

2171

2172

2173

2174

2175

2176

2177

2178

2179

2180

2181

2182

2183

2184

2185

2186

2187

2188

2189

2190

2191

2192

2193

2194

2195

2196

2197

2198

2199

2200

2201

2202

2203

2204

2205

2206

2207

2208

2209

2210

2211

2212

2213

2214

2215

2216

2217

2218

2219

2220

2221

2222

2223

2224

2225

2226

2227

2228

2229

2230

2231

2232

2233

2234

2235

2236

2237

2238

2239

2240

2241

2242

2243

2244

2245

2246

2247

2248

2249

2250

2251

2252

2253

2254

2255

2256

2257

2258

2259

2260

2261

2262

2263

2264

2265

2266

2267

2268

2269

2270

2271

2272

2273

2274

2275

2276

2277

2278

2279

2280

2281

2282

2283

2284

2285

2286

2287

2288

2289

2290

2291

2292

2293

2294

2295

2296

2297

2298

2299

2300

2301

2302

2303

2304

2305

2306

2307

2308

2309

2310

2311

2312

2313

2314

2315

2316

2317

2318

2319

2320

2321

2322

2323

2324

2325

2326

2327

2328

2329

2330

2331

2332

2333

2334

2335

2336

2337

2338

2339

2340

2341

2342

2343

2344

2345

2346

2347

2348

2349

2350

2351

2352

2353

2354

2355

2356

2357

2358

2359

2360

2361

2362

2363

2364

2365

2366

2367

2368

2369

2370

2371

2372

2373

2374

2375

2376

2377

2378

2379

2380

2381

2382

2383

2384

2385

2386

2387

2388

2389

2390

2391

2392

2393

2394

2395

2396

2397

2398

2399

2400

2401

2402

2403

2404

2405

2406

2407

2408

2409

2410

2411

2412

2413

2414

2415

2416

2417

2418

2419

2420

2421

2422

2423

2424

2425

2426

2427

2428

2429

2430

2431

2432

2433

2434

2435

2436

2437

2438

2439

2440

2441

2442

2443

2444

2445

2446

2447

2448

2449

2450

2451

2452

2453

2454

2455

2456

2457

2458

2459

2460

2461

2462

2463

2464

2465

2466

2467

2468

2469

2470

2471

2472

2473

2474

2475

2476

2477

2478

2479

2480

2481

2482

2483

2484

2485

2486

2487

2488

2489

2490

2491

2492

2493

2494

2495

2496

2497

2498

2499

2500

2501

2502

2503

2504

2505

2506

2507

2508

2509

2510

2511

2512

2513

2514

2515

2516

2517

2518

2519

2520

2521

2522

2523

2524

2525

2526

2527

2528

2529

2530

2531

2532

2533

2534

2535

2536

2537

2538

2539

2540

2541

2542

2543

2544

2545

2546

2547

2548

2549

2550

2551

2552

2553

2554

2555

2556

2557

2558

2559

2560

2561

2562

2563

2564

2565

2566

2567

2568

2569

2570

2571

2572

2573

2574

2575

2576

2577

2578

2579

2580

2581

2582

2583

2584

2585

2586

2587

2588

2589

2590

2591

2592

2593

2594

2595

2596

2597

2598

2599

2600

2601

2602

2603

2604

2605

2606

2607

2608

2609

2610

2611

2612

2613

2614

2615

2616

2617

2618

2619

2620

2621

2622

2623

2624

2625

2626

2627

2628

2629

2630

2631

2632

2633

2634

2635

2636

2637

2638

2639

2640

2641

2642

2643

2644

2645

2646

2647

2648

2649

2650

2651

2652

2653

2654

2655

2656

2657

2658

2659

2660

2661

2662

2663

2664

2665

2666

2667

2668

2669

2670

2671

2672

2673

2674

2675

2676

2677

2678

2679

2680

2681

2682

2683

2684

2685

2686

2687

2688

2689

2690

2691

2692

2693

2694

2695

2696

2697

2698

2699

2700

2701

2702

2703

2704

2705

2706

2707

2708

2709

2710

2711

2712

2713

2714

2715

2716

2717

2718

2719

2720

2721

2722

2723

2724

2725

2726

2727

2728

2729

2730

2731

2732

2733

2734

2735

2736

2737

2738

2739

2740

2741

2742

2743

2744

2745

2746

2747

2748

2749

2750

2751

2752

2753

2754

2755

2756

2757

2758

2759

2760

2761

2762

2763

2764

2765

2766

2767

2768

2769

2770

2771

2772

2773

2774

2775

2776

2777

2778

2779

2780

2781

2782

2783

2784

2785

2786

2787

2788

2789

2790

2791

2792

2793

2794

2795

2796

2797

2798

2799

2800

2801

2802

2803

2804

2805

2806

2807

2808

2809

2810

2811

2812

2813

2814

2815

2816

2817

2818

2819

2820

2821

2822

2823

2824

2825

2826

2827

2828

2829

2830

2831

2832

2833

2834

2835

2836

2837

2838

2839

2840

2841

2842

2843

2844

2845

2846

2847

2848

2849

2850

2851

2852

2853

2854

2855

2856

2857

2858

2859

2860

2861

2862

2863

2864

2865

2866

2867

2868

2869

2870

2871

2872

2873

2874

2875

2876

2877

2878

2879

2880

2881

2882

2883

2884

2885

2886

2887

2888

2889

2890

2891

2892

2893

2894

2895

2896

2897

2898

2899

2900

2901

2902

2903

2904

2905

2906

2907

2908

2909

2910

2911

2912

2913

2914

2915

2916

2917

2918

2919

2920

2921

2922

2923

2924

2925

2926

2927

2928

2929

2930

2931

2932

2933

2934

2935

2936

2937

2938

2939

2940

2941

2942

2943

2944

2945

2946

2947

2948

2949

2950

2951

2952

2953

2954

2955

2956

2957

2958

2959

2960

2961

2962

2963

2964

2965

2966

2967

2968

2969

2970

2971

2972

2973

2974

2975

2976

2977

2978

2979

2980

2981

2982

2983

2984

2985

2986

2987

2988

2989

2990

2991

2992

2993

2994

2995

2996

2997

2998

2999

3000

3001

3002

3003

3004

3005

3006

3007

3008

3009

3010

3011

3012

3013

3014

3015

3016

3017

3018

3019

3020

3021

3022

3023

3024

3025

3026

3027

3028

3029

3030

3031

3032

3033

3034

3035

3036

3037

3038

3039

3040

3041

3042

3043

3044

3045

3046

3047

3048

3049

3050

3051

3052

3053

3054

3055

3056

3057

3058

3059

3060

3061

3062

3063

3064

3065

3066

3067

3068

3069

3070

3071

3072

3073

3074

3075

3076

3077

3078

3079

3080

3081

3082

3083

3084

3085

3086

3087

3088

3089

3090

3091

3092

3093

3094

3095

3096

3097

3098

3099

3100

3101

3102

3103

3104

3105

3106

3107

3108

3109

3110

3111

3112

3113

3114

3115

3116

3117

3118

3119

3120

3121

3122

3123

3124

3125

3126

3127

3128

3129

3130

3131

3132

3133

3134

3135

3136

3137

3138

3139

3140

3141

3142

3143

3144

3145

3146

3147

3148

3149

3150

3151

3152

3153

3154

3155

3156

3157

3158

3159

3160

3161

3162

3163

3164

3165

3166

3167

3168

3169

3170

3171

3172

3173

3174

3175

3176

3177

3178

3179

3180

3181

3182

3183

3184

3185

3186

3187

3188

3189

3190

3191

3192

3193

3194

3195

3196

3197

3198

3199

3200

3201

3202

3203

3204

3205

3206

3207

3208

3209

3210

3211

3212

3213

3214

3215

3216

3217

3218

3219

3220

3221

3222

3223

3224

3225

3226

3227

3228

3229

3230

3231

3232

3233

3234

3235

3236

3237

3238

3239

3240

3241

3242

3243

3244

3245

3246

3247

3248

3249

3250

3251

3252

3253

3254

3255

3256

3257

3258

3259

3260

3261

3262

3263

3264

3265

3266

3267

3268

3269

3270

3271

3272

3273

3274

3275

3276

3277

3278

3279

3280

3281

3282

3283

3284

3285

3286

3287

3288

3289

3290

3291

3292

3293

3294

3295

3296

3297

3298

3299

3300

3301

3302

3303

3304

3305

3306

3307

3308

3309

3310

3311

3312

3313

3314

3315

3316

3317

3318

3319

3320

3321

3322

3323

3324

3325

3326

3327

3328

3329

3330

3331

3332

3333

3334

3335

3336

3337

3338

3339

3340

3341

3342

3343

3344

3345

3346

3347

3348

3349

3350

3351

3352

3353

3354

3355

3356

3357

3358

3359

3360

3361

3362

3363

3364

3365

3366

3367

3368

3369

3370

3371

3372

3373

3374

3375

3376

3377

3378

3379

3380

3381

3382

3383

3384

3385

3386

3387

3388

3389

3390

3391

3392

3393

3394

3395

3396

3397

3398

3399

3400

3401

3402

3403

3404

3405

3406

3407

3408

3409

3410

3411

3412

3413

3414

3415

3416

3417

3418

3419

3420

3421

3422

3423

3424

3425

3426

3427

3428

3429

3430

3431

3432

3433

3434

3435

3436

3437

3438

3439

3440

3441

3442

3443

3444

3445

3446

3447

3448

3449

3450

3451

3452

3453

3454

3455

3456

3457

3458

3459

3460

3461

3462

3463

3464

3465

3466

3467

3468

3469

3470

3471

3472

3473

3474

3475

3476

3477

3478

3479

3480

3481

3482

3483

3484

3485

3486

3487

3488

3489

3490

3491

3492

3493

3494

3495

3496

3497

3498

3499

3500

3501

3502

3503

3504

3505

3506

3507

3508

3509

3510

3511

3512

3513

3514

3515

3516

3517

3518

3519

3520

3521

3522

3523

3524

3525

3526

3527

3528

3529

3530

3531

3532

3533

3534

3535

3536

3537

3538

3539

3540

3541

3542

3543

3544

3545

3546

3547

3548

3549

3550

3551

3552

3553

3554

3555

3556

3557

3558

3559

3560

3561

3562

3563

3564

3565

3566

3567

3568

3569

3570

3571

3572

3573

3574

3575

3576

3577

3578

3579

3580

3581

3582

3583

3584

3585

3586

3587

3588

3589

3590

3591

3592

3593

3594

3595

3596

3597

3598

3599

3600

3601

3602

3603

3604

3605

3606

3607

3608

3609

3610

3611

3612

3613

3614

3615

3616

3617

3618

3619

3620

3621

3622

3623

3624

3625

3626

3627

3628

3629

3630

3631

3632

3633

3634

3635

3636

3637

3638

3639

3640

3641

3642

3643

3644

3645

3646

3647

3648

3649

3650

3651

3652

3653

3654

3655

3656

3657

3658

3659

3660

3661

3662

3663

3664

3665

3666

3667

3668

3669

3670

3671

3672

3673

3674

3675

3676

3677

3678

3679

3680

3681

3682

3683

3684

3685

3686

3687

3688

3689

3690

3691

3692

3693

3694

3695

3696

3697

3698

3699

3700

3701

3702

3703

3704

3705

3706

3707

3708

3709

3710

3711

3712

3713

3714

3715

3716

3717

3718

3719

3720

3721

3722

3723

3724

3725

3726

3727

3728

3729

3730

3731

3732

3733

3734

3735

3736

3737

3738

3739

3740

3741

3742

3743

3744

3745

3746

3747

3748

3749

3750

3751

3752

3753

3754

3755

3756

3757

3758

3759

3760

3761

3762

3763

3764

3765

3766

3767

3768

3769

3770

3771

3772

3773

3774

3775

3776

3777

3778

3779

3780

3781

3782

3783

3784

3785

3786

3787

3788

3789

3790

3791

3792

3793

3794

3795

3796

3797

3798

3799

3800

3801

3802

3803

3804

3805

3806

3807

3808

3809

3810

3811

3812

3813

3814

3815

3816

3817

3818

3819

3820

3821

3822

3823

3824

3825

3826

3827

3828

3829

3830

3831

3832

3833

3834

3835

3836

3837

3838

3839

3840

3841

3842

3843

3844

3845

3846

3847

3848

3849

3850

3851

3852

3853

3854

3855

3856

3857

3858

3859

3860

3861

3862

3863

3864

3865

3866

3867

3868

3869

3870

3871

3872

3873

3874

3875

3876

3877

3878

3879

3880

3881

3882

3883

3884

3885

3886

3887

3888

3889

3890

3891

3892

3893

3894

3895

3896

3897

3898

3899

3900

3901

3902

3903

3904

3905

3906

3907

3908

3909

3910

3911

3912

3913

3914

3915

3916

3917

3918

3919

3920

3921

3922

3923

3924

3925

3926

3927

3928

3929

3930

3931

3932

3933

3934

3935

3936

3937

3938

3939

3940

3941

3942

3943

3944

3945

3946

3947

3948

3949

3950

3951

3952

3953

3954

3955

3956

3957

3958

3959

3960

3961

3962

3963

3964

3965

3966

3967

3968

3969

3970

3971

3972

3973

3974

3975

3976

3977

3978

3979

3980

3981

3982

3983

3984

3985

3986

3987

3988

3989

3990

3991

3992

3993

3994

3995

3996

3997

3998

3999

4000

4001

4002

4003

4004

4005

4006

4007

4008

4009

4010

4011

4012

4013

4014

4015

4016

4017

4018

4019

4020

4021

4022

4023

4024

4025

4026

4027

4028

4029

4030

4031

4032

4033

4034

4035

4036

4037

4038

4039

4040

4041

4042

4043

4044

4045

4046

4047

4048

4049

4050

4051

4052

4053

4054

4055

4056

4057

4058

4059

4060

4061

4062

4063

4064

4065

4066

4067

4068

4069

4070

4071

4072

4073

4074

4075

4076

4077

4078

4079

4080

4081

4082

4083

4084

4085

4086

4087

4088

4089

4090

4091

4092

4093

4094

4095

4096

4097

4098

4099

4100

4101

4102

4103

4104

4105

4106

4107

4108

4109

4110

4111

4112

4113

4114

4115

4116

4117

4118

4119

4120

4121

4122

4123

4124

4125

4126

4127

4128

4129

4130

4131

4132

4133

4134

4135

4136

4137

4138

4139

4140

4141

4142

4143

4144

4145

4146

4147

4148

4149

4150

4151

4152

4153

4154

4155

4156

4157

4158

4159

4160

4161

4162

4163

4164

4165

4166

4167

4168

4169

4170

4171

4172

4173

4174

4175

4176

4177

4178

4179

4180

4181

4182

4183

4184

4185

4186

4187

4188

4189

4190

4191

4192

4193

4194

4195

4196

4197

4198

4199

4200

4201

4202

4203

4204

4205

4206

4207

# http://pyrocko.org - GPLv3 

# 

# The Pyrocko Developers, 21st Century 

# ---|P------/S----------~Lg---------- 

 

'''Classical seismic ray theory for layered earth models (*layer cake* models). 

 

This module can be used to e.g. calculate arrival times, ray paths, reflection 

and transmission coefficients, take-off and incidence angles and geometrical 

spreading factors for arbitrary seismic phases. Computations are done for a 

spherical earth, even though the module name may suggests something flat. 

 

The main classes defined in this module are: 

 

* :py:class:`Material` - Defines an isotropic elastic material. 

* :py:class:`PhaseDef` - Defines a seismic phase arrival / wave propagation 

history. 

* :py:class:`Leg` - Continuous propagation in a :py:class:`PhaseDef`. 

* :py:class:`Knee` - Conversion/reflection in a :py:class:`PhaseDef`. 

* :py:class:`LayeredModel` - Representation of a layer cake model. 

* :py:class:`Layer` - A layer in a :py:class:`LayeredModel`. 

 

* :py:class:`HomogeneousLayer` - A homogeneous :py:class:`Layer`. 

* :py:class:`GradientLayer` - A gradient :py:class:`Layer`. 

 

* :py:class:`Discontinuity` - A discontinuity in a :py:class:`LayeredModel`. 

 

* :py:class:`Interface` - A :py:class:`Discontinuity` between two 

:py:class:`Layer` instances. 

* :py:class:`Surface` - The surface :py:class:`Discontinuity` on top of 

a :py:class:`LayeredModel`. 

 

* :py:class:`RayPath` - A fan of rays running through a common sequence of 

layers / interfaces. 

* :py:class:`Ray` - A specific ray with a specific (ray parameter, distance, 

arrival time) choice. 

* :py:class:`RayElement` - An element of a :py:class:`RayPath`. 

 

* :py:class:`Straight` - A ray segment representing propagation through 

one :py:class:`Layer`. 

* :py:class:`Kink` - An interaction of a ray with a 

:py:class:`Discontinuity`. 

''' 

 

from __future__ import absolute_import 

from functools import reduce 

 

import os 

import logging 

import copy 

import math 

import cmath 

import operator 

try: 

from StringIO import StringIO 

except ImportError: 

from io import StringIO 

 

import glob 

import numpy as num 

from scipy.optimize import bisect, brentq 

 

from . import util, config 

 

try: 

newstr = unicode 

except NameError: 

newstr = str 

 

logger = logging.getLogger('cake') 

 

ZEPS = 0.01 

P = 1 

S = 2 

DOWN = 4 

UP = -4 

 

DEFAULT_BURGERS = (0., 0., 1.) 

 

earthradius = config.config().earthradius 

 

r2d = 180./math.pi 

d2r = 1./r2d 

km = 1000. 

d2m = d2r*earthradius 

m2d = 1./d2m 

sprad2spm = 1.0/(r2d*d2m) 

sprad2spkm = 1.0/(r2d*d2m/km) 

spm2sprad = 1.0/sprad2spm 

spkm2sprad = 1.0/sprad2spkm 

 

 

class CakeError(Exception): 

pass 

 

 

class InvalidArguments(CakeError): 

pass 

 

 

class Material(object): 

'''Isotropic elastic material. 

 

:param vp: P-wave velocity [m/s] 

:param vs: S-wave velocity [m/s] 

:param rho: density [kg/m^3] 

:param qp: P-wave attenuation Qp 

:param qs: S-wave attenuation Qs 

:param poisson: Poisson ratio 

:param lame: tuple with Lame parameter `lambda` and `shear modulus` [Pa] 

:param qk: bulk attenuation Qk 

:param qmu: shear attenuation Qmu 

 

:param burgers: Burgers rheology paramerters as `tuple`. 

`transient viscosity` [Pa], <= 0 means infinite value, 

`steady-state viscosity` [Pa] and `alpha`, the ratio between the 

effective and unreleaxed shear modulus, mu1/(mu1 + mu2). 

:type burgers: tuple 

 

If no velocities and no lame parameters are given, standard crustal values 

of vp = 5800 m/s and vs = 3200 m/s are used. If no Q values are given, 

standard crustal values of qp = 1456 and qs = 600 are used. If no Burgers 

material parameters are given, transient and steady-state viscosities are 

0 and alpha=1. 

 

Everything is in SI units (m/s, Pa, kg/m^3) unless explicitly stated. 

 

The main material properties are considered independant and are accessible 

as attributes (it is allowed to assign to these): 

 

.. py:attribute:: vp, vs, rho, qp, qs 

 

Other material properties are considered dependant and can be queried by 

instance methods. 

''' 

 

def __init__( 

self, vp=None, vs=None, rho=2600., qp=None, qs=None, poisson=None, 

lame=None, qk=None, qmu=None, burgers=None): 

 

parstore_float(locals(), self, 'vp', 'vs', 'rho', 'qp', 'qs') 

 

if vp is not None and vs is not None: 

if poisson is not None or lame is not None: 

raise InvalidArguments( 

'If vp and vs are given, poisson ratio and lame paramters ' 

'should not be given.') 

 

elif vp is None and vs is None and lame is None: 

self.vp = 5800. 

if poisson is None: 

poisson = 0.25 

self.vs = self.vp / math.sqrt(2.0*(1.0-poisson)/(1.0-2.0*poisson)) 

 

elif vp is None and vs is None and lame is not None: 

if poisson is not None: 

raise InvalidArguments( 

'Poisson ratio should not be given, when lame parameters ' 

'are given.') 

 

lam, mu = float(lame[0]), float(lame[1]) 

self.vp = math.sqrt((lam + 2.0*mu)/rho) 

self.vs = math.sqrt(mu/rho) 

 

elif vp is not None and vs is None: 

if poisson is None: 

poisson = 0.25 

 

if lame is not None: 

raise InvalidArguments( 

'If vp is given, Lame parameters should not be given.') 

 

poisson = float(poisson) 

self.vs = vp / math.sqrt(2.0*(1.0-poisson)/(1.0-2.0*poisson)) 

 

elif vp is None and vs is not None: 

if poisson is None: 

poisson = 0.25 

if lame is not None: 

raise InvalidArguments( 

'If vs is given, Lame parameters should not be given.') 

 

poisson = float(poisson) 

self.vp = vs * math.sqrt(2.0*(1.0-poisson)/(1.0-2.0*poisson)) 

 

else: 

raise InvalidArguments( 

'Invalid combination of input parameters in material ' 

'definition.') 

 

if qp is not None or qs is not None: 

if not (qk is None and qmu is None): 

raise InvalidArguments( 

'if qp or qs are given, qk and qmu should not be given.') 

 

if qp is None: 

if self.vs != 0.0: 

s = (4.0/3.0)*(self.vs/self.vp)**2 

self.qp = self.qs / s 

else: 

self.qp = 1456. 

 

if qs is None: 

if self.vs != 0.0: 

s = (4.0/3.0)*(self.vs/self.vp)**2 

self.qs = self.qp * s 

else: 

self.vs = 600. 

 

elif qp is None and qs is None and qk is None and qmu is None: 

if self.vs == 0.: 

self.qs = 0. 

self.qp = 5782e4 

else: 

self.qs = 600. 

s = (4.0/3.0)*(self.vs/self.vp)**2 

self.qp = self.qs/s 

 

elif qp is None and qs is None and qk is not None and qmu is not None: 

s = (4.0/3.0)*(self.vs/self.vp)**2 

if qmu == 0. and self.vs == 0.: 

self.qp = qk 

else: 

if num.isinf(qk): 

self.qp = qmu/s 

else: 

self.qp = 1.0 / (s/qmu + (1.0-s)/qk) 

self.qs = qmu 

else: 

raise InvalidArguments( 

'Invalid combination of input parameters in material ' 

'definition.') 

 

if burgers is None: 

burgers = DEFAULT_BURGERS 

 

self.burger_eta1 = burgers[0] 

self.burger_eta2 = burgers[1] 

self.burger_valpha = burgers[2] 

 

def astuple(self): 

'''Get independant material properties as a tuple. 

 

Returns a tuple with ``(vp, vs, rho, qp, qs)``. 

''' 

return self.vp, self.vs, self.rho, self.qp, self.qs 

 

def __eq__(self, other): 

return self.astuple() == other.astuple() 

 

def lame(self): 

'''Get Lame's parameter lambda and shear modulus.''' 

mu = self.vs**2 * self.rho 

lam = self.vp**2 * self.rho - 2.0*mu 

return lam, mu 

 

def lame_lambda(self): 

'''Get Lame's parameter lambda. 

 

Returned units are [Pa]. 

''' 

lam, _ = self.lame() 

return lam 

 

def shear_modulus(self): 

'''Get shear modulus. 

 

Returned units are [Pa]. 

''' 

return self.vs**2 * self.rho 

 

def poisson(self): 

'''Get Poisson's ratio.''' 

lam, mu = self.lame() 

return lam / (2.0*(lam+mu)) 

 

def bulk(self): 

'''Get bulk modulus.''' 

lam, mu = self.lame() 

return lam + 2.0*mu/3.0 

 

def youngs(self): 

'''Get Young's modulus.''' 

lam, mu = self.lame() 

return mu * (3.0*lam + 2.0*mu) / (lam+mu) 

 

def vp_vs_ratio(self): 

'''Get vp/vs ratio.''' 

return self.vp/self.vs 

 

def qmu(self): 

'''Get shear attenuation coefficient Qmu.''' 

return self.qs 

 

def qk(self): 

'''Get bulk attenuation coefficient Qk.''' 

if self.vs == 0. and self.qs == 0.: 

return self.qp 

else: 

s = (4.0/3.0)*(self.vs/self.vp)**2 

denom = (1/self.qp - s/self.qs) 

if denom <= 0.0: 

return num.inf 

else: 

return (1.-s)/(1.0/self.qp - s/self.qs) 

 

def burgers(self): 

'''Get Burger parameters.''' 

return self.burger_eta1, self.burger_eta2, self.burger_valpha 

 

def _rayleigh_equation(self, cr): 

cr_a = (cr/self.vp)**2 

cr_b = (cr/self.vs)**2 

if cr_a > 1.0 or cr_b > 1.0: 

return None 

 

return (2.0-cr_b)**2 - 4.0 * math.sqrt(1.0-cr_a) * math.sqrt(1.0-cr_b) 

 

def rayleigh(self): 

'''Get rayleigh velocity assuming a homogenous halfspace. 

 

Returned units are [m/s].''' 

return bisect(self._rayleigh_equation, 0.001*self.vs, self.vs) 

 

def _has_default_burgers(self): 

if self.burger_eta1 == DEFAULT_BURGERS[0] and \ 

self.burger_eta2 == DEFAULT_BURGERS[1] and \ 

self.burger_valpha == DEFAULT_BURGERS[2]: 

return True 

return False 

 

def describe(self): 

'''Get a readable listing of the material properties.''' 

template = ''' 

P wave velocity [km/s] : %12g 

S wave velocity [km/s] : %12g 

P/S wave vel. ratio : %12g 

Lame lambda [GPa] : %12g 

Lame shear modulus [GPa] : %12g 

Poisson ratio : %12g 

Bulk modulus [GPa] : %12g 

Young's modulus [GPa] : %12g 

Rayleigh wave vel. [km/s] : %12g 

Density [g/cm**3] : %12g 

Qp P-wave attenuation : %12g 

Qs S-wave attenuation (Qmu) : %12g 

Qk bulk attenuation : %12g 

transient viscos., eta1 [GPa] : %12g 

st.-state viscos., eta2 [GPa] : %12g 

relaxation: valpha : %12g 

'''.strip() 

 

return template % ( 

self.vp/km, 

self.vs/km, 

self.vp/self.vs, 

self.lame_lambda()*1e-9, 

self.shear_modulus()*1e-9, 

self.poisson(), 

self.bulk()*1e-9, 

self.youngs()*1e-9, 

self.rayleigh()/km, 

self.rho/km, 

self.qp, 

self.qs, 

self.qk(), 

self.burger_eta1*1e-9, 

self.burger_eta2*1e-9, 

self.burger_valpha) 

 

def __str__(self): 

vp, vs, rho, qp, qs = self.astuple() 

return '%10g km/s %10g km/s %10g g/cm^3 %10g %10g' % ( 

vp/km, vs/km, rho/km, qp, qs) 

 

def __repr__(self): 

return 'Material(vp=%s, vs=%s, rho=%s, qp=%s, qs=%s)' % \ 

tuple(repr(x) for x in ( 

self.vp, self.vs, self.rho, self.qp, self.qs)) 

 

 

class Leg(object): 

''' 

Represents a continuous piece of wave propagation in a phase definition 

 

**Attributes:** 

 

To be considered as read-only. 

 

.. py:attribute:: departure 

 

One of the constants :py:const:`UP` or :py:const:`DOWN` indicating 

upward or downward departure. 

 

.. py:attribute:: mode 

 

One of the constants :py:const:`P` or :py:const:`S`, indicating the 

propagation mode. 

 

.. py:attribute:: depthmin 

 

``None``, a number (a depth in [m]) or a string (an interface name), 

minimum depth. 

 

.. py:attribute:: depthmax 

 

``None``, a number (a depth in [m]) or a string (an interface name), 

maximum depth. 

 

''' 

 

def __init__(self, departure=None, mode=None): 

self.departure = departure 

self.mode = mode 

self.depthmin = None 

self.depthmax = None 

 

def set_depthmin(self, depthmin): 

self.depthmin = depthmin 

 

def set_depthmax(self, depthmax): 

self.depthmax = depthmax 

 

def __str__(self): 

def sd(d): 

if isinstance(d, float): 

return '%g km' % (d/km) 

else: 

return 'interface %s' % d 

 

s = '%s mode propagation, departing %s' % ( 

smode(self.mode).upper(), { 

UP: 'upward', DOWN: 'downward'}[self.departure]) 

 

sc = [] 

if self.depthmax is not None: 

sc.append('deeper than %s' % sd(self.depthmax)) 

if self.depthmin is not None: 

sc.append('shallower than %s' % sd(self.depthmin)) 

 

if sc: 

s = s + ' (may not propagate %s)' % ' or '.join(sc) 

 

return s 

 

 

class InvalidKneeDef(CakeError): 

pass 

 

 

class Knee(object): 

'''Represents a change in wave propagation within a :py:class:`PhaseDef`. 

 

**Attributes:** 

 

To be considered as read-only. 

 

.. py:attribute:: depth 

 

Depth at which the conversion/reflection should happen. this can be 

a string or a number. 

 

.. py:attribute:: direction 

 

One of the constants :py:const:`UP` or :py:const:`DOWN` to indicate 

the incoming direction. 

 

.. py:attribute:: in_mode 

 

One of the constants :py:const:`P` or :py:const:`S` to indicate the 

type of mode of the incoming wave. 

 

.. py:attribute:: out_mode 

 

One of the constants :py:const:`P` or :py:const:`S` to indicate the 

type of mode of the outgoing wave. 

 

.. py:attribute:: conversion 

 

Boolean, whether there is a mode conversion involved. 

 

.. py:attribute:: reflection 

 

Boolean, whether there is a reflection involved. 

 

.. py:attribute:: headwave 

 

Boolean, whether there is headwave propagation involved. 

 

''' 

 

defaults = dict( 

depth='surface', 

direction=UP, 

conversion=True, 

reflection=False, 

headwave=False, 

in_setup_state=True) 

 

defaults_surface = dict( 

depth='surface', 

direction=UP, 

conversion=False, 

reflection=True, 

headwave=False, 

in_setup_state=True) 

 

def __init__(self, *args): 

if args: 

(self.depth, self.direction, self.reflection, self.in_mode, 

self.out_mode) = args 

 

self.conversion = self.in_mode != self.out_mode 

self.in_setup_state = False 

 

def default(self, k): 

depth = self.__dict__.get('depth', 'surface') 

if depth == 'surface': 

return Knee.defaults_surface[k] 

else: 

return Knee.defaults[k] 

 

def __setattr__(self, k, v): 

if self.in_setup_state and k in self.__dict__: 

raise InvalidKneeDef('%s has already been set' % k) 

else: 

self.__dict__[k] = v 

 

def __getattr__(self, k): 

if k.startswith('__'): 

raise AttributeError(k) 

 

if k not in self.__dict__: 

return self.default(k) 

 

def set_modes(self, in_leg, out_leg): 

 

if out_leg.departure == UP and ( 

(self.direction == UP) == self.reflection): 

 

raise InvalidKneeDef( 

'cannot enter %s from %s and emit ray upwards' % ( 

['conversion', 'reflection'][self.reflection], 

{UP: 'below', DOWN: 'above'}[self.direction])) 

 

if out_leg.departure == DOWN and ( 

(self.direction == DOWN) == self.reflection): 

 

raise InvalidKneeDef( 

'cannot enter %s from %s and emit ray downwards' % ( 

['conversion', 'reflection'][self.reflection], 

{UP: 'below', DOWN: 'above'}[self.direction])) 

 

self.in_mode = in_leg.mode 

self.out_mode = out_leg.mode 

 

def at_surface(self): 

return self.depth == 'surface' 

 

def matches(self, discontinuity, mode, direction): 

''' 

Check whether it is relevant to a given combination of interface, 

propagation mode, and direction. 

''' 

 

if isinstance(self.depth, float): 

if abs(self.depth - discontinuity.z) > ZEPS: 

return False 

else: 

if discontinuity.name != self.depth: 

return False 

 

return self.direction == direction and self.in_mode == mode 

 

def out_direction(self): 

'''Get outgoing direction. 

 

Returns one of the constants :py:const:`UP` or :py:const:`DOWN`. 

''' 

 

if self.reflection: 

return - self.direction 

else: 

return self.direction 

 

def __str__(self): 

x = [] 

if self.reflection: 

if self.at_surface(): 

x.append('surface') 

else: 

if not self.headwave: 

if self.direction == UP: 

x.append('underside') 

else: 

x.append('upperside') 

 

if self.headwave: 

x.append('headwave propagation along') 

elif self.reflection and self.conversion: 

x.append('reflection with conversion from %s to %s' % ( 

smode(self.in_mode).upper(), smode(self.out_mode).upper())) 

if not self.at_surface(): 

x.append('at') 

elif self.reflection: 

x.append('reflection') 

if not self.at_surface(): 

x.append('at') 

elif self.conversion: 

x.append('conversion from %s to %s at' % ( 

smode(self.in_mode).upper(), smode(self.out_mode).upper())) 

else: 

x.append('passing through') 

 

if isinstance(self.depth, float): 

x.append('interface in %g km depth' % (self.depth/1000.)) 

else: 

if not self.at_surface(): 

x.append('%s' % self.depth) 

 

if not self.reflection: 

if self.direction == UP: 

x.append('on upgoing path') 

else: 

x.append('on downgoing path') 

 

return ' '.join(x) 

 

 

class Head(Knee): 

def __init__(self, *args): 

if args: 

z, in_direction, mode = args 

Knee.__init__(self, z, in_direction, True, mode, mode) 

else: 

Knee.__init__(self) 

 

def __str__(self): 

x = ['propagation as headwave'] 

if isinstance(self.depth, float): 

x.append('at interface in %g km depth' % (self.depth/1000.)) 

else: 

x.append('at %s' % self.depth) 

 

return ' '.join(x) 

 

 

class UnknownClassicPhase(CakeError): 

def __init__(self, phasename): 

self.phasename = phasename 

 

def __str__(self): 

return 'Unknown classic phase name: %s' % self.phasename 

 

 

class PhaseDefParseError(CakeError): 

''' 

Exception raised when an error occures during parsing of a phase 

definition string. 

