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 depthdependent 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.

S U M M A R YThis 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 (v s ) structure. This approach accounts for the limited knowledge of the optimal earth model parametrization (e.g. the number of layers in the v s profile) and of the dataerror statistics in the resulting v s 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 transdimensional 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, v s uncertainty estimates are obtained by marginalizing the trans-dimensional posterior distribution in terms of v s -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.

This paper examines joint inversion of acoustic scattering and reflection data to resolve seabed interface roughness parameters (spectral strength, exponent, and cutoff) and geoacoustic profiles. Trans-dimensional (trans-D) Bayesian sampling is applied with both the number of sediment layers and the order (zeroth or first) of auto-regressive parameters in the error model treated as unknowns. A prior distribution that allows fluid sediment layers over an elastic basement in a trans-D inversion is derived and implemented. Three cases are considered: Scattering-only inversion, joint scattering and reflection inversion, and joint inversion with the trans-D auto-regressive error model. Including reflection data improves the resolution of scattering and geoacoustic parameters. The trans-D auto-regressive model further improves scattering resolution and correctly differentiates between strongly and weakly correlated residual errors.

Measurements made of the acoustical characteristics of, and occupied noise levels in, ten eating establishments are described. Levels to which diners and employees were exposed varied from 45 to 82 dB(A). From these levels and diner questionnaire responses, the number of customers present and average noise levels to which individual diners were exposed during their visits were estimated. These data, assumptions about the number of talkers per customer, and classical room-acoustical theory were used to deduce talker voice output levels. These varied from slightly above "casual" to "loud." An iterative model for predicting speech and noise levels in eating establishments, including the Lombard effect as described by a new, proposed model, was developed. With the measured noise levels as the target for prediction, optimization techniques were used to find best estimates of unknown prediction parameters--such as those defining the Lombard effect, the number of talkers per customer, and the average absorption per customer--with highly credible results. The prediction algorithm and optimal parameters constitute a novel model for predicting speech and noise levels--and thus speech intelligibility--in eating establishments, as a function of the number of customers, including a proven, realistic model of the Lombard effect.

This paper considers sampling efficiency of trans-dimensional (trans-D) Bayesian inversion based on the reversible-jump Markov-chain Monte Carlo (rjMCMC) algorithm, with application to seabed acoustic reflectivity inversion. Trans-D inversion is applied to sample the posterior probability density over geoacoustic parameters for an unknown number of seabed 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 applied to perturb existing parameters and to assign values for parameters added to the model. Several proposal schemes are examined, some of which appear new for trans-D geoacoustic 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 considered. Parallel tempering, which employs a sequence of interacting Markov chains with successively relaxed likelihoods, is also considered 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.

This paper presents seabed interface-scattering and geoacoustic parameters estimated on the Malta Plateau, Mediterranean Sea, by joint Bayesian inversion of monostatic backscatter and spherical-wave reflection-coefficient data. The data are modeled assuming a stack of homogeneous fluid sediment layers overlying an elastic basement. The scattering model also assumes a randomly rough water-sediment interface with a von Karman roughness power spectrum. Scattering and reflection data are inverted simultaneously using a population of interacting Markov-chains to sample roughness and geoacoustic parameters as well as residual error parameters. Trans-dimensional sampling is applied to treat the unknown number of sediment layers and unknown autoregressive order of the errors (to represent residual correlation). Results are considered in terms of marginal posterior probability profiles and distributions, which quantify the effective data information content to resolve scattering∕geoacoustic parameters and structure. Results indicate well-defined scattering (roughness) parameters in good agreement with existing measurements, and a multi-layer sediment profile over a high-speed (elastic) basement, consistent with independent knowledge of sand layers over limestone. [Work supported by ONR.]

This paper presents estimates of seabed roughness and geoacoustic parameters and uncertainties on the Malta Plateau, Mediterranean Sea, by joint Bayesian inversion of mono-static backscatter and spherical wave reflection-coefficient data. The data are modeled using homogeneous fluid sediment layers overlying an elastic basement. The scattering model assumes a randomly rough water-sediment interface with a von Karman roughness power spectrum. Scattering and reflection data are inverted simultaneously using a population of interacting Markov chains to sample roughness and geoacoustic parameters as well as residual error parameters. Trans-dimensional sampling is applied to treat the number of sediment layers and the order (zeroth or first) of an autoregressive error model (to represent potential residual correlation) as unknowns. Results are considered in terms of marginal posterior probability profiles and distributions, which quantify the effective data information content to resolve scattering/geoacoustic structure. Results indicate well-defined scattering (roughness) parameters in good agreement with existing measurements, and a multi-layer sediment profile over a high-speed (elastic) basement, consistent with independent knowledge of sand layers over limestone.

A quantitative inversion procedure is developed and applied to determine the dominant scattering mechanism (surface roughness and/or volume scattering) from seabed scattering-strength data. The classification system is based on trans-dimensional Bayesian inversion with the deviance information criterion used to select the dominant scattering mechanism. Scattering is modeled using first-order perturbation theory as due to one of three mechanisms: Interface scattering from a rough seafloor, volume scattering from a heterogeneous sediment layer, or mixed scattering combining both interface and volume scattering. The classification system is applied to six simulated test cases where it correctly identifies the true dominant scattering mechanism as having greater support from the data in five cases; the remaining case is indecisive. The approach is also applied to measured backscatter-strength data where volume scattering is determined as the dominant scattering mechanism. Comparison of inversion results with core data indicates the method yields both a reasonable volume heterogeneity size distribution and a good estimate of the sub-bottom depths at which scatterers occur.