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Receiver Function Inversion (Shibutani)

Receiver Function Inversion (Shibutani)#

“Receiver functions” are a class of seismic data used to study discontinuities (layering) in the Earth’s crust. At each discontinuity, P-to-S conversions occur, introducing complexity in the waveform. By deconvolving horizontal- and vertical-channel waveforms from earthquakes at teleseismic distances, we can isolate information about these conversions, and hence learn about the crustal structure. This deconvolved signal is the receiver function, and has a highly non-linear dependence on the local crustal properties.

More explanation on the log likelihood function and covariance matrix#

To measure the fit of the observed and calculationed Receiver function we specify a Likelihood function \(p({\mathbf d}_{obs}| {\mathbf m})\)

\[ p({\mathbf d}_{obs} | {\mathbf m}) \propto \exp \left\{- \frac{1}{2} ({\mathbf d}_{obs}-{\mathbf d}_{pred}({\mathbf m}))^T C_D^{-1} ({\mathbf d}_{obs}-{\mathbf d}_{pred}({\mathbf m})) \right\} \]

where \({\mathbf d}_{obs}\) is the observed receiver function and \({\mathbf d}_{pred}({\mathbf m})\) is the one predicted by model \(({\mathbf m})\). The Likelihood is defined as the probability of observing the data actually observed, given an Earth model.

In practice we usually need to evaluate the negative log of the Likelihood which involves the inverse data covariance matrix describing correlated noise in the receiver function, \(C_D^{-1}\) . For this problem you can evaluate \(C_D^{-1}\) using the supplied routine InvDataCov(\(\sigma_t\),\(\sigma_a\),\(n\)), which builds an \(nxn\) inverse data covariance matrix for a time signal with amplitude variance \(\sigma_a^2\) and time correlation length \(\sigma_t\), for receiver length of \(n\) time samples. In this example you can use \(\sigma_t = 2.5\), \(\sigma_a=0.01\).

For a toy inference run we can also temper the Likelihood by rescaling the data covariance matrix by a temperature value, \(T\), i.e. use \(C_D^{-1}/T\) as the data covariance matrix in all calculations. This effectively reduces the size of `peakiness’ of the posterior PDF to a simpler more manageable distribution. For example we can set \(T=100\).

Contribution Metadata for Receiver function (Fortran)

‘Receiver functions’ are a class of seismic data used to study discontinuities (layering) in the Earth’s crust

Author:

Malcolm Sambridge, Takuo Shibutani

Contact:

Malcolm Sambridge (Malcolm.Sambridge@anu.edu.au)

Citation:

  • Langston, C. A., Structure under Mount Rainer, Washington, inferred from teleseismic body waves, J. Geophys. Res., vol 84, 4749-4762, 1979.

  • Shibutani, T., Kennett, B. and Sambridge, M., Genetic algorithm inversion for receiver functions with application to crust and uppermost mantle structure beneath Eastern Australia, Geophys. Res. Lett., 23 , No. 4, 1829-1832, 1996

Example usage for ReceiverFunctionInversionShibutani#

Code block below is generated automatically based on our tests and inspection of the example objects. We’d like to also refer you to the API Reference for more about the Espresso API.

 1import espresso
 2
 3# Create a ReceiverFunctionInversionShibutani object
 4my_receiver_function_inversion_shibutani = espresso.ReceiverFunctionInversionShibutani(example_number=1)
 5
 6# Guaranteed API
 7model_size     = my_receiver_function_inversion_shibutani.model_size
 8data_size      = my_receiver_function_inversion_shibutani.data_size
 9null_model     = my_receiver_function_inversion_shibutani.starting_model
10good_model     = my_receiver_function_inversion_shibutani.good_model
11given_data     = my_receiver_function_inversion_shibutani.data
12synthetic_data = my_receiver_function_inversion_shibutani.forward(good_model)
13
14# Optional API
15example_desc       = my_receiver_function_inversion_shibutani.description
16cov_matrix         = my_receiver_function_inversion_shibutani.covariance_matrix
17inv_cov_matrix     = my_receiver_function_inversion_shibutani.inverse_covariance_matrix
18model_fig          = my_receiver_function_inversion_shibutani.plot_model(good_model)
19data_fig           = my_receiver_function_inversion_shibutani.plot_data(given_data)
20misfit_val         = my_receiver_function_inversion_shibutani.misfit(given_data, synthetic_data)
21log_likelihood_val = my_receiver_function_inversion_shibutani.log_likelihood(given_data, synthetic_data)
22log_prior_val      = my_receiver_function_inversion_shibutani.log_prior(good_model)

Additional attributes to explore: [list_capabilities, rf].

LICENCE#

Copyright 2021-2022 The Contributors All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

  1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

  2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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