Targets

Targets is what Grond tries to match: The misfit between an observed target and forward model is minimized. Targets are derived from observables or synthesised data. A target can be, a filtered waveform, a spectrum, InSAR or GNSS displacement. Each target has properties which can be tuned. These can be frequency filters, selected observed components and essentially a Green’s functions store which is responsible for the synthetics at a particular target.

Different TargetGroups can be combined to solve the inverse problem, leading to a joint optimisation from different data sources and observables.

Note

Weighting between different targets is described in the Method section.

General Target configuration

Parameters valid for all types of MisfitTargets are:

normalisation_family

Normalisation family (see the Grond documentation for how it works). Use distinct normalisation families when mixing misfit contributors with different magnitude scaling, like e.g. cross-correlation based misfit and time-domain \(L^p\) norm.

weight

How to weight contributions from this group in the global misfit.

path

Just a name used to identify targets from this group. Use dot-separated path notation to group related contributors.

interpolation

Green’s function store interpolation, choose from:

• multilinear

  Performs a linear interpolation between discrete Green’s function for improved resolution of synthetic data. This option is computationally more expensive.

• nearest_neighbor

  Uses the Green’s function calculation for the forward model.

Choices other than ‘nearest_neighbor’ may require dense GF stores to avoid aliasing artefacts in the forward modelling.

store_id

Name of the GF Store to use.

store_id_selector

GF Store selectors can be used to select different stores, based on station location, source location or other characteristics. The Crust2StoreIDSelector can be used to pick CRUST 2.0 models based on the event location. A manual station to store ID mapping can be achieved using the StationDictStoreIDSelector with keys NET.STAT.

Waveform targets

See the dataset configuration for loading waveforms and response information.

Example WaveformTarget configuration

- !grond.WaveformTargetGroup
    normalisation_family: time_domain
    path: all
    weight: 1.0
    distance_min: 10000.0
    distance_max: 1000000.0
    channels: [Z, R, T]
    misfit_config: !grond.WaveformMisfitConfig
      fmin: 0.01
      fmax: 0.1
      ffactor: 1.5
      tmin: vel_surface:5.5
      tmax: vel_surface:3.0
      domain: time_domain
      norm_exponent: 2
      tautoshift_max: 0.0
      autoshift_penalty_max: 0.0
    interpolation: multilinear
    store_id: crust2_ib

    # store_id_selector: !grond.Crust2StoreIDSelector
    #   template: crust2_${id}

Tapering

tmin and tmax define time-windows around phase arrivals of interest, those are cut out and tapered.

\({\bf d}_{raw, synth}\) and the restituted observed waveforms. Only these parts are used in the misfit calculation. The taper window duration is configured for each seismic station individually by phase arrivals.

The tapering is source-model dependent, since the tapering time is given with respect to the theoretic phase arrival time. This arrival time depends on the source location, which is often part of the optimisation itself and therefore may change continuously with each iteration. Therefore, restitution, tapering and filtering are done for each misfit calculation anew. Grond uses the Pyrocko CosTaper taper. The fade_out time can be configured or it is calculated as the inverse of the minimum frequency of the chosen bandpass filter.

Frequency filtering

fmin and fmax in Hz define the desired bandpass filter.

norm_exponent

The Lp normalisation for calculating the waveform misfit.

domain

Can be selection from

• time_domain

  Misfit calculated in time domain, here it is useful to configure the tautoshift_max and autoshift_penalty_max to allow for small time shifts of the synthetic data.

• frequency_domain

  Waveform misfit is calculated in the frequency domain.

• log_frequency_domain

  Waveform misfit is calculated in the logarithmic frequency domain.

• envelope

  Waveform envelops are compared.

• absolute

  The absolute amplitudes are used to calculate the misfit

• cc_max_norm

  Misfit is calculated from cross-correlation of the traces.

tautoshift_max

defines the maximum allowed time uin seconds the observed and synthetic trace may be shifted during the inversion.

autoshift_penalty_max

is the misfit penalty for autoshifting seismic traces.

Example WaveformTargetGroup configuration section:

Satellite targets

Example SatelliteTarget configuration

- !grond.SatelliteTargetGroup
  normalisation_family: insar_target
  path: all
  weight: 1.0
  kite_scenes: ['*all']
  misfit_config: !grond.SatelliteMisfitConfig
    optimise_orbital_ramp: true
    ranges:
      offset: -0.5 .. 0.5
      ramp_east: -1e-4 .. 1e-4
      ramp_north: -1e-4 .. 1e-4
  interpolation: multilinear
  store_id: crust2_ib_static

Observations of spatial surface displacements as derived from unwrapped InSAR data. These data must be hold in a special container format and prepared using the kite software package.

Prior to optimisation we have to parametrise a quadtree of the surface displacements (spatial sub-sampling) and pre-calculate the data’s covariance matrix with kite’s spool tool:

spool events/<event_name>/data/insar/scene_ascending.yml

Please see kite’s documentation for insights into the pre-processing methods.

kite_scenes

The InSAR scenes are identified by their kite scene_id. Scenes can be explicitly selected, or the wildcard *all can be used.

optimise_orbital_ramp:

Optimisation for a 2D offset plane in each InSAR scene. This will compensate tradeoffs between the earthquake signal and uncorrected trends in the unwrapped surface displacements. The slopes of ramp_north and ramp_east are given in \(\frac{m}{m}\), the offset in \(m\) - these parameters have to be tuned with touch.

Example SatelliteTargetGroup configuration section:

GNSS campaign targets

Example GNSSTarget configuration

- !grond.GNSSCampaignTargetGroup
  normalisation_family: gnss_target
  path: all
  weight: 1.0
  gnss_campaigns: ['*all']
  misfit_config: !grond.GNSSCampaignMisfitConfig {}
  interpolation: multilinear
  store_id: crust2_ib_static

True 3D surface displacement as measured by GNSS stations can be included in the inversion process by defining a GNSSCampaignTargetGroup. The station’s displacement data has to be stored according to gnss_campaign. Please refer to Pyrocko’s documentation of the GNSS model (See example)

gnss_campaigns

The campaigns are identified by their campaign_name. Campaigns can be explicitly selected, or the wildcard *all can be used.

Example GNSSCampaignTargetGroup configuration section: