How do body-wave traveltimes constrain the Earth's radial (1-D) seismic structure? Existing 1-D s... more How do body-wave traveltimes constrain the Earth's radial (1-D) seismic structure? Existing 1-D seismological models underpin 3-D seismic tomography and earthquake location algorithms. It is therefore crucial to assess the quality of such 1-D models, yet quantifying uncertainties in seismological models is challenging and thus often ignored. Ideally, quality assessment should be an integral part of the inverse method. Our aim in this study is twofold: (i) we show how to solve a general Bayesian non-linear inverse problem and quantify model uncertainties, and (ii) we investigate the constraint on spherically symmetric P-wave velocity (Vp) structure provided by body-wave traveltimes from the EHB bulletin (phases Pn, P, PP and PKP). Our approach is based on artificial neural networks, which are very common in pattern recognition problems and can be used to approximate an arbitrary function. We use a Mixture Density Network to obtain 1-D marginal posterior probability density functions (pdfs), which provide a quantitative description of our knowledge on the individual Earth parameters. No linearization or model damping is required, which allows us to infer a model which is constrained purely by the data.
We present 1-D marginal posterior pdfs for the 22 Vp parameters and seven discontinuity depths in our model. P-wave velocities in the inner core, outer core and lower mantle are resolved well, with standard deviations of ~0.2 to 1 per cent with respect to the mean of the posterior pdfs. The maximum likelihoods of Vp are in general similar to the corresponding ak135 values, which lie within one or two standard deviations from the posterior means, thus providing an independent validation of ak135 in this part of the radial model. Conversely, the data contain little or no information on P-wave velocity in the D'' layer, the upper mantle and the homogeneous crustal layers. Further, the data do not constrain the depth of the discontinuities in our model. Using additional phases available in the ISC bulletin, such as PcP, PKKP and the converted phases SP and ScP, may enhance the resolvability of these parameters. Finally, we show how the method can be extended to obtain a posterior pdf for a multidimensional model space. This enables us to investigate correlations between model parameters.
Identifying and cataloguing occurrences of particular topographic features are important but time... more Identifying and cataloguing occurrences of particular topographic features are important but time-consuming tasks. Typically, automation is challenging, as simple models do not fully describe the complexities of natural features. We propose a new approach, where a particular class of neural network (the 'autoencoder') is used to assimilate the characteristics of the feature to be catalogued, and then applied to a systematic search for new examples. To demonstrate the feasibility of this method, we construct a network that may be used to find seamounts in global bathymetric data. We show results for two test regions, which compare favourably with results from traditional algorithms.
We describe a method for determining an optimal Centroid—Moment Tensor (CMT) solution of an earth... more We describe a method for determining an optimal Centroid—Moment Tensor (CMT) solution of an earthquake from a set of static displacements measured using a network of Global Positioning System (GPS) receivers. Using static displacements observed after the 4 April 2010, Mw 7.2 El Mayor-Cucapah, Mexico, earthquake, we perform an iterative inversion to obtain the source mechanism and location which minimise the least-squares difference between data and synthetics. The efficiency of our algorithm for forward modelling static displacements in a layered elastic medium allows the inversion to be performed in real-time, without the need for precomputed libraries of excitation kernels; we present simulated real-time results for the El Mayor-Cucapah earthquake. The only a priori information that our inversion scheme needs is a crustal model and approximate source location, and so the method proposed here may represent an improvement on existing early warning approaches which rely on foreknowledge of fault locations and geometries.