''' 

 

def __init__(self, definition, position, exception): 

self.definition = definition 

self.position = position 

self.exception = exception 

 

def __str__(self): 

return 'Invalid phase definition: "%s" (at character %i: %s)' % ( 

self.definition, self.position+1, str(self.exception)) 

 

 

class PhaseDef(object): 

 

'''Definition of a seismic phase arrival, based on ray propagation path. 

 

:param definition: string representation of the phase in Cake's phase 

syntax 

 

Seismic phases are conventionally named e.g. P, Pn, PP, PcP, etc. In Cake, 

a slightly different terminology is adapted, which allows to specify 

arbitrary conversion/reflection histories for seismic ray paths. The 

conventions used here are inspired by those used in the TauP toolkit, but 

are not completely compatible with those. 

 

The definition of a seismic ray propagation path in Cake's phase syntax is 

a string consisting of an alternating sequence of *legs* and *knees*. 

 

A *leg* represents seismic wave propagation without any conversions, 

encountering only super-critical reflections. Legs are denoted by ``P``, 

``p``, ``S``, or ``s``. The capital letters are used when the take-off of 

the *leg* is in downward direction, while the lower case letters indicate a 

take-off in upward direction. 

 

A *knee* is an interaction with an interface. It can be a mode conversion, 

a reflection, or propagation as a headwave or diffracted wave. 

 

* conversion is simply denoted as: ``(INTERFACE)`` or ``DEPTH`` 

* upperside reflection: ``v(INTERFACE)`` or ``vDEPTH`` 

* underside reflection: ``^(INTERFACE)`` or ``^DEPTH`` 

* normal kind headwave or diffracted wave: ``v_(INTERFACE)`` or 

``v_DEPTH`` 

 

The interface may be given by name or by depth: INTERFACE is the name of an 

interface defined in the model, DEPTH is the depth of an interface in 

[km] (the interface closest to that depth is chosen). If two legs appear 

consecutively without an explicit *knee*, surface interaction is assumed. 

 

The phase definition may end with a backslash ``\\``, to indicate that the 

ray should arrive at the receiver from above instead of from below. It is 

possible to restrict the maximum and minimum depth of a *leg* by appending 

``<(INTERFACE)`` or ``<DEPTH`` or ``>(INTERFACE)`` or ``>DEPTH`` after the 

leg character, respectively. 

 

**Examples:** 

 

* ``P`` - like the classical P, but includes PKP, PKIKP, Pg 

* ``P<(moho)`` - like classical Pg, but must leave source downwards 

* ``pP`` - leaves source upward, reflects at surface, then travels as P 

* ``P(moho)s`` - conversion from P to S at the Moho on upgoing path 

* ``P(moho)S`` - conversion from P to S at the Moho on downgoing path 

* ``Pv12p`` - P with reflection at 12 km deep interface (or the 

interface closest to that) 

* ``Pv_(moho)p`` - classical Pn 

* ``Pv_(cmb)p`` - classical Pdiff 

* ``P^(conrad)P`` - underside reflection of P at the Conrad 

discontinuity 

 

**Usage:** 

 

>>> from pyrocko.cake import PhaseDef 

# must escape the backslash 

>>> my_crazy_phase = PhaseDef('pPv(moho)sP\\\\') 

>>> print my_crazy_phase 

Phase definition "pPv(moho)sP\": 

- P mode propagation, departing upward 

- surface reflection 

- P mode propagation, departing downward 

- upperside reflection with conversion from P to S at moho 

- S mode propagation, departing upward 

- surface reflection with conversion from S to P 

- P mode propagation, departing downward 

- arriving at target from above 

 

.. note:: 

 

(1) These conventions might be extended in a way to allow to fix wave 

propagation to SH mode, possibly by specifying SH, or a single 

character (e.g. H) instead of S. This would be benificial for the 

selection of conversion and reflection coefficients, which 

currently only deal with the P-SV case. 

''' 

 

allowed_characters_pattern = r'[0-9a-zA-Z_()<>^v\\.]+' 

allowed_characters_pattern_classic = r'[a-zA-Z0-9]+' 

 

@staticmethod 

def classic_definitions(): 

defs = {} 

# PmP, PmS, PcP, PcS, SmP, ... 

for r in 'mc': 

for a, b in 'PP PS SS SP'.split(): 

defs[a+r+b] = [ 

'%sv(%s)%s' % (a, {'m': 'moho', 'c': 'cmb'}[r], b.lower())] 

 

# Pg, P, S, Sg 

for a in 'PS': 

defs[a+'g'] = ['%s<(moho)' % x for x in (a, a.lower())] 

defs[a] = ['%s<(cmb)(moho)%s' % (x, x.lower()) for x in ( 

a, a.lower())] 

 

defs[a.lower()] = [a.lower()] 

 

for a, b in 'PP PS SS SP'.split(): 

defs[a+'K'+b] = ['%s(cmb)P<(icb)(cmb)%s' % (a, b.lower())] 

defs[a+'KIK'+b] = ['%s(cmb)P(icb)P(icb)p(cmb)%s' % (a, b.lower())] 

defs[a+'KJK'+b] = ['%s(cmb)P(icb)S(icb)p(cmb)%s' % (a, b.lower())] 

defs[a+'KiK'+b] = ['%s(cmb)Pv(icb)p(cmb)%s' % (a, b.lower())] 

 

# PP, SS, PS, SP, PPP, ... 

for a in 'PS': 

for b in 'PS': 

for c in 'PS': 

defs[a+b+c] = [''.join(defs[x][0] for x in a+b+c)] 

 

defs[a+b] = [''.join(defs[x][0] for x in a+b)] 

 

# Pc, Pdiff, Sc, ... 

for x in 'PS': 

defs[x+'c'] = defs[x+'diff'] = [x+'v_(cmb)'+x.lower()] 

defs[x+'n'] = [x+'v_(moho)'+x.lower()] 

 

# depth phases 

for k in list(defs.keys()): 

if k not in 'ps': 

for x in 'ps': 

defs[x+k] = [x + defs[k][0]] 

 

return defs 

 

@staticmethod 

def classic(phasename): 

'''Get phase definitions based on classic phase name. 

 

:param phasename: classic name of a phase 

:returns: list of PhaseDef objects 

 

This returns a list of PhaseDef objects, because some classic phases 

(like e.g. Pg) can only be represented by two Cake style PhaseDef 

objects (one with downgoing and one with upgoing first leg). 

''' 

 

defs = PhaseDef.classic_definitions() 

if phasename not in defs: 

raise UnknownClassicPhase(phasename) 

 

return [PhaseDef(d, classicname=phasename) for d in defs[phasename]] 

 

def __init__(self, definition=None, classicname=None): 

 

state = 0 

sdepth = '' 

sinterface = '' 

depthmax = depthmin = None 

depthlim = None 

depthlimtype = None 

sdepthlim = '' 

events = [] 

direction_stop = UP 

need_leg = True 

ic = 0 

if definition is not None: 

knee = Knee() 

try: 

for ic, c in enumerate(definition): 

 

if state in (0, 1): 

 

if c in '0123456789.': 

need_leg = True 

state = 1 

sdepth += c 

continue 

 

elif state == 1: 

knee.depth = float(sdepth)*1000. 

state = 0 

 

if state == 2: 

if c == ')': 

knee.depth = sinterface 

state = 0 

else: 

sinterface += c 

 

continue 

 

if state in (3, 4): 

 

if state == 3: 

if c in '0123456789.': 

sdepthlim += c 

continue 

elif c == '(': 

state = 4 

continue 

else: 

depthlim = float(sdepthlim)*1000. 

if depthlimtype == '<': 

depthmax = depthlim 

else: 

depthmin = depthlim 

state = 0 

 

elif state == 4: 

if c == ')': 

depthlim = sdepthlim 

if depthlimtype == '<': 

depthmax = depthlim 

else: 

depthmin = depthlim 

state = 0 

continue 

else: 

sdepthlim += c 

continue 

 

if state == 0: 

 

if c == '(': 

need_leg = True 

state = 2 

continue 

 

elif c in '<>': 

state = 3 

depthlim = None 

sdepthlim = '' 

depthlimtype = c 

continue 

 

elif c in 'psPS': 

leg = Leg() 

if c in 'ps': 

leg.departure = UP 

else: 

leg.departure = DOWN 

leg.mode = imode(c) 

 

if events: 

in_leg = events[-1] 

if depthmin is not None: 

in_leg.set_depthmin(depthmin) 

depthmin = None 

if depthmax is not None: 

in_leg.set_depthmax(depthmax) 

depthmax = None 

 

if in_leg.mode != leg.mode: 

knee.conversion = True 

else: 

knee.conversion = False 

 

if not knee.reflection: 

if c in 'ps': 

knee.direction = UP 

else: 

knee.direction = DOWN 

 

knee.set_modes(in_leg, leg) 

knee.in_setup_state = False 

events.append(knee) 

knee = Knee() 

sdepth = '' 

sinterface = '' 

 

events.append(leg) 

need_leg = False 

continue 

 

elif c == '^': 

need_leg = True 

knee.direction = UP 

knee.reflection = True 

continue 

 

elif c == 'v': 

need_leg = True 

knee.direction = DOWN 

knee.reflection = True 

continue 

 

elif c == '_': 

need_leg = True 

knee.headwave = True 

continue 

 

elif c == '\\': 

direction_stop = DOWN 

continue 

 

else: 

raise PhaseDefParseError( 

definition, ic, 'invalid character: "%s"' % c) 

 

if state == 3: 

depthlim = float(sdepthlim)*1000. 

if depthlimtype == '<': 

depthmax = depthlim 

else: 

depthmin = depthlim 

state = 0 

 

except (ValueError, InvalidKneeDef) as e: 

raise PhaseDefParseError(definition, ic, e) 

 

if state != 0 or need_leg: 

raise PhaseDefParseError( 

definition, ic, 'unfinished expression') 

 

if events and depthmin is not None: 

events[-1].set_depthmin(depthmin) 

if events and depthmax is not None: 

events[-1].set_depthmax(depthmax) 

 

self._definition = definition 

self._classicname = classicname 

self._events = events 

self._direction_stop = direction_stop 

 

def __iter__(self): 

for ev in self._events: 

yield ev 

 

def append(self, ev): 

self._events.append(ev) 

 

def first_leg(self): 

'''Get the first leg in phase definition.''' 

return self._events[0] 

 

def last_leg(self): 

'''Get the last leg in phase definition.''' 

return self._events[-1] 

 

def legs(self): 

''' 

Iterate over the continuous pieces of wave propagation (legs) defined 

within this phase definition. 

''' 

 

return (leg for leg in self if isinstance(leg, Leg)) 

 

def knees(self): 

''' 

Iterate over conversions and reflections (knees) defined within this 

phase definition. 

''' 

return (knee for knee in self if isinstance(knee, Knee)) 

 

def definition(self): 

'''Get original definition of the phase.''' 

return self._definition 

 

def given_name(self): 

''' 

Get entered classic name if any, or original definition of the phase. 

''' 

 

if self._classicname: 

return self._classicname 

else: 

return self._definition 

 

def direction_start(self): 

return self.first_leg().departure 

 

def direction_stop(self): 

return self._direction_stop 

 

def headwave_knee(self): 

for el in self: 

if type(el) == Knee and el.headwave: 

return el 

return None 

 

def used_repr(self): 

'''Translate into textual representation (cake phase syntax).''' 

def strdepth(x): 

if isinstance(x, float): 

return '%g' % (x/1000.) 

else: 

return '(%s)' % x 

 

x = [] 

for el in self: 

if type(el) == Leg: 

if el.departure == UP: 

x.append(smode(el.mode).lower()) 

else: 

x.append(smode(el.mode).upper()) 

 

if el.depthmax is not None: 

x.append('<'+strdepth(el.depthmax)) 

 

if el.depthmin is not None: 

x.append('>'+strdepth(el.depthmin)) 

 

elif type(el) == Knee: 

if el.reflection and not el.at_surface(): 

if el.direction == DOWN: 

x.append('v') 

else: 

x.append('^') 

if el.headwave: 

x.append('_') 

if not el.at_surface(): 

x.append(strdepth(el.depth)) 

 

elif type(el) == Head: 

x.append('_') 

x.append(strdepth(el.depth)) 

 

if self._direction_stop == DOWN: 

x.append('\\') 

 

return ''.join(x) 

 

def __repr__(self): 

if self._definition is not None: 

return "PhaseDef('%s')" % self._definition 

else: 

return "PhaseDef('%s')" % self.used_repr() 

 

def __str__(self): 

orig = '' 

used = self.used_repr() 

if self._definition != used: 

orig = ' (entered as "%s")' % self._definition 

 

sarrive = '\n - arriving at target from %s' % ('below', 'above')[ 

self._direction_stop == DOWN] 

 

return 'Phase definition "%s"%s:\n - ' % (used, orig) + \ 

'\n - '.join(str(ev) for ev in self) + sarrive 

 

def copy(self): 

'''Get a deep copy of it.''' 

return copy.deepcopy(self) 

 

 

def to_phase_defs(phases): 

if isinstance(phases, (str, newstr, PhaseDef)): 

phases = [phases] 

 

phases_out = [] 

for phase in phases: 

if isinstance(phase, (str, newstr)): 

phases_out.extend(PhaseDef(x.strip()) for x in phase.split(',')) 

elif isinstance(phase, PhaseDef): 

phases_out.append(phase) 

else: 

raise PhaseDefParseError('invalid phase definition') 

 

return phases_out 

 

 

def csswap(x): 

return cmath.sqrt(1.-x**2) 

 

 

def psv_surface_ind(in_mode, out_mode): 

''' 

Get indices to select the appropriate element from scatter matrix for free 

surface. 

''' 

 

return (int(in_mode == S), int(out_mode == S)) 

 

 

def psv_surface(material, p, energy=False): 

'''Scatter matrix for free surface reflection/conversions. 

 

:param material: material, object of type :py:class:`Material` 

:param p: flat ray parameter [s/m] 

:param energy: bool, when ``True`` energy normalized coefficients are 

returned 

:returns: Scatter matrix 

 

The scatter matrix is ordered as follows:: 

 

[[PP, PS], 

[SP, SS]] 

 

The formulas given in Aki & Richards are used. 

''' 

 

vp, vs, rho = material.vp, material.vs, material.rho 

sinphi = p * vp 

sinlam = p * vs 

cosphi = csswap(sinphi) 

coslam = csswap(sinlam) 

 

if vs == 0.0: 

scatter = num.array([[-1.0, 0.0], [0.0, 1.0]]) 

 

else: 

vsp_term = (1.0/vs**2 - 2.0*p**2) 

pcc_term = 4.0 * p**2 * cosphi/vp * coslam/vs 

denom = vsp_term**2 + pcc_term 

 

scatter = num.array([ 

[- vsp_term**2 + pcc_term, 4.0*p*coslam/vp*vsp_term], 

[4.0*p*cosphi/vs*vsp_term, vsp_term**2 - pcc_term]], 

dtype=num.complex) / denom 

 

if not energy: 

return scatter 

else: 

eps = 1e-16 

normvec = num.array([vp*rho*cosphi+eps, vs*rho*coslam+eps]) 

escatter = scatter*num.conj(scatter) * num.real( 

(normvec[:, num.newaxis]) / (normvec[num.newaxis, :])) 

return num.real(escatter) 

 

 

def psv_solid_ind(in_direction, out_direction, in_mode, out_mode): 

''' 

Get indices to select the appropriate element from scatter matrix for 

solid-solid interface. 

''' 

 

return ( 

(out_direction == DOWN)*2 + (out_mode == S), 

(in_direction == UP)*2 + (in_mode == S)) 

 

 

def psv_solid(material1, material2, p, energy=False): 

'''Scatter matrix for solid-solid interface. 

 

:param material1: material above, object of type :py:class:`Material` 

:param material2: material below, object of type :py:class:`Material` 

:param p: flat ray parameter [s/m] 

:param energy: bool, when ``True`` energy normalized coefficients are 

returned 

:returns: Scatter matrix 

 

The scatter matrix is ordered as follows:: 

 

[[P1P1, S1P1, P2P1, S2P1], 

[P1S1, S1S1, P2S1, S2S1], 

[P1P2, S1P2, P2P2, S2P2], 

[P1S2, S1S2, P2S2, S2S2]] 

 

The formulas given in Aki & Richards are used. 

''' 

 

vp1, vs1, rho1 = material1.vp, material1.vs, material1.rho 

vp2, vs2, rho2 = material2.vp, material2.vs, material2.rho 

 

sinphi1 = p * vp1 

cosphi1 = csswap(sinphi1) 

sinlam1 = p * vs1 

coslam1 = csswap(sinlam1) 

sinphi2 = p * vp2 

cosphi2 = csswap(sinphi2) 

sinlam2 = p * vs2 

coslam2 = csswap(sinlam2) 

 

# from aki and richards 

M = num.array([ 

[-vp1*p, -coslam1, vp2*p, coslam2], 

[cosphi1, -vs1*p, cosphi2, -vs2*p], 

[2.0*rho1*vs1**2*p*cosphi1, rho1*vs1*(1.0-2.0*vs1**2*p**2), 

2.0*rho2*vs2**2*p*cosphi2, rho2*vs2*(1.0-2.0*vs2**2*p**2)], 

[-rho1*vp1*(1.0-2.0*vs1**2*p**2), 2.0*rho1*vs1**2*p*coslam1, 

rho2*vp2*(1.0-2.0*vs2**2*p**2), -2.0*rho2*vs2**2*p*coslam2]], 

dtype=num.complex) 

N = M.copy() 

N[0] *= -1.0 

N[3] *= -1.0 

 

scatter = num.dot(num.linalg.inv(M), N) 

 

if not energy: 

return scatter 

else: 

eps = 1e-16 

if vs1 == 0.: 

vs1 = vp1*1e-16 

if vs2 == 0.: 

vs2 = vp2*1e-16 

normvec = num.array([ 

vp1*rho1*(cosphi1+eps), vs1*rho1*(coslam1+eps), 

vp2*rho2*(cosphi2+eps), vs2*rho2*(coslam2+eps)], dtype=num.complex) 

escatter = scatter*num.conj(scatter) * num.real( 

normvec[:, num.newaxis] / normvec[num.newaxis, :]) 

 

return num.real(escatter) 

 

 

class BadPotIntCoefs(CakeError): 

pass 

 

 

def potint_coefs(c1, c2, r1, r2): # r2 > r1 

eps = r2*1e-9 

if c1 == 0. and c2 == 0.: 

c1c2 = 1. 

else: 

c1c2 = c1/c2 

b = math.log(c1c2)/math.log((r1+eps)/r2) 

if abs(b) > 10.: 

raise BadPotIntCoefs() 

a = c1/(r1+eps)**b 

return a, b 

 

 

def imode(s): 

if s.lower() == 'p': 

return P 

elif s.lower() == 's': 

return S 

 

 

def smode(i): 

if i == P: 

return 'p' 

elif i == S: 

return 's' 

 

 

class PathFailed(CakeError): 

pass 

 

 

class SurfaceReached(PathFailed): 

pass 

 

 

class BottomReached(PathFailed): 

pass 

 

 

class MaxDepthReached(PathFailed): 

pass 

 

 

class MinDepthReached(PathFailed): 

pass 

 

 

class Trapped(PathFailed): 

pass 

 

 

class NotPhaseConform(PathFailed): 

pass 

 

 

class CannotPropagate(PathFailed): 

def __init__(self, direction, ilayer): 

PathFailed.__init__(self) 

self._direction = direction 

self._ilayer = ilayer 

 

def __str__(self): 

return 'Cannot enter layer %i from %s' % ( 

self._ilayer, { 

UP: 'below', 

DOWN: 'above'}[self._direction]) 

 

 

class Layer(object): 

'''Representation of a layer in a layered earth model. 

 

:param ztop: depth of top of layer 

:param zbot: depth of bottom of layer 

:param name: name of layer (optional) 

 

Subclasses are: :py:class:`HomogeneousLayer` and :py:class:`GradientLayer`. 

''' 

 

def __init__(self, ztop, zbot, name=None): 

self.ztop = ztop 

self.zbot = zbot 

self.zmid = (self.ztop + self.zbot) * 0.5 

self.name = name 

self.ilayer = None 

 

def _update_potint_coefs(self): 

potint_p = potint_s = False 

try: 

self._ppic = potint_coefs( 

self.mbot.vp, self.mtop.vp, 

radius(self.zbot), radius(self.ztop)) 

potint_p = True 

except BadPotIntCoefs: 

pass 

 

potint_s = False 

try: 

self._spic = potint_coefs( 

self.mbot.vs, self.mtop.vs, 

radius(self.zbot), radius(self.ztop)) 

potint_s = True 

except BadPotIntCoefs: 

pass 

 

assert P == 1 and S == 2 

self._use_potential_interpolation = (None, potint_p, potint_s) 

 

def potint_coefs(self, mode): 

'''Get coefficients for potential interpolation. 

 

:param mode: mode of wave propagation, :py:const:`P` or :py:const:`S` 

:returns: coefficients ``(a, b)`` 

''' 

 

if mode == P: 

return self._ppic 

else: 

return self._spic 

 

def contains(self, z): 

''' 

Tolerantly check if a given depth is within the layer 

(including boundaries). 

''' 

 

return self.ztop <= z <= self.zbot or \ 

self.at_bottom(z) or self.at_top(z) 

 

def inner(self, z): 

''' 

Tolerantly check if a given depth is within the layer 

(not including boundaries). 

''' 

 

return self.ztop <= z <= self.zbot and not \ 

self.at_bottom(z) and not \ 

self.at_top(z) 

 

def at_bottom(self, z): 

'''Tolerantly check if given depth is at the bottom of the layer.''' 

 

return abs(self.zbot - z) < ZEPS 

 

def at_top(self, z): 

'''Tolerantly check if given depth is at the top of the layer.''' 

return abs(self.ztop - z) < ZEPS 

 

def pflat_top(self, p): 

''' 

Convert spherical ray parameter to local flat ray parameter for top of 

layer. 

''' 

return p / (earthradius-self.ztop) 

 

def pflat_bottom(self, p): 

''' 

Convert spherical ray parameter to local flat ray parameter for bottom 

of layer. 

''' 

return p / (earthradius-self.zbot) 

 

def pflat(self, p, z): 

''' 

Convert spherical ray parameter to local flat ray parameter for 

given depth. 

''' 

return p / (earthradius-z) 

 

def v_potint(self, mode, z): 

a, b = self.potint_coefs(mode) 

return a*(earthradius-z)**b 

 

def u_potint(self, mode, z): 

a, b = self.potint_coefs(mode) 

return 1./(a*(earthradius-z)**b) 

 

def xt_potint(self, p, mode, zpart=None): 

''' 

Get travel time and distance for for traversal with given mode and ray 

parameter. 

 

:param p: ray parameter (spherical) 

:param mode: mode of propagation (:py:const:`P` or :py:const:`S`) 

:param zpart: if given, tuple with two depths to restrict computation 

to a part of the layer 

 

This implementation uses analytic formulas valid for a spherical earth 

in the case where the velocity c within the layer is given by potential 

interpolation of the form 

 

c(z) = a*z^b 

''' 

utop, ubot = self.u_top_bottom(mode) 

a, b = self.potint_coefs(mode) 

ztop = self.ztop 

zbot = self.zbot 

if zpart is not None: 

utop = self.u(mode, zpart[0]) 

ubot = self.u(mode, zpart[1]) 

ztop, zbot = zpart 

utop = 1./(a*(earthradius-ztop)**b) 

ubot = 1./(a*(earthradius-zbot)**b) 

 

r1 = radius(zbot) 

r2 = radius(ztop) 

burger_eta1 = r1 * ubot 

burger_eta2 = r2 * utop 

if b != 1: 

def cpe(eta): 

return num.arccos(num.minimum(p/num.maximum(eta, p/2), 1.0)) 

 

def sep(eta): 

return num.sqrt(num.maximum(eta**2 - p**2, 0.0)) 

 

x = (cpe(burger_eta2)-cpe(burger_eta1))/(1.0-b) 

t = (sep(burger_eta2)-sep(burger_eta1))/(1.0-b) 

else: 

lr = math.log(r2/r1) 

sap = num.sqrt(1.0/a**2 - p**2) 

x = p/sap * lr 

t = 1./(a**2 * sap) 

 

x *= r2d 

 

return x, t 

 

def test(self, p, mode, z): 

''' 

Check if wave mode can exist for given ray parameter at given depth 

within the layer. 

''' 

return (self.u(mode, z)*radius(z) - p) > 0. 

 

def tests(self, p, mode): 

utop, ubot = self.u_top_bottom(mode) 

return ( 

(utop * radius(self.ztop) - p) > 0., 

(ubot * radius(self.zbot) - p) > 0.) 

 

def zturn_potint(self, p, mode): 

'''Get turning depth for given ray parameter and propagation mode.''' 

 

a, b = self.potint_coefs(mode) 

r = num.exp(num.log(a*p)/(1.0-b)) 

return earthradius-r 

 

def propagate(self, p, mode, direction): 

'''Propagate ray through layer. 

 

:param p: ray parameter 

:param mode: propagation mode 

:param direction: in direction (:py:const:`UP` or :py:const:`DOWN`''' 

if direction == DOWN: 

zin, zout = self.ztop, self.zbot 

else: 

zin, zout = self.zbot, self.ztop 

 

if self.v(mode, zin) == 0.0 or not self.test(p, mode, zin): 

raise CannotPropagate(direction, self.ilayer) 

 

if not self.test(p, mode, zout): 

return -direction 

else: 

return direction 

 

def resize(self, depth_min=None, depth_max=None): 

'''Change layer thinkness and interpolate :py:class:`Material` if 

required.''' 

if depth_min: 

mtop = self.material(depth_min) 

 

if depth_max: 

mbot = self.material(depth_max) 

 

self.mtop = mtop if depth_min else self.mtop 

self.mbot = mbot if depth_max else self.mbot 

self.ztop = depth_min if depth_min else self.ztop 

self.zbot = depth_max if depth_max else self.zbot 

self.zmid = self.ztop + (self.zbot - self.ztop)/2. 

 

 

class DoesNotTurn(CakeError): 

pass 

 

 

def radius(z): 

return earthradius - z 

 

 

class HomogeneousLayer(Layer): 

'''Representation of a homogeneous layer in a layered earth model. 

 

Base class: :py:class:`Layer`. 

''' 

 

def __init__(self, ztop, zbot, m, name=None): 

Layer.__init__(self, ztop, zbot, name=name) 

self.m = m 

self.mtop = m 

self.mbot = m 

self._update_potint_coefs() 

 

def copy(self, ztop=None, zbot=None): 

if ztop is None: 

ztop = self.ztop 

 

if zbot is None: 

zbot = self.zbot 

 

return HomogeneousLayer(ztop, zbot, self.m, name=self.name) 

 

def material(self, z): 

return self.m 

 

def u(self, mode, z=None): 

if self._use_potential_interpolation[mode] and z is not None: 

return self.u_potint(mode, z) 

 

if mode == P: 

return 1./self.m.vp 

if mode == S: 

return 1./self.m.vs 

 

def u_top_bottom(self, mode): 

u = self.u(mode) 

return u, u 

 

def v(self, mode, z=None): 

if self._use_potential_interpolation[mode] and z is not None: 

return self.v_potint(mode, z) 

 

if mode == P: 

v = self.m.vp 

if mode == S: 

v = self.m.vs 

 

if num.isscalar(z): 

return v 

else: 

return filled(v, len(z)) 

 

def v_top_bottom(self, mode): 

v = self.v(mode) 

return v, v 

 

def xt(self, p, mode, zpart=None): 

if self._use_potential_interpolation[mode]: 

return self.xt_potint(p, mode, zpart) 

 

u = self.u(mode) 

pflat = self.pflat_bottom(p) 

if zpart is None: 

dz = (self.zbot - self.ztop) 

else: 

dz = abs(zpart[1]-zpart[0]) 

 

u = self.u(mode) 

eps = u*0.001 

denom = num.sqrt(u**2 - pflat**2) + eps 

 

x = r2d*pflat/(earthradius-self.zmid) * dz / denom 

t = u**2 * dz / denom 

return x, t 

 

def zturn(self, p, mode): 

if self._use_potential_interpolation[mode]: 

return self.zturn_potint(p, mode) 

 

raise DoesNotTurn() 

 

def split(self, z): 

upper = HomogeneousLayer(self.ztop, z, self.m, name=self.name) 

lower = HomogeneousLayer(z, self.zbot, self.m, name=self.name) 

upper.ilayer = self.ilayer 

lower.ilayer = self.ilayer 

return upper, lower 

 

def __str__(self): 

if self.name: 

name = self.name + ' ' 

else: 

name = '' 

 

calcmode = ''.join('HP'[self._use_potential_interpolation[mode]] 

for mode in (P, S)) 

 

return ' (%i) homogeneous layer %s(%g km - %g km) [%s]\n %s' % ( 

self.ilayer, name, self.ztop/km, self.zbot/km, calcmode, self.m) 

 

 

class GradientLayer(Layer): 

'''Representation of a gradient layer in a layered earth model. 

 

Base class: :py:class:`Layer`. 

''' 

 

def __init__(self, ztop, zbot, mtop, mbot, name=None): 

Layer.__init__(self, ztop, zbot, name=name) 

self.mtop = mtop 

self.mbot = mbot 

self._update_potint_coefs() 

 

def copy(self, ztop=None, zbot=None): 

if ztop is None: 

ztop = self.ztop 

 

if zbot is None: 

zbot = self.zbot 

 

return GradientLayer(ztop, zbot, self.mtop, self.mbot, name=self.name) 

 

def interpolate(self, z, ptop, pbot): 

return ptop + (z - self.ztop)*(pbot - ptop)/(self.zbot-self.ztop) 

 

def material(self, z): 

dtop = self.mtop.astuple() 

dbot = self.mbot.astuple() 

d = [ 

self.interpolate(z, ptop, pbot) 

for (ptop, pbot) in zip(dtop, dbot)] 

 

return Material(*d) 

 

def u_top_bottom(self, mode): 

if mode == P: 

return 1./self.mtop.vp, 1./self.mbot.vp 

if mode == S: 

return 1./self.mtop.vs, 1./self.mbot.vs 

 

def u(self, mode, z): 

if self._use_potential_interpolation[mode]: 

return self.u_potint(mode, z) 

 

if mode == P: 

return 1./self.interpolate(z, self.mtop.vp, self.mbot.vp) 

if mode == S: 

return 1./self.interpolate(z, self.mtop.vs, self.mbot.vs) 

 

def v_top_bottom(self, mode): 

if mode == P: 

return self.mtop.vp, self.mbot.vp 

if mode == S: 

return self.mtop.vs, self.mbot.vs 

 

def v(self, mode, z): 

if self._use_potential_interpolation[mode]: 

return self.v_potint(mode, z) 

 

if mode == P: 

return self.interpolate(z, self.mtop.vp, self.mbot.vp) 

if mode == S: 

return self.interpolate(z, self.mtop.vs, self.mbot.vs) 

 

def xt(self, p, mode, zpart=None): 

if self._use_potential_interpolation[mode]: 

return self.xt_potint(p, mode, zpart) 

 

utop, ubot = self.u_top_bottom(mode) 

b = (1./ubot - 1./utop)/(self.zbot - self.ztop) 

 

pflat = self.pflat_bottom(p) 

if zpart is not None: 

utop = self.u(mode, zpart[0]) 

ubot = self.u(mode, zpart[1]) 

 

peps = 1e-16 

pdp = pflat + peps 

 

def func(u): 

eta = num.sqrt(num.maximum(u**2 - pflat**2, 0.0)) 

xx = eta/u 

tt = num.where( 

pflat <= u, 

num.log(u+eta) - num.log(pdp) - eta/u, 

0.0) 

 

return xx, tt 

 

xxtop, tttop = func(utop) 

xxbot, ttbot = func(ubot) 

 

x = (xxtop - xxbot) / (b*pdp) 

t = (tttop - ttbot) / b + pflat*x 

 

x *= r2d/(earthradius - self.zmid) 

return x, t 

 

def zturn(self, p, mode): 

if self._use_potential_interpolation[mode]: 

return self.zturn_potint(p, mode) 

pflat = self.pflat_bottom(p) 

vtop, vbot = self.v_top_bottom(mode) 

return (1./pflat - vtop) * (self.zbot - self.ztop) / \ 

(vbot-vtop) + self.ztop 

 

def split(self, z): 

mmid = self.material(z) 

upper = GradientLayer(self.ztop, z, self.mtop, mmid, name=self.name) 

lower = GradientLayer(z, self.zbot, mmid, self.mbot, name=self.name) 

upper.ilayer = self.ilayer 

lower.ilayer = self.ilayer 

return upper, lower 

 

def __str__(self): 

if self.name: 

name = self.name + ' ' 

else: 

name = '' 

 

calcmode = ''.join('HP'[self._use_potential_interpolation[mode]] 

for mode in (P, S)) 

 

return ''' (%i) gradient layer %s(%g km - %g km) [%s] 

%s 

%s''' % ( 

self.ilayer, 

name, 

self.ztop/km, 

self.zbot/km, 

calcmode, 

self.mtop, 

self.mbot) 

 

 

class Discontinuity(object): 

'''Base class for discontinuities in layered earth model. 

 

Subclasses are: :py:class:`Interface` and :py:class:`Surface`. 

''' 

 

def __init__(self, z, name=None): 

self.z = z 

self.zbot = z 

self.ztop = z 

self.name = name 

 

def change_depth(self, z): 

self.z = z 

self.zbot = z 

self.ztop = z 

 

def copy(self): 

return copy.deepcopy(self) 

 

 

class Interface(Discontinuity): 

'''Representation of an interface in a layered earth model. 

 

Base class: :py:class:`Discontinuity`. 

