Waveform Double Difference Misfit

Warning

Please refer to the papers [Tromp2005] and [Yuan2016] for mathematical derivations of the waveform misfit function and cross correlation double difference measurement, from which this misfit function is derived.

For two stations, i and j, the waveform double difference misfit is defined as the squared difference of differences of observed and synthetic data. The misfit \(\chi(\mathbf{m})\) for a given Earth model \(\mathbf{m}\) at a given component is:

\[\chi (\mathbf{m}) = \frac{1}{2} \int_0^T \left| \Delta{s}(t, \mathbf{m})_{ij} - \Delta{d}(t)_{ij} \right| ^ 2 dt,\]

where \(\Delta{s}(t, \mathbf{m})_{ij}\) is the difference of synthetic waveforms s_i and s_j,

\[\Delta{s}(t, \mathbf{m})_{ij} = s_{j}(t, \mathbf{m}) - s_{i}(t, \mathbf{m}),\]

and \(\Delta{d}(t)\) is the difference of observed waveforms d_i and d_j,

\[\Delta{d}(t)_{ij} = d_{j}(t) - d_{i}(t).\]

The corresponding adjoint sources for the misfit function \(\chi(\mathbf{m})\) are defined as the difference of the differential waveform misfits,

\[ \begin{align}\begin{aligned}f_{i}^{\dagger}(t) = + (\Delta{s}(t, \mathbf{m})_{ij} - \Delta{d}(t)_{ij})\\f_{j}^{\dagger}(t) = - (\Delta{s}(t, \mathbf{m})_{ij} - \Delta{d}(t)_{ij})\end{aligned}\end{align} \]

Note

For the sake of simplicity we omit the spatial Kronecker delta and define the adjoint source as acting solely at the receiver’s location. For more details, please see [Tromp2005] and [Yuan2016].

Note

This particular implementation uses Simpson’s rule to evaluate the definite integral.

Usage

adjsrc_type = "waveform_dd"

The following code snippet illustrates the basic usage of the cross correlation traveltime misfit function. See the corresponding Config object for additional configuration parameters.

import pyadjoint

obs, syn = pyadjoint.get_example_data()
obs = obs.select(component="Z")[0]
syn = syn.select(component="Z")[0]

obs_2, syn_2 = pyadjoint.get_example_data()
obs_2 = obs_2.select(component="R")[0]
syn_2 = syn_2.select(component="R")[0]

config = pyadjoint.get_config(adjsrc_type="waveform_dd", min_period=20.,
                              max_period=100.)

# Calculating double-difference adjoint source returns two adjoint sources
adj_src, adj_src_2 = pyadjoint.calculate_adjoint_source(
    config=config, observed=obs, synthetic=syn, windows=[(800., 900.)],
    observed_2=obs_2, synthetic_2=syn_2, windows_2=[(800., 900.)]
    )