The centroid-moment-tensor (CMT) algorithm provides a straightforward, rapid method for the deter... more The centroid-moment-tensor (CMT) algorithm provides a straightforward, rapid method for the determination of seismic source parameters from waveform data. As such, it has found widespread application, and catalogues of CMT solutions – particularly the catalogue maintained by the Global CMT Project – are routinely used by geoscientists. However, there have been few attempts to quantify the uncertainties associated with any given CMT determination: whilst catalogues typically quote a ‘standard error’ for each source parameter, these are generally accepted to significantly underestimate the true scale of uncertainty, as all systematic effects are ignored. This prevents users of source parameters from properly assessing possible impacts of this uncertainty upon their own analysis.The CMT algorithm determines the best-fitting source parameters within a particular modelling framework, but any deficiencies in this framework may lead to systematic errors. As a result, the minimum-misfit source may not be equivalent to the ‘true’ source. We suggest a pragmatic solution to uncertainty assessment, based on accepting that any ‘low-misfit’ source may be a plausible model for a given event. The definition of ‘low-misfit’ should be based upon an assessment of the scale of potential systematic effects. We set out how this can be used to estimate the range of values that each parameter might take, by considering the curvature of the misfit function as minimised by the CMT algorithm. This approach is computationally efficient, with cost similar to that of performing an additional iteration during CMT inversion for each source parameter to be considered.The source inversion process is sensitive to the various choices that must be made regarding dataset, earth model and inversion strategy, and for best results, uncertainty assessment should be performed using the same choices. Unfortunately, this information is rarely available when sources are obtained from catalogues. As already indicated by Valentine and Woodhouse (2010), researchers conducting comparisons between data and synthetic waveforms must ensure that their approach to forward-modelling is consistent with the source parameters used; in practice, this suggests that they should consider performing their own source inversions. However, it is possible to obtain rough estimates of uncertainty using only forward-modelling.► Proposes method to assess uncertainties in centroid-moment tensor determination. ► Systematic errors from earth model and modelling framework may bias results. ► ‘Standard errors’ given in catalogues severely underestimate the true uncertainty. ► Users of catalogue source parameters must ensure determination and use are consistent. ► Self-consistency may require users to perform their own source determinations.
Displacement time-series recorded by Global Positioning System (GPS) receivers are a new type of ... more Displacement time-series recorded by Global Positioning System (GPS) receivers are a new type of near-field waveform observation of the seismic source. We have developed an inversion method which enables the recovery of an earthquake's mechanism and centroid coordinates from such data. Our approach is identical to that of the 'classical' Centroid–Moment Tensor (CMT) algorithm, except that we forward model the seismic wavefield using a method that is amenable to the efficient computation of synthetic GPS seismograms and their partial derivatives. We demonstrate the validity of our approach by calculating CMT solutions using 1 Hz GPS data for two recent earthquakes in Japan. These results are in good agreement with independently determined source models of these events. With wider availability of data, we envisage the CMT algorithm providing a tool for the systematic inversion of GPS waveforms, as is already the case for teleseismic data. Furthermore, this general inversion method could equally be applied to other near-field earthquake observations such as those made using accelerometers.
What makes a seismogram look like a seismogram? Seismic data sets generally contain waveforms sha... more What makes a seismogram look like a seismogram? Seismic data sets generally contain waveforms sharing some set of visual characteristics and features—indeed, seismologists routinely exploit this when performing quality control ‘by hand’. Understanding and harnessing these characteristics offers the prospect of a deeper understanding of seismic waveforms, and opens up many potential new techniques for processing and working with data. In addition, the fact that certain features are shared between waveforms suggests that it may be possible to transform the data away from the time domain, and represent the same information using fewer parameters. If so, this would be a significant step towards making fully non-linear tomographic inversions computationally tractable.Hinton & Salakhutdinov showed that a particular class of neural network, termed ‘autoencoder networks’, may be used to find lower-dimensional encodings of complex binary data sets. Here, we adapt their work to the continuous case to allow the use of autoencoders for seismic waveforms, and offer a demonstration in which we compress 512-point waveforms to 32-element encodings. We also demonstrate that the mapping from data to encoding space, and its inverse, are well behaved, as required for many applications. Finally, we sketch a number of potential applications of the technique, which we hope will be of practical interest across all seismological disciplines, and beyond.
The ability to handle large amounts of data automatically is essential for any major tomographic ... more The ability to handle large amounts of data automatically is essential for any major tomographic inversion. As part of this process, it is necessary to differentiate between high-quality seismograms, and those that are unusable due to noise or other errors. This quality assessment is traditionally made visually; however, the sheer quantity of data in a modern tomographic data set makes this approach unfeasible. It is therefore necessary to develop techniques for automating this quality assessment process.