''' 

 

def __init__(self, z, mabove, mbelow, name=None): 

Discontinuity.__init__(self, z, name) 

self.mabove = mabove 

self.mbelow = mbelow 

 

def __str__(self): 

if self.name is None: 

return 'interface' 

else: 

return 'interface "%s"' % self.name 

 

def u_top_bottom(self, mode): 

if mode == P: 

return reci_or_none(self.mabove.vp), reci_or_none(self.mbelow.vp) 

if mode == S: 

return reci_or_none(self.mabove.vs), reci_or_none(self.mbelow.vs) 

 

def critical_ps(self, mode): 

uabove, ubelow = self.u_top_bottom(mode) 

return ( 

mult_or_none(uabove, radius(self.z)), 

mult_or_none(ubelow, radius(self.z))) 

 

def propagate(self, p, mode, direction): 

uabove, ubelow = self.u_top_bottom(mode) 

if direction == DOWN: 

if ubelow is not None and ubelow*radius(self.z) - p >= 0: 

return direction 

else: 

return -direction 

if direction == UP: 

if uabove is not None and uabove*radius(self.z) - p >= 0: 

return direction 

else: 

return -direction 

 

def pflat(self, p): 

return p / (earthradius-self.z) 

 

def efficiency(self, in_direction, out_direction, in_mode, out_mode, p): 

scatter = psv_solid( 

self.mabove, self.mbelow, self.pflat(p), energy=True) 

return scatter[ 

psv_solid_ind(in_direction, out_direction, in_mode, out_mode)] 

 

 

class Surface(Discontinuity): 

'''Representation of the surface discontinuity in a layered earth model. 

 

Base class: :py:class:`Discontinuity`. 

''' 

 

def __init__(self, z, mbelow): 

Discontinuity.__init__(self, z, 'surface') 

self.z = z 

self.mbelow = mbelow 

 

def propagate(self, p, mode, direction): 

return direction # no implicit reflection at surface 

 

def u_top_bottom(self, mode): 

if mode == P: 

return None, reci_or_none(self.mbelow.vp) 

if mode == S: 

return None, reci_or_none(self.mbelow.vs) 

 

def critical_ps(self, mode): 

_, ubelow = self.u_top_bottom(mode) 

return None, mult_or_none(ubelow, radius(self.z)) 

 

def pflat(self, p): 

return p / (earthradius-self.z) 

 

def efficiency(self, in_direction, out_direction, in_mode, out_mode, p): 

if in_direction == DOWN or out_direction == UP: 

return 0.0 

else: 

return psv_surface( 

self.mbelow, self.pflat(p), energy=True)[ 

psv_surface_ind(in_mode, out_mode)] 

 

def __str__(self): 

return 'surface' 

 

 

class Walker(object): 

def __init__(self, elements): 

self._elements = elements 

self._i = 0 

 

def current(self): 

return self._elements[self._i] 

 

def go(self, direction): 

if direction == UP: 

self.up() 

else: 

self.down() 

 

def down(self): 

if self._i < len(self._elements)-1: 

self._i += 1 

else: 

raise BottomReached() 

 

def up(self): 

if self._i > 0: 

self._i -= 1 

else: 

raise SurfaceReached() 

 

def goto_layer(self, layer): 

self._i = self._elements.index(layer) 

 

 

class RayElement(object): 

'''An element of a :py:class:`RayPath`.''' 

 

def __eq__(self, other): 

return type(self) == type(other) and self.__dict__ == other.__dict__ 

 

def is_straight(self): 

return isinstance(self, Straight) 

 

def is_kink(self): 

return isinstance(self, Kink) 

 

 

class Straight(RayElement): 

''' 

A ray segment representing wave propagation through one :py:class:`Layer`. 

''' 

 

def __init__(self, direction_in, direction_out, mode, layer): 

self.mode = mode 

self._direction_in = direction_in 

self._direction_out = direction_out 

self.layer = layer 

 

def angle_in(self, p, endgaps=None): 

z = self.z_in(endgaps) 

dir = self.eff_direction_in(endgaps) 

v = self.layer.v(self.mode, z) 

pf = self.layer.pflat(p, z) 

 

if dir == DOWN: 

return num.arcsin(v*pf)*r2d 

else: 

return 180.-num.arcsin(v*pf)*r2d 

 

def angle_out(self, p, endgaps=None): 

z = self.z_out(endgaps) 

dir = self.eff_direction_out(endgaps) 

v = self.layer.v(self.mode, z) 

pf = self.layer.pflat(p, z) 

 

if dir == DOWN: 

return 180.-num.arcsin(v*pf)*r2d 

else: 

return num.arcsin(v*pf)*r2d 

 

def pflat_in(self, p, endgaps=None): 

return p / (earthradius-self.z_in(endgaps)) 

 

def pflat_out(self, p, endgaps=None): 

return p / (earthradius-self.z_out(endgaps)) 

 

def test(self, p, z): 

return self.layer.test(p, self.mode, z) 

 

def z_in(self, endgaps=None): 

if endgaps is not None: 

return endgaps[0] 

else: 

lyr = self.layer 

return (lyr.ztop, lyr.zbot)[self._direction_in == UP] 

 

def z_out(self, endgaps=None): 

if endgaps is not None: 

return endgaps[1] 

else: 

lyr = self.layer 

return (lyr.ztop, lyr.zbot)[self._direction_out == DOWN] 

 

def turns(self): 

return self._direction_in != self._direction_out 

 

def eff_direction_in(self, endgaps=None): 

if endgaps is None: 

return self._direction_in 

else: 

return endgaps[2] 

 

def eff_direction_out(self, endgaps=None): 

if endgaps is None: 

return self._direction_out 

else: 

return endgaps[3] 

 

def zturn(self, p): 

lyr = self.layer 

return lyr.zturn(p, self.mode) 

 

def u_in(self, endgaps=None): 

return self.layer.u(self.mode, self.z_in(endgaps)) 

 

def u_out(self, endgaps=None): 

return self.layer.u(self.mode, self.z_out(endgaps)) 

 

def critical_p_in(self, endgaps=None): 

z = self.z_in(endgaps) 

return self.layer.u(self.mode, z)*radius(z) 

 

def critical_p_out(self, endgaps=None): 

z = self.z_out(endgaps) 

return self.layer.u(self.mode, z)*radius(z) 

 

def xt(self, p, zpart=None): 

x, t = self.layer.xt(p, self.mode, zpart=zpart) 

if self._direction_in != self._direction_out and zpart is None: 

x *= 2. 

t *= 2. 

return x, t 

 

def xt_gap(self, p, zstart, zstop, samedir): 

z1, z2 = zstart, zstop 

if z1 > z2: 

z1, z2 = z2, z1 

 

x, t = self.layer.xt(p, self.mode, zpart=(z1, z2)) 

 

if samedir: 

return x, t 

else: 

xfull, tfull = self.xt(p) 

return xfull-x, tfull-t 

 

def __hash__(self): 

return hash(( 

self._direction_in, 

self._direction_out, 

self.mode, 

id(self.layer))) 

 

 

class HeadwaveStraight(Straight): 

def __init__(self, direction_in, direction_out, mode, interface): 

Straight.__init__(self, direction_in, direction_out, mode, None) 

 

self.interface = interface 

 

def z_in(self, zpart=None): 

return self.interface.z 

 

def z_out(self, zpart=None): 

return self.interface.z 

 

def zturn(self, p): 

return filled(self.interface.z, len(p)) 

 

def xt(self, p, zpart=None): 

return 0., 0. 

 

def x2t_headwave(self, xstretch): 

xstretch_m = xstretch*d2r*radius(self.interface.z) 

return min_not_none(*self.interface.u_top_bottom(self.mode))*xstretch_m 

 

 

class Kink(RayElement): 

'''An interaction of a ray with a :py:class:`Discontinuity`.''' 

 

def __init__( 

self, 

in_direction, 

out_direction, 

in_mode, 

out_mode, 

discontinuity): 

 

self.in_direction = in_direction 

self.out_direction = out_direction 

self.in_mode = in_mode 

self.out_mode = out_mode 

self.discontinuity = discontinuity 

 

def reflection(self): 

return self.in_direction != self.out_direction 

 

def conversion(self): 

return self.in_mode != self.out_mode 

 

def efficiency(self, p, out_direction=None, out_mode=None): 

 

if out_direction is None: 

out_direction = self.out_direction 

 

if out_mode is None: 

out_mode = self.out_mode 

 

return self.discontinuity.efficiency( 

self.in_direction, out_direction, self.in_mode, out_mode, p) 

 

def __str__(self): 

r, c = self.reflection(), self.conversion() 

if r and c: 

return '|~' 

if r: 

return '|' 

if c: 

return '~' 

return '_' 

 

def __hash__(self): 

return hash(( 

self.in_direction, 

self.out_direction, 

self.in_mode, 

self.out_mode, 

id(self.discontinuity))) 

 

 

class PRangeNotSet(CakeError): 

pass 

 

 

class RayPath(object): 

''' 

Representation of a fan of rays running through a common sequence of 

layers / interfaces. 

''' 

 

def __init__(self, phase): 

self.elements = [] 

self.phase = phase 

self._pmax = None 

self._pmin = None 

self._p = None 

self._is_headwave = False 

 

def set_is_headwave(self, is_headwave): 

self._is_headwave = is_headwave 

 

def copy(self): 

'''Get a copy of it.''' 

 

c = copy.copy(self) 

c.elements = list(self.elements) 

return c 

 

def endgaps(self, zstart, zstop): 

'''Get information needed for end point adjustments.''' 

 

return ( 

zstart, 

zstop, 

self.phase.direction_start(), 

self.phase.direction_stop()) 

 

def append(self, element): 

self.elements.append(element) 

 

def _check_have_prange(self): 

if self._pmax is None: 

raise PRangeNotSet() 

 

def set_prange(self, pmin, pmax, dp): 

self._pmin, self._pmax = pmin, pmax 

self._prange_dp = dp 

 

def used_phase(self, p=None, eps=1.): 

'''Calculate phase definition from ray path.''' 

 

used = PhaseDef() 

fleg = self.phase.first_leg() 

used.append(Leg(fleg.departure, fleg.mode)) 

n_elements_n = [None] + self.elements + [None] 

for before, element, after in zip( 

n_elements_n[:-2], 

n_elements_n[1:-1], 

n_elements_n[2:]): 

 

if element.is_kink() and HeadwaveStraight not in ( 

type(before), 

type(after)): 

 

if element.reflection() or element.conversion(): 

z = element.discontinuity.z 

used.append(Knee( 

z, 

element.in_direction, 

element.out_direction != element.in_direction, 

element.in_mode, 

element.out_mode)) 

 

used.append(Leg(element.out_direction, element.out_mode)) 

 

elif type(element) is HeadwaveStraight: 

z = element.interface.z 

k = Knee( 

z, 

before.in_direction, 

after.out_direction != before.in_direction, 

before.in_mode, 

after.out_mode) 

 

k.headwave = True 

used.append(k) 

used.append(Leg(after.out_direction, after.out_mode)) 

 

if (p is not None and before and after 

and element.is_straight() 

and before.is_kink() 

and after.is_kink() 

and element.turns() 

and not before.reflection() and not before.conversion() 

and not after.reflection() and not after.conversion()): 

 

ai = element.angle_in(p) 

if 90.0-eps < ai and ai < 90+eps: 

used.append( 

Head( 

before.discontinuity.z, 

before.out_direction, 

element.mode)) 

used.append( 

Leg(-before.out_direction, element.mode)) 

 

used._direction_stop = self.phase.direction_stop() 

used._definition = self.phase.definition() 

 

return used 

 

def pmax(self): 

'''Get maximum valid ray parameter.''' 

self._check_have_prange() 

return self._pmax 

 

def pmin(self): 

'''Get minimum valid ray parameter.''' 

self._check_have_prange() 

return self._pmin 

 

def xmin(self): 

'''Get minimal distance.''' 

self._analyse() 

return self._xmin 

 

def xmax(self): 

'''Get maximal distance.''' 

self._analyse() 

return self._xmax 

 

def kinks(self): 

''' 

Iterate over propagation mode changes (reflections/transmissions). 

''' 

return (k for k in self.elements if isinstance(k, Kink)) 

 

def straights(self): 

'''Iterate over ray segments.''' 

return (s for s in self.elements if isinstance(s, Straight)) 

 

def headwave_straight(self): 

for s in self.elements: 

if type(s) is HeadwaveStraight: 

return s 

 

def first_straight(self): 

'''Get first ray segment.''' 

for s in self.elements: 

if isinstance(s, Straight): 

return s 

 

def last_straight(self): 

'''Get last ray segment.''' 

for s in reversed(self.elements): 

if isinstance(s, Straight): 

return s 

 

def efficiency(self, p): 

''' 

Get product of all conversion/reflection coefficients encountered on 

path. 

''' 

return reduce( 

operator.mul, (k.efficiency(p) for k in self.kinks()), 1.) 

 

def spreading(self, p, endgaps): 

'''Get geometrical spreading factor.''' 

if self._is_headwave: 

return 0.0 

 

self._check_have_prange() 

dp = self._prange_dp * 0.01 

assert self._pmax - self._pmin > dp 

 

if p + dp > self._pmax: 

p = p-dp 

 

x0, t = self.xt(p, endgaps) 

x1, t = self.xt(p+dp, endgaps) 

x0 *= d2r 

x1 *= d2r 

if x1 == x0: 

return num.nan 

 

dp_dx = dp/(x1-x0) 

 

x = x0 

if x == 0.: 

x = x1 

p = dp 

 

first = self.first_straight() 

last = self.last_straight() 

return num.abs(dp_dx) * first.pflat_in(p, endgaps) / ( 

4.0 * math.pi * num.sin(x) * 

(earthradius-first.z_in(endgaps)) * 

(earthradius-last.z_out(endgaps))**2 * 

first.u_in(endgaps)**2 * 

num.abs(num.cos(first.angle_in(p, endgaps)*d2r)) * 

num.abs(num.cos(last.angle_out(p, endgaps)*d2r))) 

 

def make_p(self, dp=None, n=None, nmin=None): 

assert dp is None or n is None 

 

if self._pmin == self._pmax: 

return num.array([self._pmin]) 

 

if dp is None: 

dp = self._prange_dp 

 

if n is None: 

n = int(round((self._pmax-self._pmin)/dp)) + 1 

 

if nmin is not None: 

n = max(n, nmin) 

 

ppp = num.linspace(self._pmin, self._pmax, n) 

return ppp 

 

def xt_endgaps(self, p, endgaps, which='both'): 

''' 

Get amount of distance/traveltime to be subtracted at the generic ray 

path's ends. 

''' 

 

zstart, zstop, dirstart, dirstop = endgaps 

firsts = self.first_straight() 

lasts = self.last_straight() 

xs, ts = firsts.xt_gap( 

p, zstart, firsts.z_in(), dirstart == firsts._direction_in) 

xe, te = lasts.xt_gap( 

p, zstop, lasts.z_out(), dirstop == lasts._direction_out) 

 

if which == 'both': 

return xs + xe, ts + te 

elif which == 'left': 

return xs, ts 

elif which == 'right': 

return xe, te 

 

def xt_endgaps_ptest(self, p, endgaps): 

'''Check if ray parameter is valid at source and receiver.''' 

 

zstart, zstop, dirstart, dirstop = endgaps 

firsts = self.first_straight() 

lasts = self.last_straight() 

return num.logical_and(firsts.test(p, zstart), lasts.test(p, zstop)) 

 

def xt(self, p, endgaps): 

'''Calculate distance and traveltime for given ray parameter.''' 

 

if isinstance(p, num.ndarray): 

sx = num.zeros(p.size) 

st = num.zeros(p.size) 

else: 

sx = 0.0 

st = 0.0 

 

for s in self.straights(): 

x, t = s.xt(p) 

sx += x 

st += t 

 

if endgaps: 

dx, dt = self.xt_endgaps(p, endgaps) 

sx -= dx 

st -= dt 

 

return sx, st 

 

def xt_limits(self, p): 

''' 

Calculate limits of distance and traveltime for given ray parameter. 

''' 

 

if isinstance(p, num.ndarray): 

sx = num.zeros(p.size) 

st = num.zeros(p.size) 

sxe = num.zeros(p.size) 

ste = num.zeros(p.size) 

else: 

sx = 0.0 

st = 0.0 

sxe = 0.0 

ste = 0.0 

 

sfirst = self.first_straight() 

slast = self.last_straight() 

 

for s in self.straights(): 

if s is not sfirst and s is not slast: 

x, t = s.xt(p) 

sx += x 

st += t 

 

sends = [sfirst] 

if sfirst is not slast: 

sends.append(slast) 

 

for s in sends: 

x, t = s.xt(p) 

sxe += x 

ste += t 

 

return sx, (sx + sxe), st, (st + ste) 

 

def iter_zxt(self, p): 

''' 

Iterate over (depth, distance, traveltime) at each layer interface on 

ray path. 

''' 

 

sx = num.zeros(p.size) 

st = num.zeros(p.size) 

ok = False 

for s in self.straights(): 

yield s.z_in(), sx.copy(), st.copy() 

 

x, t = s.xt(p) 

sx += x 

st += t 

ok = True 

 

if ok: 

yield s.z_out(), sx.copy(), st.copy() 

 

def zxt_path_subdivided( 

self, p, endgaps, 

points_per_straight=20, 

x_for_headwave=None): 

 

'''Get geometrical representation of ray path.''' 

