
 The spectral ratio between horizontal and vertical components (H/V ratio) of microtremors measured at the ground surface has been used to estimate fundamental periods and amplification factors of a site, although this technique lacks theoretical background. The aim of this article is to formulate the H/V technique in terms of the characteristics of Rayleigh and Love waves, and to contribute to improve the technique. The improvement includes use of not only peaks but also troughs in the H/V ratio for reliable estimation of the period and use of a newly proposed smoothing function for better estimation of the amplification factor. The formulation leads to a simple formula for the amplification factor expressed with the H/V ratio. With microtremor data measured at 546 junior high schools in 23 wards of Tokyo, the improved technique is applied to mapping site periods and amplification factors in the area.

Ground Motion Characteristics Estimated from Spectral Ratio between Horizontal and Verticcl Components of Mietremors.

Problem Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. SECTION 1 Definition 1. Several events are inconsistent, when if one of them happens, none of the rest can. 2. Two events are contrary when one, or other of them must; and both together cannot happen. 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened. 4. An event is said to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon it’s 2 happening.

An essay towards solving a problem in the doctrine of chances. [Facsimil]

We present a novel method for joint inversion of receiver functions and surface wave dispersion data, using a transdimensional Bayesian formulation. This class of algorithm treats the number of model parameters (e.g. number of layers) as an unknown in the problem. The dimension of the model space is variable and a Markov chain Monte Carlo (McMC) scheme is used to provide a parsimonious solution that fully quantifies the degree of knowledge one has about seismic structure (i.e constraints on the model, resolution, and trade-offs). The level of data noise (i.e. the covariance matrix of data errors) effectively controls the information recoverable from the data and here it naturally determines the complexity of the model (i.e. the number of model parameters). However, it is often difficult to quantify the data noise appropriately, particularly in the case of seismic waveform inversion where data errors are correlated. Here we address the issue of noise estimation using an extended Hierarchical Bayesian formulation, which allows both the variance and covariance of data noise to be treated as unknowns in the inversion. In this way it is possible to let the data infer the appropriate level of data fit. In the context of joint inversions, assessment of uncertainty for different data types becomes crucial in the evaluation of the misfit function. We show that the Hierarchical Bayes procedure is a powerful tool in this situation, because it is able to evaluate the level of information brought by different data types in the misfit, thus removing the arbitrary choice of weighting factors. After illustrating the method with synthetic tests, a real data application is shown where teleseismic receiver functions and ambient noise surface wave dispersion measurements from the WOMBAT array (South-East Australia) are jointly inverted to provide a probabilistic 1D model of shear-wave velocity beneath a given station.

/pdf/transdimensional-inversion-of-receiver-functions-and-surface-53opuqe9gb.pdf

Transdimensional inversion of receiver functions and surface wave dispersion

S U M M A R Y Non-linear inverse problems in the geosciences often involve probabilistic sampling of multimodal density functions or global optimization and sometimes both. Efficient algorithmic tools for carrying out sampling or optimization in challenging cases are of major interest. Here results are presented of some numerical experiments with a technique, known as Parallel Tempering, which originated in the field of computational statistics but is finding increasing numbers of applications in fields ranging from Chemical Physics to Astronomy. To date, experience in use of Parallel Tempering within earth sciences problems is very limited. In this paper, we describe Parallel Tempering and compare it to related methods of Simulated Annealing and Simulated Tempering for optimization and sampling, respectively. A key feature of Parallel Tempering is that it satisfies the detailed balance condition required for convergence of Markov chain Monte Carlo (McMC) algorithms while improving the efficiency of probabilistic sampling. Numerical results are presented on use of Parallel Tempering for trans-dimensional inversion of synthetic seismic receiver functions and also the simultaneous fitting of multiple receiver functions using global optimization. These suggest that its use can significantly accelerate sampling algorithms and improve exploration of parameter space in optimization. Parallel Tempering is a meta-algorithm which may be used together with many existing McMC sampling and direct search optimization techniques. It’s generality and demonstrated performance suggests that there is significant potential for applications to both sampling and optimization problems in the geosciences.

/pdf/a-parallel-tempering-algorithm-for-probabilistic-sampling-2zoldbezbz.pdf

A Parallel Tempering algorithm for probabilistic sampling and multimodal optimization

