Fully nonlinear inversion of fundamental mode surface waves for a global crustal model

We use neural networks to find 1-dimensional marginal probability density functions (pdfs) of global crustal parameters. The information content of the full posterior and prior pdfs can quantify the extent to which a parameter is constrained by the data. We inverted fundamental mode Love and Rayleigh wave phase and group velocity maps for pdfs of crustal thickness and independently of vertically averaged crustal shear wave velocity. Using surface wave data with periods T > 35 s for phase velocities and T > 18 s for group velocities, Moho depth and vertically averaged shear wave velocity of continental crust are well constrained, but vertically averaged shear wave velocity of oceanic crust is not resolvable. The latter is a priori constrained by CRUST2.0. We show that the resulting model allows to compute global crustal corrections for surface wave tomography for periods T > 50 s for phase velocities and T > 60 s for group velocities

How do we understand and visualize uncertainty?

Geophysicists are often concerned with reconstructing subsurface properties using observations collected at or near the surface. For example, in seismic migration, we attempt to reconstruct subsurface geometry from surface seismic recordings, and in potential field inversion, observations are used to map electrical conductivity or density variations in geologic layers. The procedure of inferring information from indirect observations is called an inverse problem by mathematicians, and such problems are common in many areas of the physical sciences. The inverse problem of inferring the subsurface using surface observations has a corresponding forward problem, which consists of determining the data that would be recorded for a given subsurface configuration. In the seismic case, forward modeling involves a method for calculating a synthetic seismogram, for gravity data it consists of a computer code to compute gravity fields from an assumed subsurface density model. Note that forward modeling often involves assumptions about the appropriate physical relationship between unknowns (at depth) and observations on the surface, and all attempts to solve the problem at hand are limited by the accuracy of those assumptions. In the broadest sense then, exploration geophysicists have been engaged in inversion since the dawn of the profession and indeed algorithms often applied in processing centers can all be viewed as procedures to invert geophysical data

Trans-dimensional inverse problems, model comparison and the evidence

In most geophysical inverse problems the properties of interest are parametrized using a fixed number of unknowns. In some cases arguments can be used to bound the maximum number of parameters that need to be considered. In others the number of unknowns is set at some arbitrary value and regularization is used to encourage simple, non-extravagant models. In recent times variable or self-adaptive parametrizations have gained in popularity. Rarely, however, is the number of unknowns itself directly treated as an unknown. This situation leads to a transdimensional inverse problem, that is, one where the dimension of the parameter space is a variable to be solved for. This paper discusses trans-dimensional inverse problems from the Bayesian viewpoint. A particular type of Markov chain Monte Carlo (MCMC) sampling algorithm is highlighted which allows probabilistic sampling in variable dimension spaces. A quantity termed the evidence or marginal likelihood plays a key role in this type of problem. It is shown that once evidence calculations are performed, the results of complex variable dimension sampling algorithms can be replicated with simple and more familiar fixed dimensional MCMC sampling techniques. Numerical examples are used to illustrate the main points. The evidence can be difficult to calculate, especially in high-dimensional non-linear inverse problems. Nevertheless some general strategies are discussed and analytical expressions given for certain linear problem

Global crustal thickness from neural network inversion of surface wave data

We present a neural network approach to invert surface wave data for a global model of crustal thickness with corresponding uncertainties. We model the a posteriori probability distribution of Moho depth as a mixture of Gaussians and let the various parameters of the mixture model be given by the outputs of a conventional neural network. We show how such a network can be trained on a set of random samples to give a continuous approximation to the inverse relation in a compact and computationally efficient form. The trained networks are applied to real data consisting of fundamental mode Love and Rayleigh phase and group velocity maps. For each inversion, performed on a 2° × 2° grid globally, we obtain the a posteriori probability distribution of Moho depth. From this distribution any desired statistic such as mean and variance can be computed. The obtained results are compared with current knowledge of crustal structure. Generally our results are in good agreement with other crustal models. However in certain regions such as central Africa and the backarc of the Rocky Mountains we observe a thinner crust than the other models propose. We also see evidence for thickening of oceanic crust with increasing age. In applications, characterized by repeated inversion of similar data, the neural network approach proves to be very efficient. In particular, the speed of the individual inversions and the possibility of modelling the whole a posteriori probability distribution of the model parameters make neural networks a promising tool in seismic tomography

An objective rationale for the choice of regularisation parameter with application to global multiple-frequency S-wave tomography

International audienceIn a linear ill-posed inverse problem, the regularisation parameter (damping) controls the balance between minimising both the residual data misfit and the model norm. Poor knowledge of data uncertainties often makes the selection of damping rather arbitrary. To go beyond that subjectivity, an objective rationale for the choice of damping is presented, which is based on the coherency of delay-time estimates in different frequency bands. Our method is tailored to the problem of global multiple-frequency tomography (MFT), using a data set of 287 078 S-wave delay times measured in five frequency bands (10, 15, 22, 34, and 51 s central periods). Whereas for each ray path the delay-time estimates should vary coherently from one period to the other, the noise most likely is not coherent. Thus, the lack of coherency of the information in different frequency bands is exploited, using an analogy with the cross-validation method, to identify models dominated by noise. In addition, a sharp change of behaviour of the model ℓ∞-norm, as the damping becomes lower than a threshold value, is interpreted as the signature of data noise starting to significantly pollute at least one model component. Models with damping larger than this threshold are diagnosed as being constructed with poor data exploitation. Finally, a preferred model is selected from the remaining range of permitted model solutions. This choice is quasi-objective in terms of model interpretation, as the selected model shows a high degree of similarity with almost all other permitted models (correlation superior to 98% up to spherical harmonic degree 80). The obtained tomographic model is displayed in the mid lower-mantle (660-1910 km depth), and is shown to be compatible with three other recent global shear-velocity models. A wider application of the presented rationale should permit us to converge towards more objective seismic imaging of Earth's mantle

