Michael J. Tompkins

Scalable uncertainty estimation for nonlinear inverse problems using parameter reduction, constraint mapping, and geometric sampling: Marine controlled-source electromagnetic examples

We have developed a new uncertainty estimation method that accounts for nonlinearity inherent in most geophysical problems, allows for the explicit search of model posterior space, is scalable, and maintains computational efficiencies on the order of deterministic inverse solutions. We accomplish this by combining an efficient parameter reduction technique, a parameter constraint mapping routine, a sparse geometric sampling scheme, and an efficient forward solver. In order to reduce our model domain and determine an independent basis, we implement both a typical principal component analysis, which factorizes the model covariance matrix, and an alternative compression method, based on singular-value decomposition, which acts on training models, directly, and is storage efficient. Once we have a reduced base, we map parameter constraints, from our original model domain, to this reduced domain to define a bounded geometric region of feasible model space. We utilize an optimal scheme to sample this reduced mo...

Effects of pore pressure on compressional wave attenuation in a young oceanic basalt

Laboratory data are reported for ultrasonic compressional wave attenuation (αp) as a function of pore pressure in a Juan de Fuca Ridge dredge basalt. Four experiments have been made to determine the relationships between attenuation and quality factor (Qp) and confining and pore pressures in the shallow ocean crust. Attenuation was measured at 1) a constant differential pressure of 40 MPa; 2) confining pressures to 120 MPa and atmospheric pore pressures; 3) a constant confining pressure of 50 MPa while varying pore pressures; 4) a constant confining pressure of 100 MPa while varying pore pressures. For atmospheric pore pressures, αp ranges from 1.64 dB/cm to 7.08 dB/cm (Qp = 24 to 10). In addition, attenuation increases systematically with increasing pore pressure and decreasing differential pressure (confining pressure - pore pressure). Results from DSDP and ODP downhole packer experiments suggest that the hydrostatic pore pressure regime may best approximate in situ conditions for young oceanic crust. Hydrostatic pore pressures (@ 5000 meters depth) reduce Qp as much as 35% from normal atmospheric pressure conditions; therefore, pore pressures generated in the upper oceanic crust may be responsible in part for the observed low seismic Qp within layer 2A. Qp measurements at elevated pore pressures agree well with seismic Qp data.

Numerical analysis of near-borehole and anisotropic layer effects on the response of multicomponent induction logging tools

We present finite-difference simulation results that lend new insight into the behavior of multicomponent induction logging tools when in the presence of anisotropic layers, boreholes, and invasion zones. We use four independent models to investigate multicomponent tool properties as well as typical magnetic field responses. In addition, model variations with respect to formation dip angle, layer geometry, and conductivity provide data about the effects of geological variation on the multicomponent responses. Simulations suggest a coaxial tool configuration senses a depth of twice the source–receiver offset, although this depth is reduced to the source–receiver offset with coplanar configurations. Numerical responses in the presence of transversely isotropic layers provide evidence that anisotropy can have a measurable effect on both coaxial and coplanar magnetic fields; these effects increase as layer dip increases. Sensitivity analyses substantiate these numerical results. An investigation of tool responses to varying borehole and invasion zone conductivities and diameters demonstrates that the coplanar tool orientation is much more sensitive to near-borehole variations than the coaxial configuration. A frequency-differencing technique is presented to mitigate unwanted borehole-induced bias in multicomponent data; however, drawbacks include decreased signal strength and possible geological signal destruction.

Geometric Sampling: An Approach to Uncertainty in High Dimensional Spaces

Uncertainty is always present in inverse problems. The main reasons for that are noise in data and measurement error, solution non-uniqueness, data coverage and bandwidth limitations, physical assumptions and numerical approximations. In the context of nonlinear inversion, the uncertainty problem is that of quantifying the variability in the model space supported by prior information and the observed data. In this paper we outline a general nonlinear inverse uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining computational efficiencies similar to deterministic inversions. Integral to this method is the combination of model reduction techniques, a constrained mapping approach and a sparse sampling scheme. This approach allows for uncertainty quantification in inverse problems in high dimensional spaces and very costly forward evaluations. We show some results in non linear geophysical inversion (electromagnetic data).