SUMMARY

A meaningful interpretation of seismic measurements requires a rigorous quantification of the uncertainty. In an inverse problem, the data noise determines how accurately observations should be fit, and ultimately the level of detail contained in the recovered model. A common problem in seismic tomography is the difficulty in quantifying data uncertainties, and thus the required level of data fit. Traditionally, the complexity of the solution model (defined by both the number of basis functions and the regularization) is defined arbitrarily by the user prior to inversion with only limited use of data errors. In the context of multiscale problems, dealing with multiple data sets that are characterized by different noise variances and that span the Earth at different scales is a major challenge. Practitioners are usually required to arbitrarily weigh the contribution of each data type into the final solution. Furthermore, the basis functions are usually spatially uniform across the velocity field and regularization procedures are global, which prevents the solution model from accounting for the uneven spatial distribution of information. In this work we propose to address these issues with a Hierarchical Bayesian inversion. The new algorithm represents an extension of the transdimensional tomography to account for uncertainties in data noise. This approach has the advantage of treating the level of noise in each data set, as well as the number of model parameters, as unknowns in the inversion. It provides a parsimonious solution that fully represents the degree of knowledge one has about seismic structure (i.e. constraints, resolution and trade-offs). Rather than being forced to make decisions on parametrization, level of data fit and weights between data types in advance, as is often the case in an optimization framework, these choices are relaxed and instead constrained by the data themselves. The new methodology is presented in a synthetic example where both the data density and the underlying structure contain multiple length scales. Three ambient noise data sets that span the Australian continent at different scales are then simultaneously inverted to infer a multiscale tomographic image of Rayleigh wave group velocity for the Australian continent. The procedure turns out to be particularly useful when dealing with multiple data types with different unknown levels of noise as the algorithm is able to naturally adjust the fit to the different data sets and provide a velocity map with a spatial resolution adapted to the spatially variable information present in the data.

/pdf/transdimensional-tomography-with-unknown-data-noise-x51dtkaybk.pdf

Transdimensional tomography with unknown data noise

This paper develops a general trans-dimensional Bayesian methodology for geoacoustic inversion. Trans-dimensional inverse problems are a generalization of fixed-dimensional inversion that includes the number and type of model parameters as unknowns in the problem. By extending the inversion state space to multiple subspaces of different dimensions, the posterior probability density quantifies the state of knowledge regarding inversion parameters, including effects due to limited knowledge about appropriate parametrization of the environment and error processes. The inversion is implemented here using a reversible-jump Markov chain Monte Carlo algorithm and the seabed is parametrized using a partition model. Unknown data errors are addressed by including a data-error model. Jumps between dimensions are implemented with a birth–death methodology that allows transitions between dimensions by adding or removing interfaces while maintaining detailed balance in the Markov chain. Trans-dimensional inversion results in an inherently parsimonious solution while partition modeling provides a naturally self-regularizing algorithm based on data information content, not on subjective regularization functions. Together, this results in environmental estimates that quantify appropriate seabed structure as supported by the data, allowing sharp discontinuities while approximating smooth transitions where needed. This approach applies generally to geoacoustic inversion and is illustrated here with seabed reflection-coefficient data.

/pdf/trans-dimensional-geoacoustic-inversion-24h2cppums.pdf

Trans-dimensional geoacoustic inversion

This paper develops a trans-dimensional approach to matched-field geoacoustic inversion, including interacting Markov chains to improve efficiency and an autoregressive model to account for correlated errors. The trans-dimensional approach and hierarchical seabed model allows inversion without assuming any particular parametrization by relaxing model specification to a range of plausible seabed models (e.g., in this case, the number of sediment layers is an unknown parameter). Data errors are addressed by sampling statistical error-distribution parameters, including correlated errors (covariance), by applying a hierarchical autoregressive error model. The well-known difficulty of low acceptance rates for trans-dimensional jumps is addressed with interacting Markov chains, resulting in a substantial increase in efficiency. The trans-dimensional seabed model and the hierarchical error model relax the degree of prior assumptions required in the inversion, resulting in substantially improved (more realistic) uncertainty estimates and a more automated algorithm. In particular, the approach gives seabed parameter uncertainty estimates that account for uncertainty due to prior model choice (layering and data error statistics). The approach is applied to data measured on a vertical array in the Mediterranean Sea.

/pdf/trans-dimensional-matched-field-geoacoustic-inversion-with-3p95rgadez.pdf

Trans-dimensional matched-field geoacoustic inversion with hierarchical error models and interacting Markov chains

This paper considers the efficiency of trans-dimensional (trans-D) Bayesian inversion based on reversible-jump Markov-chain Monte Carlo (rjMCMC) sampling, with application to geophysical inverse problems for a depth-dependent earth or seabed model of an unknown number of layers (seabed acoustic reflectivity inversion is the specific example). Trans-D inversion is applied to sample the posterior probability density over geoacoustic/geophysical parameters for a variable number of layers, providing profile estimates with uncertainties that include the uncertainty in the model parameterization. However, the approach is computationally intensive. The efficiency of rjMCMC sampling is largely determined by the proposal schemes which are applied to generate perturbed values for existing parameters and new values for parameters assigned to layers added to the model. Several proposal schemes are considered here, some of which appear new for trans-D geophysical inversion. Perturbations of existing parameters are considered in a principal-component space based on an eigen-decomposition of the unit-lag parameter covariance matrix (computed from successive models along the Markov chain, a diminishing adaptation). The relative efficiency of proposing new parameters from the prior versus a Gaussian distribution focused near existing values is examined. Parallel tempering, which employs a sequence of interacting Markov chains in which the likelihood function is successively relaxed, is also considered as a means to increase the acceptance rate of new layers. The relative efficiency of various proposal schemes is compared through repeated inversions with a pragmatic convergence criterion.