Inference of abrupt changes in noisy geochemical records using transdimensional changepoint models

International audienceWe present a method to quantify abrupt changes (or changepoints) in data series, represented as a function of depth or time. These changes are often the result of climatic or environmental variations and can be manifested inmultiple datasets as different responses, but all datasets can have the same changepoint locations/timings. The method we present uses transdimensional Markov chain Monte Carlo to infer probability distributions on the number and locations (in depth or time) of changepoints, the mean values between changepoints and, if required, the noise variance associated with each dataset being considered. This latter point is important as we generally will have limited information on the noise, such as estimates only of measurement uncertainty, and in most cases it is not practical to make repeat sampling/measurement to assess other contributions to the variation in the data.Wedescribe themain features of the approach (and describe themathematical formulation in supplementary material), and demonstrate its validity using synthetic datasets, with known changepoint structure (number and locations of changepoints) and distribution of noise variance for each dataset.We show that when using multiple data, we expect to achieve better resolution of the changepoint structure than when we use each dataset individually. This is conditional on the validity of the assumption of common changepoints between different datasets.We then apply themethod to two sets of real geochemical data, both from peat cores, taken from NE Australia and eastern Tibet. Under the assumption that changes occur at the same time for all datasets, we recover solutions consistent with those previously inferred qualitatively from independent data and interpretations. However, our approach provides a quantitative estimate of the relative probability of the inferred changepoints, allowing an objective assessment of the significance of each change

Transdimensional inversion of receiver functions and surface wave dispersion

International audienceWe 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

Inversion of massive surface wave data sets: Model construction and resolution assessment

International audience[1] A new scheme is proposed for the inversion of surface waves using a continuous formulation of the inverse problem and the least squares criterion. Like some earlier schemes a Gaussian a priori covariance function controls the horizontal degree of smoothing in the inverted model, which minimizes some artifacts observed with spherical harmonic parameterizations. Unlike earlier schemes the new approach incorporates some sophisticated geometrical algorithms which dramatically increase computational efficiency and render possible the inversion of several tens of thousands of seismograms in few hours on a typical workstation. The new algorithm is also highly suited to parallelization which makes practical the inversion of data sets with more than 50,000 ray paths. The constraint on structural and anisotropic parameters is assessed using a new geometric approach based on Voronoi diagrams, polygonal cells covering the Earth's surface. The size of the Voronoi cells is used to give an indication of the length scale of the structures that can be resolved, while their shape provides information on the variation of azimuthal resolution. The efficiency of the scheme is illustrated with realistic uneven ray path configurations. A preliminary global tomographic model has been built for SV wave heterogeneities and azimuthal variations through the inversion of 24,124 fundamental and higher-mode Rayleigh waveforms. Our results suggest that the use of relatively short paths (<10,000 km) in a global inversion should minimize multipathing, or focusing/defocusing effects and provide lateral resolution of a few hundred kilometers across the globe

Location and mechanism of the Little Skull Mountain earthquake as constrained by satellite radar interferometry and seismic waveform modeling

We use interferometric synthetic aperture radar (InSAR) and broadband seismic waveform data to estimate source parameters of the 29 June 1992, M_s 5.4 Little Skull Mountain (LSM) earthquake. This event occurred within a geodetic network designed to measure the strain rate across the region around Yucca Mountain. The LSM earthquake complicates interpretation of the existing GPS and trilateration data, as the earthquake magnitude is sufficiently small that seismic data do not tightly constrain the epicenter but large enough to potentially affect the geodetic observations. We model the InSAR data using a finite dislocation in a layered elastic space. We also invert regional seismic waveforms both alone and jointly with the InSAR data. Because of limitations in the existing data set, InSAR data alone cannot determine the area of the fault plane independent of magnitude of slip nor the location of the fault plane independent of the earthquake mechanism. Our seismic waveform data tightly constrain the mechanism of the earthquake but not the location. Together, the two complementary data types can be used to determine the mechanism and location but cannot distinguish between the two potential conjugate fault planes. Our preferred model has a moment of ∼3.2 × 10^(17) N m (M_w 5.6) and predicts a line length change between the Wahomie and Mile geodetic benchmarks of ∼5 mm

Law of the leading digits and the ideological struggle for numbers

Benford's law states that the occurrence of significant digits in many data sets is not uniform but tends to follow a logarithmic distribution such that the smaller digits appear as first significant digits more frequently than the larger ones. We investigate here numerical data on the country-wise adherent distribution of seven major world religions i.e. Christianity, Islam, Buddhism, Hinduism, Sikhism, Judaism and Baha'ism to see if the proportion of the leading digits occurring in the distribution conforms to Benford's law. We find that the adherent data of all the religions, except Christianity, excellently does conform to Benford's law. Furthermore, unlike the adherent data on Christianity, the significant digit distribution of the three major Christian denominations i.e. Catholicism, Protestantism and Orthodoxy obeys the law. Thus in spite of their complexity general laws can be established for the evolution of the religious groups.Comment: 11 pages, 11 figures, 3 tables, title changed to "The law of the leading digits and the world religions" for journal version in publicatio