 

if self._is_headwave: 

assert p.size == 1 

x, t = self.xt(p, endgaps) 

xstretch = x_for_headwave-x 

nout = xstretch.size 

else: 

nout = p.size 

 

dxl, dtl = self.xt_endgaps(p, endgaps, which='left') 

dxr, dtr = self.xt_endgaps(p, endgaps, which='right') 

 

# first create full path including the endgaps 

sx = num.zeros(nout) - dxl 

st = num.zeros(nout) - dtl 

zxt = [] 

for s in self.straights(): 

n = points_per_straight 

 

back = None 

zin, zout = s.z_in(), s.z_out() 

if type(s) is HeadwaveStraight: 

z = zin 

for i in range(n): 

xs = float(i)/(n-1) * xstretch 

ts = s.x2t_headwave(xs) 

zxt.append((filled(z, xstretch.size), sx+xs, st+ts)) 

else: 

if zin != zout: # normal traversal 

zs = num.linspace(zin, zout, n).tolist() 

for z in zs: 

x, t = s.xt(p, zpart=sorted([zin, z])) 

zxt.append((filled(z, nout), sx + x, st + t)) 

 

else: # ray turns in layer 

zturn = s.zturn(p) 

back = [] 

for i in range(n): 

z = zin + (zturn - zin) * num.sin( 

float(i)/(n-1)*math.pi/2.0) * 0.999 

 

if zturn[0] >= zin: 

x, t = s.xt(p, zpart=[zin, z]) 

else: 

x, t = s.xt(p, zpart=[z, zin]) 

zxt.append((z, sx + x, st + t)) 

back.append((z, x, t)) 

 

if type(s) is HeadwaveStraight: 

x = xstretch 

t = s.x2t_headwave(xstretch) 

else: 

x, t = s.xt(p) 

 

sx += x 

st += t 

if back: 

for z, x, t in reversed(back): 

zxt.append((z, sx - x, st - t)) 

 

# gather results as arrays with such that x[ip, ipoint] 

fanz, fanx, fant = [], [], [] 

for z, x, t in zxt: 

fanz.append(z) 

fanx.append(x) 

fant.append(t) 

 

z = num.array(fanz).T 

x = num.array(fanx).T 

t = num.array(fant).T 

 

# cut off the endgaps, add exact endpoints 

xmax = x[:, -1] - dxr 

tmax = t[:, -1] - dtr 

zstart, zstop = endgaps[:2] 

zs, xs, ts = [], [], [] 

for i in range(nout): 

t_ = t[i] 

indices = num.where(num.logical_and(0. <= t_, t_ <= tmax[i]))[0] 

n = indices.size + 2 

zs_, xs_, ts_ = [num.empty(n, dtype=num.float) for j in range(3)] 

zs_[1:-1] = z[i, indices] 

xs_[1:-1] = x[i, indices] 

ts_[1:-1] = t[i, indices] 

zs_[0], zs_[-1] = zstart, zstop 

xs_[0], xs_[-1] = 0., xmax[i] 

ts_[0], ts_[-1] = 0., tmax[i] 

zs.append(zs_) 

xs.append(xs_) 

ts.append(ts_) 

 

return zs, xs, ts 

 

def _analyse(self): 

if self._p is not None: 

return 

 

p = self.make_p(nmin=20) 

xmin, xmax, tmin, tmax = self.xt_limits(p) 

 

self._x, self._t, self._p = xmax, tmax, p 

self._xmin, self._xmax = xmin.min(), xmax.max() 

self._tmin, self._tmax = tmin.min(), tmax.max() 

 

def draft_pxt(self, endgaps): 

self._analyse() 

 

if not self._is_headwave: 

cp, cx, ct = self._p, self._x, self._t 

pcrit = min( 

self.critical_pstart(endgaps), 

self.critical_pstop(endgaps)) 

 

if pcrit < self._pmin: 

empty = num.array([], dtype=num.float) 

return empty, empty, empty 

 

elif pcrit >= self._pmax: 

dx, dt = self.xt_endgaps(cp, endgaps) 

return cp, cx-dx, ct-dt 

 

else: 

n = num.searchsorted(cp, pcrit) + 1 

rp, rx, rt = num.empty((3, n), dtype=num.float) 

rp[:-1] = cp[:n-1] 

rx[:-1] = cx[:n-1] 

rt[:-1] = ct[:n-1] 

rp[-1] = pcrit 

rx[-1], rt[-1] = self.xt(pcrit, endgaps) 

dx, dt = self.xt_endgaps(rp, endgaps) 

rx[:-1] -= dx[:-1] 

rt[:-1] -= dt[:-1] 

return rp, rx, rt 

 

else: 

dx, dt = self.xt_endgaps(self._p, endgaps) 

p, x, t = self._p, self._x - dx, self._t - dt 

p, x, t = p[0], x[0], t[0] 

xh = num.linspace(0., x*10-x, 10) 

th = self.headwave_straight().x2t_headwave(xh) 

return filled(p, xh.size), x+xh, t+th 

 

def interpolate_x2pt_linear(self, x, endgaps): 

'''Get approximate ray parameter and traveltime for distance.''' 

 

self._analyse() 

 

if self._is_headwave: 

dx, dt = self.xt_endgaps(self._p, endgaps) 

xmin = self._x[0] - dx[0] 

tmin = self._t[0] - dt[0] 

el = self.headwave_straight() 

xok = x[x >= xmin] 

th = el.x2t_headwave(xstretch=(xok-xmin)) + tmin 

return [ 

(x_, self._p[0], t, None) for (x_, t) in zip(xok, th)] 

 

else: 

if num.all(x < self._xmin) or num.all(self._xmax < x): 

return [] 

 

rp, rx, rt = self.draft_pxt(endgaps) 

 

xp = interp(x, rx, rp, 0) 

xt = interp(x, rx, rt, 0) 

 

if (rp.size and 

len(xp) == 0 and 

rx[0] == 0.0 and 

any(x == 0.0) and 

rp[0] == 0.0): 

 

xp = [(0.0, rp[0])] 

xt = [(0.0, rt[0])] 

 

return [ 

(x_, p, t, (rp, rx, rt)) for ((x_, p), (_, t)) in zip(xp, xt)] 

 

def __eq__(self, other): 

if len(self.elements) != len(other.elements): 

return False 

 

return all(a == b for a, b in zip(self.elements, other.elements)) 

 

def __hash__(self): 

return hash( 

tuple(hash(x) for x in self.elements) + 

(self.phase.definition(), )) 

 

def __str__(self, p=None, eps=1.): 

x = [] 

start_i = None 

end_i = None 

turn_i = None 

 

def append_layers(si, ei, ti): 

if si == ei and (ti is None or ti == si): 

x.append('%i' % si) 

else: 

if ti is not None: 

x.append('(%i-%i-%i)' % (si, ti, ei)) 

else: 

x.append('(%i-%i)' % (si, ei)) 

 

for el in self.elements: 

if type(el) is Straight: 

if start_i is None: 

start_i = el.layer.ilayer 

if el._direction_in != el._direction_out: 

turn_i = el.layer.ilayer 

end_i = el.layer.ilayer 

 

elif isinstance(el, Kink): 

if start_i is not None: 

append_layers(start_i, end_i, turn_i) 

start_i = None 

turn_i = None 

 

x.append(str(el)) 

 

if start_i is not None: 

append_layers(start_i, end_i, turn_i) 

 

su = '(%s)' % self.used_phase(p=p, eps=eps).used_repr() 

 

return '%-15s %-17s %s' % (self.phase.definition(), su, ''.join(x)) 

 

def critical_pstart(self, endgaps): 

'''Get critical ray parameter for source depth choice.''' 

 

return self.first_straight().critical_p_in(endgaps) 

 

def critical_pstop(self, endgaps): 

'''Get critical ray parameter for receiver depth choice.''' 

 

return self.last_straight().critical_p_out(endgaps) 

 

def ranges(self, endgaps): 

'''Get valid ranges of ray parameter, distance, and traveltime.''' 

p, x, t = self.draft_pxt(endgaps) 

return p.min(), p.max(), x.min(), x.max(), t.min(), t.max() 

 

def describe(self, endgaps=None, as_degrees=False): 

'''Get textual representation.''' 

 

self._analyse() 

 

if as_degrees: 

xunit = 'deg' 

xfact = 1. 

else: 

xunit = 'km' 

xfact = d2m/km 

 

sg = ''' Ranges for all depths in source and receiver layers: 

- x [%g, %g] %s 

- t [%g, %g] s 

- p [%g, %g] s/deg 

''' % ( 

self._xmin*xfact, 

self._xmax*xfact, 

xunit, 

self._tmin, 

self._tmax, 

self._pmin/r2d, 

self._pmax/r2d) 

 

if endgaps is not None: 

pmin, pmax, xmin, xmax, tmin, tmax = self.ranges(endgaps) 

ss = ''' Ranges for given source and receiver depths: 

\n - x [%g, %g] %s 

\n - t [%g, %g] s 

\n - p [%g, %g] s/deg 

\n''' % (xmin*xfact, xmax*xfact, xunit, tmin, tmax, pmin/r2d, pmax/r2d) 

 

else: 

ss = '' 

 

return '%s\n' % self + ss + sg 

 

 

class RefineFailed(CakeError): 

pass 

 

 

class Ray(object): 

''' 

Representation of a ray with a specific (path, ray parameter, distance, 

arrival time) choice. 

 

**Attributes:** 

 

.. py:attribute:: path 

 

:py:class:`RayPath` object containing complete propagation history. 

 

.. py:attribute:: p 

 

Ray parameter (spherical) [s/rad] 

 

.. py:attribute:: x 

 

Radial distance [deg] 

 

.. py:attribute:: t 

 

Traveltime [s] 

 

.. py:attribute:: endgaps 

 

Needed for source/receiver depth adjustments in many 

:py:class:`RayPath` methods. 

''' 

 

def __init__(self, path, p, x, t, endgaps, draft_pxt): 

self.path = path 

self.p = p 

self.x = x 

self.t = t 

self.endgaps = endgaps 

self.draft_pxt = draft_pxt 

 

def given_phase(self): 

'''Get phase definition which was used to create the ray. 

 

:returns: :py:class:`PhaseDef` object 

''' 

 

return self.path.phase 

 

def used_phase(self): 

'''Compute phase definition from propagation path. 

 

:returns: :py:class:`PhaseDef` object 

''' 

 

return self.path.used_phase(self.p) 

 

def refine(self): 

if self.path._is_headwave: 

return 

 

if self.t == 0.0 and self.p == 0.0 and self.x == 0.0: 

return 

 

cp, cx, ct = self.draft_pxt 

ip = num.searchsorted(cp, self.p) 

if not (0 < ip < cp.size): 

raise RefineFailed() 

 

pl, ph = cp[ip-1], cp[ip] 

p_to_t = {} 

i = [0] 

 

def f(p): 

i[0] += 1 

x, t = self.path.xt(p, self.endgaps) 

p_to_t[p] = t 

return self.x - x 

 

try: 

self.p = brentq(f, pl, ph) 

self.t = p_to_t[self.p] 

 

except ValueError: 

raise RefineFailed() 

 

def takeoff_angle(self): 

'''Get takeoff angle of ray. 

 

The angle is returned in [degrees]. 

''' 

 

return self.path.first_straight().angle_in(self.p, self.endgaps) 

 

def incidence_angle(self): 

'''Get incidence angle of ray. 

 

The angle is returned in [degrees]. 

''' 

 

return self.path.last_straight().angle_out(self.p, self.endgaps) 

 

def efficiency(self): 

'''Get conversion/reflection efficiency of the ray. 

 

A value between 0 and 1 is returned, reflecting the relative amount of 

energy which is transmitted along the ray and not lost by reflections 

or conversions. 

''' 

 

return self.path.efficiency(self.p) 

 

def spreading(self): 

'''Get geometrical spreading factor.''' 

 

return self.path.spreading(self.p, self.endgaps) 

 

def surface_sphere(self): 

x1, y1 = 0., earthradius - self.endgaps[0] 

r2 = earthradius - self.endgaps[1] 

x2, y2 = r2*math.sin(self.x*d2r), r2*math.cos(self.x*d2r) 

return ((x2-x1)**2 + (y2-y1)**2)*4.0*math.pi 

 

def zxt_path_subdivided(self, points_per_straight=20): 

'''Get geometrical representation of ray path. 

 

Three arrays (depth, distance, time) with points on the ray's path of 

propagation are returned. The number of points which are used in each 

ray segment (passage through one layer) may be controlled by the 

``points_per_straight`` parameter. 

''' 

return self.path.zxt_path_subdivided( 

num.atleast_1d(self.p), self.endgaps, 

points_per_straight=points_per_straight, 

x_for_headwave=num.atleast_1d(self.x)) 

 

def __str__(self, as_degrees=False): 

if as_degrees: 

sd = '%6.3g deg' % self.x 

else: 

sd = '%7.5g km' % (self.x*(d2r*earthradius/km)) 

 

return '%7.5g s/deg %s %6.4g s %5.1f %5.1f %3.0f%% %3.0f%% %s' % ( 

self.p/r2d, 

sd, 

self.t, 

self.takeoff_angle(), 

self.incidence_angle(), 

100*self.efficiency(), 

100*self.spreading()*self.surface_sphere(), 

self.path.__str__(p=self.p)) 

 

 

def anything_to_crust2_profile(crust2_profile): 

from pyrocko.dataset import crust2x2 

if isinstance(crust2_profile, tuple): 

lat, lon = [float(x) for x in crust2_profile] 

return crust2x2.get_profile(lat, lon) 

elif isinstance(crust2_profile, (str, newstr)): 

return crust2x2.get_profile(crust2_profile) 

elif isinstance(crust2_profile, crust2x2.Crust2Profile): 

return crust2_profile 

else: 

assert False, 'crust2_profile must be (lat, lon) a profile ' \ 

'key or a crust2x2 Profile object)' 

 

 

class DiscontinuityNotFound(CakeError): 

def __init__(self, depth_or_name): 

CakeError.__init__(self) 

self.depth_or_name = depth_or_name 

 

def __str__(self): 

return 'Cannot find discontinuity from given depth or name: %s' % \ 

self.depth_or_name 

 

 

class LayeredModelError(CakeError): 

pass 

 

 

class LayeredModel(object): 

'''Representation of a layer cake model. 

 

There are several ways to initialize an instance of this class. 

 

1. Use the module function :py:func:`load_model` to read a model from a 

file. 

2. Create an empty model with the default constructor and append layers and 

discontinuities with the :py:meth:`append` method (from top to bottom). 

3. Use the constructor :py:meth:`LayeredModel.from_scanlines`, to 

automatically create the :py:class:`Layer` and :py:class:`Discontinuity` 

objects from a given velocity profile. 

 

An earth model is represented by as stack of :py:class:`Layer` and 

:py:class:`Discontinuity` objects. The method :py:meth:`arrivals` returns 

:py:class:`Ray` objects which may be e.g. queried for arrival times of 

specific phases. Each ray is associated with a :py:class:`RayPath` object. 

Ray objects share common ray paths if they have the same 

conversion/reflection/propagation history. Creating the ray path objects is 

relatively expensive (this is done in :py:meth:`gather_paths`), but they 

are cached for reuse in successive invocations. 

''' 

 

def __init__(self): 

self._surface_material = None 

self._elements = [] 

self.nlayers = 0 

self._np = 10000 

self._pdepth = 5 

self._pathcache = {} 

 

def copy_with_elevation(self, elevation): 

'''Get a copy of the model with surface layer stretched to given elevation. 

 

:param elevation: new surface elevation in [m] 

 

Elevation is positiv upward, contrary to the layered models downward 

`z` axis. 

''' 

 

c = copy.deepcopy(self) 

c._pathcache = {} 

surface = c._elements[0] 

toplayer = c._elements[1] 

 

assert toplayer.zbot > -elevation 

 

surface.z = -elevation 

c._elements[1] = toplayer.copy(ztop=-elevation) 

c._elements[1].ilayer = 0 

return c 

 

def zeq(self, z1, z2): 

return abs(z1-z2) < ZEPS 

 

def append(self, element): 

'''Add a layer or discontinuity at bottom of model. 

 

:param element: object of subclass of :py:class:`Layer` or 

:py:class:`Discontinuity`. 

''' 

 

if isinstance(element, Layer): 

if element.zbot >= earthradius: 

element.zbot = earthradius - 1. 

 

if element.ztop >= earthradius: 

raise CakeError('Layer deeper than earthradius') 

 

element.ilayer = self.nlayers 

self.nlayers += 1 

 

self._elements.append(element) 

 

def elements(self, direction=DOWN): 

'''Iterate over all elements of the model. 

 

:param direction: direction of traversal :py:const:`DOWN` or 

:py:const:`UP`. 

 

Objects derived from the :py:class:`Discontinuity` and 

:py:class:`Layer` classes are yielded. 

''' 

 

if direction == DOWN: 

return iter(self._elements) 

else: 

return reversed(self._elements) 

 

def layers(self, direction=DOWN): 

'''Iterate over all layers of model. 

 

:param direction: direction of traversal :py:const:`DOWN` or 

:py:const:`UP`. 

 

Objects derived from the :py:class:`Layer` class are yielded. 