Inverse Problems and Model Reduction Techniques

Real problems come from engineering, industry, science and technology. Inverse problems for real applications usually have a large number of parameters to be reconstructed due to the accuracy needed to make accurate data predictions. This feature makes these problems highly underdetermined and ill-posed. Good prior information and regularization techniques are needed when using local optimization methods but only linear model appraisal (uncertainty) around the solution can be performed. The large number of parameters precludes the use of global sampling methods to approach inverse problem solution and appraisal. In this paper we show how to construct different kinds of reduced bases using Principal Component Analysis (PCA), Singular Value Decomposition (SVD), Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT). The use of a reduced base helps us to regularize the inverse problem and to find the set of equivalent models that fit the data within a prescribed tolerance and are compatible with the model prior.

Scalable Solutions for Nonlinear Inverse Uncertainty Using Model Reduction, Constraint Mapping, and Sparse Sampling

We present a general nonlinear inverse uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining computational efficiencies similar to deterministic inversions. Integral to this method is the combination of a parameter reduction technique, like Principal Component Analysis, a parameter bounds mapping routine, a sparse sampling scheme, and a forward solver. Parameter reduction, based on linearized model covariances, is used to reduce the model space by orders of magnitude. Parameter constraints are then mapped to this reduced space, using a linear programming scheme, defining a bounded posterior polytope. Sparse deterministic grids are employed to sample this feasible model region, while forward evaluations determine which model samples are equi-probable. The resulting ensemble represents the equivalent model space, consistent with Principal Components, that is used to infer uncertainty measures. The number of forward evaluations is determined adaptively and minimized by finding the sparsest sampling required for convergence of uncertainty measures. We demonstrate, with a surface electromagnetic example, that this method has the potential to reduce the nonlinear inverse uncertainty problem to a deterministic sampling problem in only a few dimensions, requiring limited forward solves, and resulting in an optimally sparse representation of the posterior model space.

A transversely isotropic 1‐D electromagnetic inversion scheme requiring minimal a priori information

We have developed a fast inversion algorithm for interpreting multi-component induction logging data assuming a 1-D transversely isotropic earth model. The iterative scheme implements an accurate semi-analytical forward solver, employs a quadratic program, and utilizes a sub-domain Jacobian computation to increase iteration speeds. Unlike similar 1-D algorithms, the inversion presented here accounts for depth variation, dip, and azimuth of layers as well as vertical and horizontal conductivities, which are all typically unknown parameters. In addition, the quadratic program affords the use of “global” physical bounds on model parameters allowing for accurate update estimates at each iterate. The resulting inversion requires minimal a priori knowledge of the 1-D formation properties. The scheme has achieved good results for synthetic data and we are currently testing it on field data recorded with a prototype multi-component induction logging tool.

Comparison of Sparse Polynomial and Random Sampling Methods in Electromagnetic Uncertainty Estimation

We have developed a general uncertainty estimation method that allows for the comprehensive search of model posterior space while maintaining a high degree of scalability and computational efficiency. We accomplish this by coupling parameter reduction with an optimally-sparse polynomial interpolation scheme. In contrast to Bayesian inference, which treats the posterior sampling problem as a random process, our method exploits the inherent structure of the posterior model space by estimating it with polynomial interpolation. In either case, the posterior represents the equivalent model space, consistent with our prior knowledge, and is used to estimate the inverse solution uncertainty. We demonstrate the efficiency of our method by comparing results from two EM problems with results for two commonly used Bayesian inference schemes: Gibbs and Metropolis-Hastings.

Regularized sparse-grid geometric sampling for uncertainty analysis in non-linear inverse problems

This paper introduces an efficiency improvement to the sparse-grid geometric sampling methodology for assessing uncertainty in non-linear geophysical inverse problems. Traditional sparse-grid geometric sampling works by sampling in a reduced-dimension parameter space bounded by a feasible polytope, e.g., a generalization of a polygon to dimension above two. The feasible polytope is approximated by a hypercube. When the polytope is very irregular, the hypercube can be a poor approximation leading to computational inefficiency in sampling. We show how the polytope can be regularized using a rotation and scaling based on principal component analysis. This simple regularization helps to increase the efficiency of the sampling and by extension the computational complexity of the uncertainty solution. We demonstrate this on two synthetic 1D examples related to controlled-source electromagnetic and amplitude versus offset inversion. The results show an improvement of about 50% in the performance of the proposed methodology when compared with the traditional one. However, as the amplitude versus offset example shows, the differences in the efficiency of the proposed methodology are very likely to be dependent on the shape and complexity of the original polytope. However, it is necessary to pursue further investigations on the regularization of the original polytope in order to fully understand when a simple regularization step based on rotation and scaling is enough.