We demonstrate that a simple neural network, trained to recognize the frequency-domain characteristics of high- and low-quality data, can successfully distinguish the two classes in unseen data. We demonstrate that the resulting clean data sets are of sufficient quality to allow full-waveform determination of event focal mechanisms and hypocentral parameters.
The process we outline allows the rapid creation of a high-quality data set for seismic tomography. Depending on application, this may be suitable for use without further refinement. In some circumstances, a further visual inspection may remain desirable to ensure the data set is noise-free; however, a significant benefit will still derive from the reduction in number of traces to be examined. This will enable full-waveform inversion using significantly larger data sets than has hitherto been possible. The selection strategy relies only on measurements made from the seismogram, and on rough estimates of hypocentral location — the final data set does not depend on any a priori assumptions regarding earth structure or wave propagation.
Our focus has been on data selection for seismic tomography, but the approach is general and may find application across a wide range of seismic investigations. An automated system is of interest wherever large data sets must be handled, or where time is of the essence — such as in earthquake hazard assessment.
To perform seismic tomography, accurate determinations of event locations and focal mechanisms ar... more To perform seismic tomography, accurate determinations of event locations and focal mechanisms are required. These are usually obtained prior to tomographic inversion; often, little attention is paid to the earth model used for their determination. We show that an imprint of this model is found in the model recovered after tomographic inversion. To reduce this problem, it is important to recalculate earthquake source parameters as the inversion proceeds; synthetic tests suggest that this yields a better correspondence between recovered and true models. However, alternate source and structure inversions lead to a slow rate of convergence, and significant errors remain. We therefore propose combining source and structure inversion into a single inverse problem, and present an efficient algorithm to do this. We demonstrate that this reduces the number of iterations required to achieve a given accuracy; in our experiments, we observe a fourfold improvement on alternate inversions. We focus on full-waveform inversions for both source and structure, although the methods presented are general and should be applicable to other techniques.
How do body-wave traveltimes constrain the Earth's radial (1-D) seismic structure? Existing 1-D s... more How do body-wave traveltimes constrain the Earth's radial (1-D) seismic structure? Existing 1-D seismological models underpin 3-D seismic tomography and earthquake location algorithms. It is therefore crucial to assess the quality of such 1-D models, yet quantifying uncertainties in seismological models is challenging and thus often ignored. Ideally, quality assessment should be an integral part of the inverse method. Our aim in this study is twofold: (i) we show how to solve a general Bayesian non-linear inverse problem and quantify model uncertainties, and (ii) we investigate the constraint on spherically symmetric P-wave velocity (Vp) structure provided by body-wave traveltimes from the EHB bulletin (phases Pn, P, PP and PKP). Our approach is based on artificial neural networks, which are very common in pattern recognition problems and can be used to approximate an arbitrary function. We use a Mixture Density Network to obtain 1-D marginal posterior probability density functions (pdfs), which provide a quantitative description of our knowledge on the individual Earth parameters. No linearization or model damping is required, which allows us to infer a model which is constrained purely by the data.
We present 1-D marginal posterior pdfs for the 22 Vp parameters and seven discontinuity depths in our model. P-wave velocities in the inner core, outer core and lower mantle are resolved well, with standard deviations of ~0.2 to 1 per cent with respect to the mean of the posterior pdfs. The maximum likelihoods of Vp are in general similar to the corresponding ak135 values, which lie within one or two standard deviations from the posterior means, thus providing an independent validation of ak135 in this part of the radial model. Conversely, the data contain little or no information on P-wave velocity in the D'' layer, the upper mantle and the homogeneous crustal layers. Further, the data do not constrain the depth of the discontinuities in our model. Using additional phases available in the ISC bulletin, such as PcP, PKKP and the converted phases SP and ScP, may enhance the resolvability of these parameters. Finally, we show how the method can be extended to obtain a posterior pdf for a multidimensional model space. This enables us to investigate correlations between model parameters.