''' 

 

if direction == DOWN: 

return (el for el in self._elements if isinstance(el, Layer)) 

else: 

return ( 

el for el in reversed(self._elements) if isinstance(el, Layer)) 

 

def layer(self, z, direction=DOWN): 

'''Get layer for given depth. 

 

:param z: depth [m] 

:param direction: direction of traversal :py:const:`DOWN` or 

:py:const:`UP`. 

 

Returns first layer which touches depth ``z`` (tolerant at boundaries). 

''' 

 

for layer in self.layers(direction): 

if layer.contains(z): 

return layer 

else: 

raise CakeError('Failed extracting layer at depth z=%s' % z) 

 

def walker(self): 

return Walker(self._elements) 

 

def material(self, z, direction=DOWN): 

'''Get material at given depth. 

 

:param z: depth [m] 

:param direction: direction of traversal :py:const:`DOWN` or 

:py:const:`UP` 

:returns: object of type :py:class:`Material` 

 

If given depth ``z`` happens to be at an interface, the material of the 

first layer with respect to the the traversal ordering is returned. 

''' 

 

lyr = self.layer(z, direction) 

return lyr.material(z) 

 

def discontinuities(self): 

'''Iterate over all discontinuities of the model.''' 

 

return (el for el in self._elements if isinstance(el, Discontinuity)) 

 

def discontinuity(self, name_or_z): 

'''Get discontinuity by name or depth. 

 

:param name_or_z: name of discontinuity or depth [m] as float value 

''' 

 

if isinstance(name_or_z, float): 

candi = sorted( 

self.discontinuities(), key=lambda i: abs(i.z-name_or_z)) 

else: 

candi = [i for i in self.discontinuities() if i.name == name_or_z] 

 

if not candi: 

raise DiscontinuityNotFound(name_or_z) 

 

return candi[0] 

 

def adapt_phase(self, phase): 

'''Adapt a phase definition for use with this model. 

 

This returns a copy of the phase definition, where named 

discontinuities are replaced with the actual depth of these, as defined 

in the model. 

''' 

 

phase = phase.copy() 

for knee in phase.knees(): 

if knee.depth != 'surface': 

knee.depth = self.discontinuity(knee.depth).z 

for leg in phase.legs(): 

if leg.depthmax is not None and isinstance(leg.depthmax, str): 

leg.depthmax = self.discontinuity(leg.depthmax).z 

 

return phase 

 

def path(self, p, phase, layer_start, layer_stop): 

''' 

Get ray path for given combination of ray parameter, phase definition, 

source and receiver layers. 

 

:param p: ray parameter (spherical) [s/rad] 

:param phase: phase definition (:py:class:`PhaseDef` object) 

:param layer_start: layer with source 

:param layer_stop: layer with receiver 

:returns: :py:class:`RayPath` object 

 

If it is not possible to find a solution, an exception of type 

:py:exc:`NotPhaseConform`, :py:exc:`MinDepthReached`, 

:py:exc:`MaxDepthReached`, :py:exc:`CannotPropagate`, 

:py:exc:`BottomReached` or :py:exc:`SurfaceReached` is raised. 

''' 

 

phase = self.adapt_phase(phase) 

knees = phase.knees() 

legs = phase.legs() 

next_knee = next_or_none(knees) 

leg = next_or_none(legs) 

assert leg is not None 

 

direction = leg.departure 

direction_stop = phase.direction_stop() 

mode = leg.mode 

mode_stop = phase.last_leg().mode 

 

walker = self.walker() 

walker.goto_layer(layer_start) 

current = walker.current() 

 

ttop, tbot = current.tests(p, mode) 

if not ttop and not tbot: 

raise CannotPropagate(direction, current.ilayer) 

 

if (direction == DOWN and not ttop) or (direction == UP and not tbot): 

direction = -direction 

 

path = RayPath(phase) 

trapdetect = set() 

while True: 

at_layer = isinstance(current, Layer) 

at_discontinuity = isinstance(current, Discontinuity) 

 

# detect trapped wave 

k = (id(next_knee), id(current), direction, mode) 

if k in trapdetect: 

raise Trapped() 

 

trapdetect.add(k) 

 

if at_discontinuity: 

oldmode, olddirection = mode, direction 

headwave = False 

if next_knee is not None and next_knee.matches( 

current, mode, direction): 

 

headwave = next_knee.headwave 

direction = next_knee.out_direction() 

mode = next_knee.out_mode 

next_knee = next_or_none(knees) 

leg = next(legs) 

 

else: # implicit reflection/transmission 

direction = current.propagate(p, mode, direction) 

 

if headwave: 

path.set_is_headwave(True) 

 

path.append(Kink( 

olddirection, olddirection, oldmode, oldmode, current)) 

 

path.append(HeadwaveStraight( 

olddirection, direction, oldmode, current)) 

 

path.append(Kink( 

olddirection, direction, oldmode, mode, current)) 

 

else: 

path.append(Kink( 

olddirection, direction, oldmode, mode, current)) 

 

if at_layer: 

direction_in = direction 

direction = current.propagate(p, mode, direction_in) 

 

zturn = None 

if direction_in != direction: 

zturn = current.zturn(p, mode) 

 

zmin, zmax = leg.depthmin, leg.depthmax 

if zmin is not None or zmax is not None: 

if direction_in != direction: 

if zmin is not None and zturn <= zmin: 

raise MinDepthReached() 

if zmax is not None and zturn >= zmax: 

raise MaxDepthReached() 

else: 

if zmin is not None and current.ztop <= zmin: 

raise MinDepthReached() 

if zmax is not None and current.zbot >= zmax: 

raise MaxDepthReached() 

 

path.append(Straight(direction_in, direction, mode, current)) 

 

if next_knee is None and mode == mode_stop and \ 

current is layer_stop: 

 

if zturn is None: 

if direction == direction_stop: 

break 

else: 

break 

 

walker.go(direction) 

current = walker.current() 

 

return path 

 

def gather_paths(self, phases=PhaseDef('P'), zstart=0.0, zstop=0.0): 

''' 

Get all possible ray paths for given source and receiver depths for one 

or more phase definitions. 

 

:param phases: a :py:class:`PhaseDef` object or a list of such objects. 

Comma-separated strings and lists of such strings are also accepted 

and are converted to :py:class:`PhaseDef` objects for convenience. 

:param zstart: source depth [m] 

:param zstop: receiver depth [m] 

:returns: a list of :py:class:`RayPath` objects 

 

Results of this method are cached internally. Cached results are 

returned, when a given combination of source layer, receiver layer and 

phase definition has been used before. 

''' 

 

eps = 1e-7 # num.finfo(float).eps * 1000. 

 

phases = to_phase_defs(phases) 

 

paths = [] 

for phase in phases: 

 

layer_start = self.layer(zstart, -phase.direction_start()) 

layer_stop = self.layer(zstop, phase.direction_stop()) 

 

pathcachekey = (phase.definition(), layer_start, layer_stop) 

 

if pathcachekey in self._pathcache: 

phase_paths = self._pathcache[pathcachekey] 

else: 

hwknee = phase.headwave_knee() 

if hwknee: 

name_or_z = hwknee.depth 

interface = self.discontinuity(name_or_z) 

mode = hwknee.in_mode 

in_direction = hwknee.direction 

 

pabove, pbelow = interface.critical_ps(mode) 

 

p = min_not_none(pabove, pbelow) 

 

# diffracted wave: 

if in_direction == DOWN and ( 

pbelow is None or pbelow >= pabove): 

 

p *= (1.0 - eps) 

 

path = self.path(p, phase, layer_start, layer_stop) 

path.set_prange(p, p, 1.) 

 

phase_paths = [path] 

 

else: 

try: 

pmax_start = max([ 

radius(z)/layer_start.v(phase.first_leg().mode, z) 

for z in (layer_start.ztop, layer_start.zbot)]) 

 

pmax_stop = max([ 

radius(z)/layer_stop.v(phase.last_leg().mode, z) 

for z in (layer_stop.ztop, layer_stop.zbot)]) 

 

pmax = min(pmax_start, pmax_stop) 

 

pedges = [0.] 

for layer in self.layers(): 

for z in (layer.ztop, layer.zbot): 

for mode in (P, S): 

for eps2 in [eps]: 

v = layer.v(mode, z) 

if v != 0.0: 

p = radius(z)/v 

if p <= pmax: 

pedges.append(p*(1.0-eps2)) 

pedges.append(p) 

pedges.append(p*(1.0+eps2)) 

 

pedges = num.unique(sorted(pedges)) 

 

phase_paths = {} 

cached = {} 

counter = [0] 

 

def p_to_path(p): 

if p in cached: 

return cached[p] 

 

try: 

counter[0] += 1 

path = self.path( 

p, phase, layer_start, layer_stop) 

 

if path not in phase_paths: 

phase_paths[path] = [] 

 

phase_paths[path].append(p) 

 

except PathFailed: 

path = None 

 

cached[p] = path 

return path 

 

def recurse(pmin, pmax, i=0): 

if i > self._pdepth: 

return 

path1 = p_to_path(pmin) 

path2 = p_to_path(pmax) 

if path1 is None and path2 is None and i > 0: 

return 

if path1 is None or path2 is None or \ 

hash(path1) != hash(path2): 

 

recurse(pmin, (pmin+pmax)/2., i+1) 

recurse((pmin+pmax)/2., pmax, i+1) 

 

for (pl, ph) in zip(pedges[:-1], pedges[1:]): 

recurse(pl, ph) 

 

for path, ps in phase_paths.items(): 

path.set_prange( 

min(ps), max(ps), pmax/(self._np-1)) 

 

phase_paths = list(phase_paths.keys()) 

 

except ZeroDivisionError: 

phase_paths = [] 

 

self._pathcache[pathcachekey] = phase_paths 

 

paths.extend(phase_paths) 

 

paths.sort(key=lambda x: x.pmin()) 

return paths 

 

def arrivals( 

self, 

distances=[], 

phases=PhaseDef('P'), 

zstart=0.0, 

zstop=0.0, 

refine=True): 

 

'''Compute rays and traveltimes for given distances. 

 

:param distances: list or array of distances [deg] 

:param phases: a :py:class:`PhaseDef` object or a list of such objects. 

Comma-separated strings and lists of such strings are also accepted 

and are converted to :py:class:`PhaseDef` objects for convenience. 

:param zstart: source depth [m] 

:param zstop: receiver depth [m] 

:param refine: bool flag, whether to use bisectioning to improve 

(p, x, t) estimated from interpolation 

:returns: a list of :py:class:`Ray` objects, sorted by 

(distance, arrival time) 

''' 

 

distances = num.asarray(distances, dtype=num.float) 

 

arrivals = [] 

for path in self.gather_paths(phases, zstart=zstart, zstop=zstop): 

 

endgaps = path.endgaps(zstart, zstop) 

for x, p, t, draft_pxt in path.interpolate_x2pt_linear( 

distances, endgaps): 

 

arrivals.append(Ray(path, p, x, t, endgaps, draft_pxt)) 

 

if refine: 

refined = [] 

for ray in arrivals: 

 

if ray.path._is_headwave: 

refined.append(ray) 

 

try: 

ray.refine() 

refined.append(ray) 

 

except RefineFailed: 

pass 

 

arrivals = refined 

 

arrivals.sort(key=lambda x: (x.x, x.t)) 

return arrivals 

 

@classmethod 

def from_scanlines(cls, producer): 

'''Create layer cake model from sequence of materials at depths. 

 

:param producer: iterable yielding (depth, material, name) tuples 

 

Creates a new :py:class:`LayeredModel` object and uses its 

:py:meth:`append` method to add layers and discontinuities as needed. 

''' 

 

self = cls() 

for z, material, name in producer: 

 

if not self._elements: 

self.append(Surface(z, material)) 

else: 

element = self._elements[-1] 

if self.zeq(element.zbot, z): 

assert isinstance(element, Layer) 

self.append( 

Interface(z, element.mbot, material, name=name)) 

 

else: 

if isinstance(element, Discontinuity): 

ztop = element.z 

mtop = element.mbelow 

elif isinstance(element, Layer): 

ztop = element.zbot 

mtop = element.mbot 

 

if mtop == material: 

layer = HomogeneousLayer( 

ztop, z, material, name=name) 

else: 

layer = GradientLayer( 

ztop, z, mtop, material, name=name) 

 

self.append(layer) 

 

return self 

 

def to_scanlines(self, get_burgers=False): 

def fmt(z, m): 

if not m._has_default_burgers() or get_burgers: 

return (z, m.vp, m.vs, m.rho, m.qp, m.qs, 

m.burger_eta1, m.burger_eta2, m.burger_valpha) 

return (z, m.vp, m.vs, m.rho, m.qp, m.qs) 

 

last = None 

lines = [] 

for element in self.elements(): 

if isinstance(element, Layer): 

if not isinstance(last, Layer): 

lines.append(fmt(element.ztop, element.mtop)) 

 

lines.append(fmt(element.zbot, element.mbot)) 

 

last = element 

 

if not isinstance(last, Layer): 

lines.append(fmt(last.z, last.mbelow)) 

 

return lines 

 

def iter_material_parameter(self, get): 

assert get in ('vp', 'vs', 'rho', 'qp', 'qs', 'z') 

if get == 'z': 

for layer in self.layers(): 

yield layer.ztop 

yield layer.zbot 

else: 

getter = operator.attrgetter(get) 

for layer in self.layers(): 

yield getter(layer.mtop) 

yield getter(layer.mbot) 

 

def profile(self, get): 

''' 

Get parameter profile along depth of the earthmodel. 

 

:param get: property to be queried ( 

``'vp'``, ``'vs'``, ``'rho'``, ``'qp'``, or ``'qs'``, or ``'z'``) 

:type get: string 

''' 

 

return num.array(list(self.iter_material_parameter(get))) 

 

def min(self, get='vp'): 

''' 

Find minimum value of a material property or depth. 

 

:param get: property to be queried ( 

``'vp'``, ``'vs'``, ``'rho'``, ``'qp'``, or ``'qs'``, or ``'z'``) 

''' 

 

return min(self.iter_material_parameter(get)) 

 

def max(self, get='vp'): 

''' 

Find maximum value of a material property or depth. 

 

:param get: property to be queried ( 

``'vp'``, ``'vs'``, ``'rho'``, ``'qp'``, ``'qs'``, or ``'z'``) 

''' 

 

return max(self.iter_material_parameter(get)) 

 

def simplify_layers(self, layers, max_rel_error=0.001): 

if len(layers) <= 1: 

return layers 

 

ztop = layers[0].ztop 

zbot = layers[-1].zbot 

zorigs = [layer.ztop for layer in layers] 

zorigs.append(zbot) 

zs = num.linspace(ztop, zbot, 100) 

data = [] 

for z in zs: 

if z == ztop: 

direction = UP 

else: 

direction = DOWN 

 

mat = self.material(z, direction) 

data.append(mat.astuple()) 

 

data = num.array(data, dtype=num.float) 

data_means = num.mean(data, axis=0) 

nmax = len(layers) // 2 

accept = False 

 

zcut_best = [] 

for n in range(1, nmax+1): 

ncutintervals = 20 

zdelta = (zbot-ztop)/ncutintervals 

if n == 2: 

zcuts = [ 

[ztop, ztop + i*zdelta, zbot] 

for i in range(1, ncutintervals)] 

elif n == 3: 

zcuts = [] 

for j in range(1, ncutintervals): 

for i in range(j+1, ncutintervals): 

zcuts.append( 

[ztop, ztop + j*zdelta, ztop + i*zdelta, zbot]) 

else: 

zcuts = [] 

zcuts.append(num.linspace(ztop, zbot, n+1)) 

if zcut_best: 

zcuts.append(sorted(num.linspace( 

ztop, zbot, n).tolist() + zcut_best[1])) 

zcuts.append(sorted(num.linspace( 

ztop, zbot, n-1).tolist() + zcut_best[2])) 

 

best = None 

for icut, zcut in enumerate(zcuts): 

rel_par_errors = num.zeros(5) 

mpar_nodes = num.zeros((n+1, 5)) 

 

for ipar in range(5): 

znodes, vnodes, error_rms = util.polylinefit( 

zs, data[:, ipar], zcut) 

 

mpar_nodes[:, ipar] = vnodes 

if data_means[ipar] == 0.0: 

rel_par_errors[ipar] = -1 

else: 

rel_par_errors[ipar] = error_rms/data_means[ipar] 

 

rel_error = rel_par_errors.max() 

if best is None or rel_error < best[0]: 

best = (rel_error, zcut, mpar_nodes) 

 

rel_error, zcut, mpar_nodes = best 

 

zcut_best.append(list(zcut)) 

zcut_best[-1].pop(0) 

zcut_best[-1].pop() 

 

if rel_error <= max_rel_error: 

accept = True 

break 

 

if not accept: 

return layers 

 

rel_error, zcut, mpar_nodes = best 

 

material_nodes = [] 

for i in range(n+1): 

material_nodes.append(Material(*mpar_nodes[i, :])) 

 

out_layers = [] 

for i in range(n): 

mtop = material_nodes[i] 

mbot = material_nodes[i+1] 

ztop = zcut[i] 

zbot = zcut[i+1] 

if mtop == mbot: 

lyr = HomogeneousLayer(ztop, zbot, mtop) 

else: 

lyr = GradientLayer(ztop, zbot, mtop, mbot) 

 

out_layers.append(lyr) 

return out_layers 

 

def simplify(self, max_rel_error=0.001): 

'''Get representation of model with lower resolution. 

 

Returns an approximation of the model. All discontinuities are kept, 

but layer stacks with continuous model parameters are represented, if 

possible, by a lower number of layers. Piecewise linear functions are 

fitted against the original model parameter's piecewise linear 

functions. Successively larger numbers of layers are tried, until the 

difference to the original model is below ``max_rel_error``. The 

difference is measured as the RMS error of the fit normalized by the 

mean of the input (i.e. the fitted curves should deviate, on average, 

less than 0.1% from the input curves if ``max_rel_error`` = 0.001).''' 

 

mod_simple = LayeredModel() 

 

glayers = [] 

for element in self.elements(): 

 

if isinstance(element, Discontinuity): 

for layer in self.simplify_layers( 

glayers, max_rel_error=max_rel_error): 

 

mod_simple.append(layer) 

 

glayers = [] 

mod_simple.append(element) 

else: 

glayers.append(element) 

 

for layer in self.simplify_layers( 

glayers, max_rel_error=max_rel_error): 

mod_simple.append(layer) 

 

return mod_simple 

 

def extract(self, depth_min=None, depth_max=None): 

'''Extract :py:class:`LayeredModel` from :py:class:`LayeredModel`. 

 

:param depth_min: depth of upper cut or name of :py:class:`Interface` 

:param depth_max: depth of lower cut or name of :py:class:`Interface` 

 

Interpolates a :py:class:`GradientLayer` at ``depth_min`` and/or 

``depth_max``.''' 

 

if isinstance(depth_min, (str, newstr)): 

depth_min = self.discontinuity(depth_min).z 

 

if isinstance(depth_max, (str, newstr)): 

depth_max = self.discontinuity(depth_max).z 

 

mod_extracted = LayeredModel() 

 

for element in self.elements(): 

element = element.copy() 

do_append = False 

if (depth_min is None or depth_min <= element.ztop) \ 

and (depth_max is None or depth_max >= element.zbot): 

mod_extracted.append(element) 

continue 

 

if depth_min is not None: 

if element.ztop < depth_min and depth_min < element.zbot: 

_, element = element.split(depth_min) 

do_append = True 

 

if depth_max is not None: 

if element.zbot > depth_max and depth_max > element.ztop: 

element, _ = element.split(depth_max) 

do_append = True 

 

if do_append: 

mod_extracted.append(element) 

 

return mod_extracted 

 

def replaced_crust(self, crust2_profile=None, crustmod=None): 

if crust2_profile is not None: 

profile = anything_to_crust2_profile(crust2_profile) 

crustmod = LayeredModel.from_scanlines( 

from_crust2x2_profile(profile)) 

 

newmod = LayeredModel() 

for element in crustmod.extract(depth_max='moho').elements(): 

if element.name != 'moho': 

newmod.append(element) 

else: 

moho1 = element 

 

mod = self.extract(depth_min='moho') 

first = True 

for element in mod.elements(): 

if element.name == 'moho': 

if element.z <= moho1.z: 

mbelow = mod.material(moho1.z, direction=UP) 

else: 

mbelow = element.mbelow 

 

moho = Interface(moho1.z, moho1.mabove, mbelow, name='moho') 

newmod.append(moho) 

else: 

if first: 

if isinstance(element, Layer) and element.zbot > moho.z: 

newmod.append(GradientLayer( 

moho.z, 

element.zbot, 

moho.mbelow, 

element.mbot, 

name=element.name)) 

 

first = False 

else: 

newmod.append(element) 

return newmod 

 

def perturb(self, rstate=None, keep_vp_vs=False, **kwargs): 

''' 

Create a perturbed variant of the earth model. 

 

Randomly change the thickness and material parameters of the earth 

model from a uniform distribution. 

 

:param kwargs: Maximum change fraction (e.g. 0.1) of the parameters. 

Name the parameter, prefixed by ``p``. Supported parameters are 

``ph, pvp, pvs, prho, pqs, pqp``. 

:type kwargs: dict 

:param rstate: Random state to draw from, defaults to ``None`` 

:type rstate: :class:`numpy.random.RandomState`, optional 

:param keep_vp_vs: Keep the Vp/Vs ratio, defaults to False 

:type keep_vp_vs: bool, optional 

 

:returns: A new, perturbed earth model 

:rtype: :class:`~pyrocko.cake.LayeredModel` 

 

.. code-block :: python 

 

perturbed_model = model.perturb(ph=.1, pvp=.05, prho=.1) 

''' 

_pargs = set(['ph', 'pvp', 'pvs', 'prho', 'pqs', 'pqp']) 

earthmod = copy.deepcopy(self) 

 

if rstate is None: 

rstate = num.random.RandomState() 

 

layers = earthmod.layers() 

discont = earthmod.discontinuities() 

prev_layer = None 

 

def get_change_ratios(): 

values = dict.fromkeys([p[1:] for p in _pargs], 0.) 

 

for param, pval in kwargs.items(): 

if param not in _pargs: 

continue 

values[param[1:]] = float(rstate.uniform(-pval, pval, size=1)) 

return values 

 

# skip Surface 

while True: 

disc = next(discont) 

if isinstance(disc, Surface): 

break 

 

while True: 

try: 

layer = next(layers) 

m = layer.material(None) 

h = layer.zbot - layer.ztop 

except StopIteration: 

break 

 

if not isinstance(layer, HomogeneousLayer): 

raise NotImplementedError( 

'Can only perturbate homogeneous layers!') 

 

changes = get_change_ratios() 

 

# Changing thickness 

dh = h * changes['h'] 

changes['h'] = dh 

 

layer.resize(depth_max=layer.zbot + dh, 

depth_min=prev_layer.zbot if prev_layer else None) 

 

try: 

disc = next(discont) 

disc.change_depth(disc.z + dh) 

except StopIteration: 

pass 

 

# Setting material parameters 

for param, change_ratio in changes.items(): 

if param == 'h': 

continue 

 

value = m.__getattribute__(param) 

changes[param] = value * change_ratio 

 

if keep_vp_vs and changes['vp'] != 0.: 

changes['vs'] = (m.vp + changes['vp']) / m.vp_vs_ratio() - m.vs 

 

for param, change in changes.items(): 

if param == 'h': 

continue 

value = m.__getattribute__(param) 

m.__setattr__(param, value + change) 

 

logger.info( 

'perturbating earthmodel: {}'.format( 

' '.join(['{param}: {change:{len}.2f}'.format( 

param=p, change=c, len=8) 

for p, c in changes.items()]))) 

 

prev_layer = layer 

 

return earthmod 

 

def require_homogeneous(self): 

elements = list(self.elements()) 

 

if len(elements) != 2: 

raise LayeredModelError('More than one layer in earthmodel') 

if not isinstance(elements[1], HomogeneousLayer): 

raise LayeredModelError('Layer has to be a HomogeneousLayer') 

 

return elements[1].m 

 

def __str__(self): 

return '\n'.join(str(element) for element in self._elements) 

 

 

def read_hyposat_model(fn): 

'''Reader for HYPOSAT earth model files. 

 

To be used as producer in :py:meth:`LayeredModel.from_scanlines`. 

 

Interface names are translated as follows: ``'MOHO'`` -> ``'moho'``, 

``'CONR'`` -> ``'conrad'`` 

''' 

 

with open(fn, 'r') as f: 

translate = {'MOHO': 'moho', 'CONR': 'conrad'} 

lname = None 

for iline, line in enumerate(f): 

if iline == 0: 

continue 

 

z, vp, vs, name = util.unpack_fixed('f10, f10, f10, a4', line) 

if not name: 

name = None 

material = Material(vp*1000., vs*1000.) 

 

tname = translate.get(lname, lname) 

yield z*1000., material, tname 

 

lname = name 

 

 

def read_nd_model(fn): 

'''Reader for TauP style '.nd' (named discontinuity) files. 

 

To be used as producer in :py:meth:`LayeredModel.from_scanlines`. 

 

Interface names are translated as follows: ``'mantle'`` -> ``'moho'``, 

``'outer-core'`` -> ``'cmb'``, ``'inner-core'`` -> ``'icb'``. 

 

The format has been modified to include Burgers materials parameters in 

columns 7 (burger_eta1), 8 (burger_eta2) and 9. eta(3). 

''' 

with open(fn, 'r') as f: 

for x in read_nd_model_fh(f): 

yield x 

 

 

def read_nd_model_str(s): 

f = StringIO(s) 

for x in read_nd_model_fh(f): 

yield x 

f.close() 

 

 

def read_nd_model_fh(f): 

translate = {'mantle': 'moho', 'outer-core': 'cmb', 'inner-core': 'icb'} 

name = None 

for line in f: 

toks = line.split() 

if len(toks) == 9 or len(toks) == 6 or len(toks) == 4: 

z, vp, vs, rho = [float(x) for x in toks[:4]] 

qp, qs = None, None 

burgers = None 

if len(toks) == 6 or len(toks) == 9: 

qp, qs = [float(x) for x in toks[4:6]] 

if len(toks) == 9: 

burgers = \ 

[float(x) for x in toks[6:]] 

 

material = Material( 

vp*1000., vs*1000., rho*1000., qp, qs, 

burgers=burgers) 

 

yield z*1000., material, name 

name = None 

elif len(toks) == 1: 

name = translate.get(toks[0], toks[0]) 

 

f.close() 

 

 

def from_crust2x2_profile(profile, depthmantle=50000): 

from pyrocko.dataset import crust2x2 

 

default_qp_qs = { 

'soft sed.': (50., 50.), 

'hard sed.': (200., 200.), 

'upper crust': (600., 400.), 

} 

 

z = 0. 

for i in range(8): 

dz, vp, vs, rho = profile.get_layer(i) 

name = crust2x2.Crust2Profile.layer_names[i] 

if name in default_qp_qs: 

qp, qs = default_qp_qs[name] 

else: 

qp, qs = None, None 

 

material = Material(vp, vs, rho, qp, qs) 

iname = None 

if i == 7: 

iname = 'moho' 

if dz != 0.0: 

yield z, material, iname 

if i != 7: 

yield z+dz, material, name 

else: 

yield z+depthmantle, material, name 

 

z += dz 

 

 

def write_nd_model_fh(mod, fh): 

def fmt(z, mat): 

rstr = ' '.join( 

util.gform(x, 4) 

for x in ( 

z/1000., 

mat.vp/1000., 

mat.vs/1000., 

mat.rho/1000., 

mat.qp, mat.qs)) 

if not mat._has_default_burgers(): 

rstr += ' '.join( 

util.gform(x, 4) 

for x in ( 

mat.burger_eta1, 

mat.burger_eta2, 

mat.burger_valpha)) 

return rstr.rstrip() + '\n' 

 

translate = { 

'moho': 'mantle', 

'cmb': 'outer-core', 

'icb': 'inner-core'} 

 

last = None 

for element in mod.elements(): 

if isinstance(element, Interface): 

if element.name is not None: 

n = translate.get(element.name, element.name) 

fh.write('%s\n' % n) 

 

elif isinstance(element, Layer): 

if not isinstance(last, Layer): 

fh.write(fmt(element.ztop, element.mtop)) 

 

fh.write(fmt(element.zbot, element.mbot)) 

 

last = element 

 

if not isinstance(last, Layer): 

fh.write(fmt(last.z, last.mbelow)) 

 

 

def write_nd_model_str(mod): 

f = StringIO() 

write_nd_model_fh(mod, f) 

return f.getvalue() 

 

 

def write_nd_model(mod, fn): 

with open(fn, 'w') as f: 

write_nd_model_fh(mod, f) 

 

 

def builtin_models(): 

return sorted([ 

os.path.splitext(os.path.basename(x))[0] 

for x in glob.glob(builtin_model_filename('*'))]) 

 

 

def builtin_model_filename(modelname): 

return util.data_file(os.path.join('earthmodels', modelname+'.nd')) 

 

 

def load_model(fn='ak135-f-continental.m', format='nd', crust2_profile=None): 

'''Load layered earth model from file. 

 

:param fn: filename 

:param format: format 

:param crust2_profile: ``(lat, lon)`` or 

:py:class:`pyrocko.crust2x2.Crust2Profile` object, merge model with 

crustal profile. If ``fn`` is forced to be ``None`` only the converted 

CRUST2.0 profile is returned. 

:returns: object of type :py:class:`LayeredModel` 

 

The following formats are currently supported: 

 

============== =========================================================== 

format description 

============== =========================================================== 

``'nd'`` 'named discontinuity' format used by the TauP programs 

``'hyposat'`` format used by the HYPOSAT location program 

============== =========================================================== 

 

The naming of interfaces is translated from the file format's native naming 

to Cake's own convention (See :py:func:`read_nd_model` and 

:py:func:`read_hyposat_model` for details). Cake likes the following 

internal names: ``'conrad'``, ``'moho'``, ``'cmb'`` (core-mantle boundary), 

``'icb'`` (inner core boundary). 

''' 

 

if fn is not None: 

if format == 'nd': 

if not os.path.exists(fn) and fn in builtin_models(): 

fn = builtin_model_filename(fn) 

reader = read_nd_model(fn) 

elif format == 'hyposat': 

reader = read_hyposat_model(fn) 

else: 

assert False, 'unsupported model format' 

 

mod = LayeredModel.from_scanlines(reader) 

if crust2_profile is not None: 

return mod.replaced_crust(crust2_profile) 

 

return mod 

 

else: 

assert crust2_profile is not None 

profile = anything_to_crust2_profile(crust2_profile) 

return LayeredModel.from_scanlines( 

from_crust2x2_profile(profile)) 

 

 

def castagna_vs_to_vp(vs): 

'''Calculate vp from vs using castagna's relation. 

 

Castagna's relation (the mudrock line) is an empirical relation for vp/vs 

for siliciclastic rocks (i.e. sandstones and shales). [Castagna et al., 

1985] 

 

vp = 1.16 * vs + 1360 [m/s] 

 

:param vs: S-wave velocity [m/s] 

:returns: P-wave velocity [m/s] 

''' 

 

return vs*1.16 + 1360.0 

 

 

def castagna_vp_to_vs(vp): 

'''Calculate vp from vs using castagna's relation. 

 

Castagna's relation (the mudrock line) is an empirical relation for vp/vs 

for siliciclastic rocks (i.e. sandstones and shales). [Castagna et al., 

1985] 

 

vp = 1.16 * vs + 1360 [m/s] 

 

:param vp: P-wave velocity [m/s] 

:returns: S-wave velocity [m/s] 

''' 

 

return (vp - 1360.0) / 1.16 

 

 

def evenize(x, y, minsize=10): 

if x.size < minsize: 

return x 

ry = (y.max()-y.min()) 

if ry == 0: 

return x 

dx = (x[1:] - x[:-1])/(x.max()-x.min()) 

dy = (y[1:] + y[:-1])/ry 

 

s = num.zeros(x.size) 

s[1:] = num.cumsum(num.sqrt(dy**2 + dx**2)) 

s2 = num.linspace(0, s[-1], x.size) 

x2 = num.interp(s2, s, x) 

x2[0] = x[0] 

x2[-1] = x[-1] 

return x2 

 

 

def filled(v, *args, **kwargs): 

''' 

Create NumPy array filled with given value. 

 

This works like :py:func:`numpy.ones` but initializes the array with ``v`` 

instead of ones. 

''' 

x = num.empty(*args, **kwargs) 

x.fill(v) 

return x 

 

 

def next_or_none(i): 

try: 

return next(i) 

except StopIteration: 

return None 

 

 

def reci_or_none(x): 

try: 

return 1./x 

except ZeroDivisionError: 

return None 

 

 

def mult_or_none(a, b): 

if a is None or b is None: 

return None 

return a*b 

 

 

def min_not_none(a, b): 

if a is None: 

return b 

if b is None: 

return a 

return min(a, b) 

 

 

def xytups(xx, yy): 

d = [] 

for x, y in zip(xx, yy): 

if num.isfinite(y): 

d.append((x, y)) 

return d 

 

 

def interp(x, xp, fp, monoton): 

if monoton == 1: 

return xytups( 

x, num.interp(x, xp, fp, left=num.nan, right=num.nan)) 

elif monoton == -1: 

return xytups( 

x, num.interp(x, xp[::-1], fp[::-1], left=num.nan, right=num.nan)) 

else: 

fs = [] 

for xv in x: 

indices = num.where(num.logical_or( 

num.logical_and(xp[:-1] >= xv, xv > xp[1:]), 

num.logical_and(xp[:-1] <= xv, xv < xp[1:])))[0] 

 

for i in indices: 

xr = (xv - xp[i])/(xp[i+1]-xp[i]) 

fv = xr*fp[i] + (1.-xr)*fp[i+1] 

fs.append((xv, fv)) 

 

return fs 

 

 

def float_or_none(x): 

if x is not None: 

return float(x) 

 

 

def parstore_float(thelocals, obj, *args): 

for k, v in thelocals.items(): 

if k != 'self' and (not args or k in args): 

setattr(obj, k, float_or_none(v))