https://iopscience.iop.org/article/10.1088/0266-5611/30/11/114018/pdf

Efficient trans-dimensional Bayesian inversion for geoacoustic profile estimation

SUMMARY This paper develops a probabilistic Bayesian approach to the problem of inferring the spatiotemporal evolution of earthquake rupture on a fault surface from seismic data with rigorous uncertainty estimation. To date, uncertainties of rupture parameters are poorly understood, and the effect of choices such as fault discretization on uncertainties has not been studied. We show that model choice is fundamentally linked to uncertainty estimation and can have profound effects on results. The approach developed here is based on a trans-dimensional selfparametrizationofthefault,avoidsregularizationconstraintsandprovidesrigorousuncertainty estimationthataccountsformodel-selectionambiguityassociatedwiththefaultdiscretization. In particular, the fault is parametrized using self-adapting irregular grids which intrinsically match the local resolving power of the data and provide parsimonious solutions requiring few parameters to capture complex rupture characteristics. Rupture causality is ensured by parametrizing rupture-onset time by a rupture-velocity field and obtaining first rupture times from the eikonal equation. The Bayesian sampling of the parameter space is implemented on a computer cluster with a highly efficient parallel tempering algorithm. The inversion is applied to simulated and observed W-phase waveforms from the 2010 Maule (Chile) earthquake. Simulation results show that our approach avoids both over- and underparametrization to ensure unbiased inversion results with uncertainty estimates that are consistent with data information. The simulation results also show the ability of W-phase data to resolve the spatial variability of slip magnitude and rake angles. In addition, sensitivity to spatially dependent rupture velocities exists for kinematic slip models. Application to the observed data indicates that residual errors are highly correlated and likely dominated by theory error, necessitating the iterative estimation of a non-stationary data covariance matrix. The moment magnitude for the Maule earthquake is estimated to be ∼8.9, with slip concentrated in two zones updip of and north and south of the hypocentre, respectively. While this aspect of the slip distribution is similar to previous studies, we show that the slip maximum in the southern zone is poorly resolved compared to the northern zone. Both slip maxima are higher than reported in previous studies, which we speculate may be due to the lack of bias caused by the regularization used in other studies.

/pdf/trans-dimensional-finite-fault-inversion-3wwgwhj4vw.pdf

Trans-dimensional finite-fault inversion

SUMMARY

This paper applies a general trans-dimensional Bayesian inference methodology and hierarchical autoregressive data-error models to the inversion of microtremor array dispersion data for shear wave velocity (vs) structure. This approach accounts for the limited knowledge of the optimal earth model parametrization (e.g. the number of layers in the vs profile) and of the data-error statistics in the resulting vs parameter uncertainty estimates. The assumed earth model parametrization influences estimates of parameter values and uncertainties due to different parametrizations leading to different ranges of data predictions. The support of the data for a particular model is often non-unique and several parametrizations may be supported. A trans-dimensional formulation accounts for this non-uniqueness by including a model-indexing parameter as an unknown so that groups of models (identified by the indexing parameter) are considered in the results. The earth model is parametrized in terms of a partition model with interfaces given over a depth-range of interest. In this work, the number of interfaces (layers) in the partition model represents the trans-dimensional model indexing.



In addition, serial data-error correlations are addressed by augmenting the geophysical forward model with a hierarchical autoregressive error model that can account for a wide range of error processes with a small number of parameters. Hence, the limited knowledge about the true statistical distribution of data errors is also accounted for in the earth model parameter estimates, resulting in more realistic uncertainties and parameter values. Hierarchical autoregressive error models do not rely on point estimates of the model vector to estimate data-error statistics, and have no requirement for computing the inverse or determinant of a data-error covariance matrix. This approach is particularly useful for trans-dimensional inverse problems, as point estimates may not be representative of the state space that spans multiple subspaces of different dimensionalities. The order of the autoregressive process required to fit the data is determined here by posterior residual-sample examination and statistical tests.



Inference for earth model parameters is carried out on the trans-dimensional posterior probability distribution by considering ensembles of parameter vectors. In particular, vs uncertainty estimates are obtained by marginalizing the trans-dimensional posterior distribution in terms of vs-profile marginal distributions. The methodology is applied to microtremor array dispersion data collected at two sites with significantly different geology in British Columbia, Canada. At both sites, results show excellent agreement with estimates from invasive measurements.

Jan Dettmer

Papers

Trans-dimensional geoacoustic inversion

Trans-dimensional matched-field geoacoustic inversion with hierarchical error models and interacting Markov chains

Efficient trans-dimensional Bayesian inversion for geoacoustic profile estimation

Trans-dimensional finite-fault inversion

Trans-dimensional inversion of microtremor array dispersion data with hierarchical autoregressive error models