Identifying and cataloguing occurrences of particular topographic features are important but time... more Identifying and cataloguing occurrences of particular topographic features are important but time-consuming tasks. Typically, automation is challenging, as simple models do not fully describe the complexities of natural features. We propose a new approach, where a particular class of neural network (the 'autoencoder') is used to assimilate the characteristics of the feature to be catalogued, and then applied to a systematic search for new examples. To demonstrate the feasibility of this method, we construct a network that may be used to find seamounts in global bathymetric data. We show results for two test regions, which compare favourably with results from traditional algorithms.
We describe a method for determining an optimal Centroid—Moment Tensor (CMT) solution of an earth... more We describe a method for determining an optimal Centroid—Moment Tensor (CMT) solution of an earthquake from a set of static displacements measured using a network of Global Positioning System (GPS) receivers. Using static displacements observed after the 4 April 2010, Mw 7.2 El Mayor-Cucapah, Mexico, earthquake, we perform an iterative inversion to obtain the source mechanism and location which minimise the least-squares difference between data and synthetics. The efficiency of our algorithm for forward modelling static displacements in a layered elastic medium allows the inversion to be performed in real-time, without the need for precomputed libraries of excitation kernels; we present simulated real-time results for the El Mayor-Cucapah earthquake. The only a priori information that our inversion scheme needs is a crustal model and approximate source location, and so the method proposed here may represent an improvement on existing early warning approaches which rely on foreknowledge of fault locations and geometries.
The centroid-moment-tensor (CMT) algorithm provides a straightforward, rapid method for the deter... more The centroid-moment-tensor (CMT) algorithm provides a straightforward, rapid method for the determination of seismic source parameters from waveform data. As such, it has found widespread application, and catalogues of CMT solutions – particularly the catalogue maintained by the Global CMT Project – are routinely used by geoscientists. However, there have been few attempts to quantify the uncertainties associated with any given CMT determination: whilst catalogues typically quote a ‘standard error’ for each source parameter, these are generally accepted to significantly underestimate the true scale of uncertainty, as all systematic effects are ignored. This prevents users of source parameters from properly assessing possible impacts of this uncertainty upon their own analysis.The CMT algorithm determines the best-fitting source parameters within a particular modelling framework, but any deficiencies in this framework may lead to systematic errors. As a result, the minimum-misfit source may not be equivalent to the ‘true’ source. We suggest a pragmatic solution to uncertainty assessment, based on accepting that any ‘low-misfit’ source may be a plausible model for a given event. The definition of ‘low-misfit’ should be based upon an assessment of the scale of potential systematic effects. We set out how this can be used to estimate the range of values that each parameter might take, by considering the curvature of the misfit function as minimised by the CMT algorithm. This approach is computationally efficient, with cost similar to that of performing an additional iteration during CMT inversion for each source parameter to be considered.The source inversion process is sensitive to the various choices that must be made regarding dataset, earth model and inversion strategy, and for best results, uncertainty assessment should be performed using the same choices. Unfortunately, this information is rarely available when sources are obtained from catalogues. As already indicated by Valentine and Woodhouse (2010), researchers conducting comparisons between data and synthetic waveforms must ensure that their approach to forward-modelling is consistent with the source parameters used; in practice, this suggests that they should consider performing their own source inversions. However, it is possible to obtain rough estimates of uncertainty using only forward-modelling.► Proposes method to assess uncertainties in centroid-moment tensor determination. ► Systematic errors from earth model and modelling framework may bias results. ► ‘Standard errors’ given in catalogues severely underestimate the true uncertainty. ► Users of catalogue source parameters must ensure determination and use are consistent. ► Self-consistency may require users to perform their own source determinations.
Displacement time-series recorded by Global Positioning System (GPS) receivers are a new type of ... more Displacement time-series recorded by Global Positioning System (GPS) receivers are a new type of near-field waveform observation of the seismic source. We have developed an inversion method which enables the recovery of an earthquake's mechanism and centroid coordinates from such data. Our approach is identical to that of the 'classical' Centroid–Moment Tensor (CMT) algorithm, except that we forward model the seismic wavefield using a method that is amenable to the efficient computation of synthetic GPS seismograms and their partial derivatives. We demonstrate the validity of our approach by calculating CMT solutions using 1 Hz GPS data for two recent earthquakes in Japan. These results are in good agreement with independently determined source models of these events. With wider availability of data, we envisage the CMT algorithm providing a tool for the systematic inversion of GPS waveforms, as is already the case for teleseismic data. Furthermore, this general inversion method could equally be applied to other near-field earthquake observations such as those made using accelerometers.
What makes a seismogram look like a seismogram? Seismic data sets generally contain waveforms sha... more What makes a seismogram look like a seismogram? Seismic data sets generally contain waveforms sharing some set of visual characteristics and features—indeed, seismologists routinely exploit this when performing quality control ‘by hand’. Understanding and harnessing these characteristics offers the prospect of a deeper understanding of seismic waveforms, and opens up many potential new techniques for processing and working with data. In addition, the fact that certain features are shared between waveforms suggests that it may be possible to transform the data away from the time domain, and represent the same information using fewer parameters. If so, this would be a significant step towards making fully non-linear tomographic inversions computationally tractable.Hinton & Salakhutdinov showed that a particular class of neural network, termed ‘autoencoder networks’, may be used to find lower-dimensional encodings of complex binary data sets. Here, we adapt their work to the continuous case to allow the use of autoencoders for seismic waveforms, and offer a demonstration in which we compress 512-point waveforms to 32-element encodings. We also demonstrate that the mapping from data to encoding space, and its inverse, are well behaved, as required for many applications. Finally, we sketch a number of potential applications of the technique, which we hope will be of practical interest across all seismological disciplines, and beyond.
The ability to handle large amounts of data automatically is essential for any major tomographic ... more The ability to handle large amounts of data automatically is essential for any major tomographic inversion. As part of this process, it is necessary to differentiate between high-quality seismograms, and those that are unusable due to noise or other errors. This quality assessment is traditionally made visually; however, the sheer quantity of data in a modern tomographic data set makes this approach unfeasible. It is therefore necessary to develop techniques for automating this quality assessment process.
We demonstrate that a simple neural network, trained to recognize the frequency-domain characteristics of high- and low-quality data, can successfully distinguish the two classes in unseen data. We demonstrate that the resulting clean data sets are of sufficient quality to allow full-waveform determination of event focal mechanisms and hypocentral parameters.
The process we outline allows the rapid creation of a high-quality data set for seismic tomography. Depending on application, this may be suitable for use without further refinement. In some circumstances, a further visual inspection may remain desirable to ensure the data set is noise-free; however, a significant benefit will still derive from the reduction in number of traces to be examined. This will enable full-waveform inversion using significantly larger data sets than has hitherto been possible. The selection strategy relies only on measurements made from the seismogram, and on rough estimates of hypocentral location — the final data set does not depend on any a priori assumptions regarding earth structure or wave propagation.
Our focus has been on data selection for seismic tomography, but the approach is general and may find application across a wide range of seismic investigations. An automated system is of interest wherever large data sets must be handled, or where time is of the essence — such as in earthquake hazard assessment.
To perform seismic tomography, accurate determinations of event locations and focal mechanisms ar... more To perform seismic tomography, accurate determinations of event locations and focal mechanisms are required. These are usually obtained prior to tomographic inversion; often, little attention is paid to the earth model used for their determination. We show that an imprint of this model is found in the model recovered after tomographic inversion. To reduce this problem, it is important to recalculate earthquake source parameters as the inversion proceeds; synthetic tests suggest that this yields a better correspondence between recovered and true models. However, alternate source and structure inversions lead to a slow rate of convergence, and significant errors remain. We therefore propose combining source and structure inversion into a single inverse problem, and present an efficient algorithm to do this. We demonstrate that this reduces the number of iterations required to achieve a given accuracy; in our experiments, we observe a fourfold improvement on alternate inversions. We focus on full-waveform inversions for both source and structure, although the methods presented are general and should be applicable to other techniques.
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We present 1-D marginal posterior pdfs for the 22 Vp parameters and seven discontinuity depths in our model. P-wave velocities in the inner core, outer core and lower mantle are resolved well, with standard deviations of ~0.2 to 1 per cent with respect to the mean of the posterior pdfs. The maximum likelihoods of Vp are in general similar to the corresponding ak135 values, which lie within one or two standard deviations from the posterior means, thus providing an independent validation of ak135 in this part of the radial model. Conversely, the data contain little or no information on P-wave velocity in the D'' layer, the upper mantle and the homogeneous crustal layers. Further, the data do not constrain the depth of the discontinuities in our model. Using additional phases available in the ISC bulletin, such as PcP, PKKP and the converted phases SP and ScP, may enhance the resolvability of these parameters. Finally, we show how the method can be extended to obtain a posterior pdf for a multidimensional model space. This enables us to investigate correlations between model parameters.
We demonstrate that a simple neural network, trained to recognize the frequency-domain characteristics of high- and low-quality data, can successfully distinguish the two classes in unseen data. We demonstrate that the resulting clean data sets are of sufficient quality to allow full-waveform determination of event focal mechanisms and hypocentral parameters.
The process we outline allows the rapid creation of a high-quality data set for seismic tomography. Depending on application, this may be suitable for use without further refinement. In some circumstances, a further visual inspection may remain desirable to ensure the data set is noise-free; however, a significant benefit will still derive from the reduction in number of traces to be examined. This will enable full-waveform inversion using significantly larger data sets than has hitherto been possible. The selection strategy relies only on measurements made from the seismogram, and on rough estimates of hypocentral location — the final data set does not depend on any a priori assumptions regarding earth structure or wave propagation.
Our focus has been on data selection for seismic tomography, but the approach is general and may find application across a wide range of seismic investigations. An automated system is of interest wherever large data sets must be handled, or where time is of the essence — such as in earthquake hazard assessment.
We present 1-D marginal posterior pdfs for the 22 Vp parameters and seven discontinuity depths in our model. P-wave velocities in the inner core, outer core and lower mantle are resolved well, with standard deviations of ~0.2 to 1 per cent with respect to the mean of the posterior pdfs. The maximum likelihoods of Vp are in general similar to the corresponding ak135 values, which lie within one or two standard deviations from the posterior means, thus providing an independent validation of ak135 in this part of the radial model. Conversely, the data contain little or no information on P-wave velocity in the D'' layer, the upper mantle and the homogeneous crustal layers. Further, the data do not constrain the depth of the discontinuities in our model. Using additional phases available in the ISC bulletin, such as PcP, PKKP and the converted phases SP and ScP, may enhance the resolvability of these parameters. Finally, we show how the method can be extended to obtain a posterior pdf for a multidimensional model space. This enables us to investigate correlations between model parameters.
We demonstrate that a simple neural network, trained to recognize the frequency-domain characteristics of high- and low-quality data, can successfully distinguish the two classes in unseen data. We demonstrate that the resulting clean data sets are of sufficient quality to allow full-waveform determination of event focal mechanisms and hypocentral parameters.
The process we outline allows the rapid creation of a high-quality data set for seismic tomography. Depending on application, this may be suitable for use without further refinement. In some circumstances, a further visual inspection may remain desirable to ensure the data set is noise-free; however, a significant benefit will still derive from the reduction in number of traces to be examined. This will enable full-waveform inversion using significantly larger data sets than has hitherto been possible. The selection strategy relies only on measurements made from the seismogram, and on rough estimates of hypocentral location — the final data set does not depend on any a priori assumptions regarding earth structure or wave propagation.
Our focus has been on data selection for seismic tomography, but the approach is general and may find application across a wide range of seismic investigations. An automated system is of interest wherever large data sets must be handled, or where time is of the essence — such as in earthquake hazard assessment.