NS23A-01
Pseudo GPR Full wave form inversion using stochastic tomography
A new approach to GPR tomography based on stochastic simulations and full wave form modelling is proposed. The method is applied on a synthetic field with known electrical properties. First full wave modelling is used to compute the synthetic traces. The travel time and peak-to-peak amplitudes are picked on each trace. Then, 128 different velocity and attenuation fields are computed by stochastic tomography conditioned, up to the first order, to the reference travel times and amplitudes obtained from synthetic traces and fully conditioned to velocity and attenuation data available along boreholes. According to electromagnetic theory, attenuation and velocity fields are combined to compute the electrical conductivity and permittivity fields. Full wave form simulation is performed on each of the 128 simulated fields yielding as many simulated trace sets. Average time shift and cross-correlation are calculated between the simulated trace sets and the reference trace set to identify the simulated fields most similar to the synthetic field. The best adjusted trace sets ensures that the corresponding electrical fields are the closest to the synthetic field.
NS23A-02
Finite-Frequency Traveltime Tomography for Active-Source Seismic Data
Infinite-frequency traveltime tomography/inversion is the most common approach for modeling active-source wide-angle data. This study-in-progress considers the advantages of using finite-frequency traveltime tomography for active-source data. In theory, finite-frequency methods should provide higher spatial resolution and more accurate recovery of anomaly magnitudes. For active-source data a nonlinear iterative gradient approach is necessary since there is no reference model capable of providing accurate ray or wave paths for a typical dataset. The forward step solves the acoustic wave equation using a finite-difference scheme, and the first-arrival times are determined using a limited amount of manual picking to train an automatic picking algorithm. The inverse step uses Fresnel-zone sensitivity kernels and conventional smoothing regularization. Applications to both synthetic and real high-resolution near-surface data are presented. Preliminary results suggest that regularization and appropriate data fitting act to defeat the potential advantages of the finite- frequency approach. As a result, this study suggests that the precise form of inversion methodology is the critical factor in determining the need for a finite-frequency approach.
NS23A-03 INVITED
Two-Dimensional Inversion of Marine Electromagnetic Data Using Seismic Reflection Data as Apriori Information
The use of controlled source electromagnetics (CSEM) in the marine environment has grown rapidly in the past few years from a simple anomaly hunting technique used in geologically simple environments to a modeling and inversion based technique applied in structurally and lithologically complex environments. The tool set most commonly available to interpreters includes one-, two- and three-dimensional forward and inverse modeling codes. All previous examples in the literature of inversion codes as applied to CSEM data have been cell-based regularized techniques designed to produce the smoothest possible isotropic conductivity model (in two- or three-dimensions) which fits the observed data. We report on the development of two inversion methodologies that allow for the inclusion of apriori information in the form of structural surfaces as picked from seismic reflection data. The first technique uses the same basic parameterization as the smoothed-pixel inversion technique, except that it allows for discontinuities in the smoothing matrix, thus allowing for preservation of boundaries in the isotropic conductivity model. The second approach is an �anisotropic sharp-boundary inversion' in which the model is parameterized by two-dimensional interfaces. Regularization is applied to the smoothness of the interface and the lateral variations of conductivity between interfaces. After an explanation of the basic inversion algorithms, we will provide examples where the algorithms are applied to data collected in a marine EM environment.
NS23A-04 INVITED
Error Quantification in Inverse Modeling: Perspectives and Methods
Hydrogeology, geophysics, and other areas of expertise in earth sciences rely heavily on data to infer properties of geologic formations from the data with methods of analysis that come under the general rubric of inverse methods. Each field has developed its own methods that reflect its special needs as well as the training and intellectual biases of its practitioners. Quantification of uncertainty in the solution of algebraically underdetermined and/or mathematically ill-posed problems is probably where methods differ the most. Views vary widely on how to quantify uncertainty, and even whether rational and objective quantification is at all possible. We briefly review and assess critically various viewpoints and sets of guidelines, including deterministic error bounds and various stochastic doctrines. For each approach, we highlight practical applicability, usefulness of results, and actual limitations. We propose that, in many cases, techniques grounded on an empirical Bayes approach can offer the practitioner a satisfactory combination of generality, rigor, computational efficiency, and objectivity.
NS23A-05 INVITED
A Multistage Sampling Method for Rapid Quantification of Uncertainty During Subsurface Characterization
We discuss a novel sampling method for rapid quantification of uncertainty during subsurface characterization via inverse modeling of dynamic data, specifically multiphase production response and time-lapse (4D) seismic data. Uncertainty evaluation is generally carried out in a Bayesian framework whereby multiple subsurface models can be evaluated by sampling from a posterior distribution that incorporates the observed data and the prior parameter distribution. Rigorous sampling methods such as the Markov Chain Monte Carlo (MCMC) method provide accurate sampling but at a high cost because of their high rejection rates and the need to run a full flow and transport simulation for every proposed candidate. Approximate sampling methods like the randomized maximum likelihood (RML) are often used for computational efficiency but the assumptions are too restrictive, particularly for nonlinear problems in multiphase flow and transport. We discuss here a two-stage Markov Chain Monte Carlo (MCMC) method that utilizes a combination of a fast linearized approximation of the dynamic data and the MCMC algorithm. Our proposed sampling approach is rigorous, computationally efficient and has a significantly higher acceptance rate compared to traditional MCMC algorithms. In the first stage we utilize streamlines or trajectory-based analytic sensitivities to obtain an approximation in a small neighborhood of the previously computed dynamic data. These analytic sensitivities do not require any additional flow simulations. The approximation of the dynamic data is then used to modify the instrumental proposal distribution in the MCMC. Only those proposals that pass the acceptance criterion in the first stage are then assessed by running full flow simulations to assure rigorousness in sampling and are either accepted or rejected using the Metropolis- Hastings criterion. It is shown that the modified Markov chain converges to a stationary state corresponding to the posterior distribution. Both two dimensional synthetic examples and three dimensional field applications will be used to demonstrate the power and utility of the two-stage sampling method for dynamic data integration and uncertainty analysis.
http://www.pe.tamu.edu/datta- gupta/public_html
NS23A-06 INVITED
Stochastic Information Fusion for Monitor Water Movement in the Vadose Zone
Electrical resistivity tomography (ERT) has emerged as a potentially cost-effective, least-invasive tool for imaging changes of moisture content in the vadose zone. The accuracy of ERT surveys, however, has been the subject of debate because of its non-unique inverse solution and spatial variability in the constitutive relation between resistivity and moisture content. In this paper, an estimator for ERT based on a stochastic information fusion concept was developed to derive the best unbiased estimate of the moisture content distribution and to quantify its uncertainty. Unlike classical ERT inversion approaches, this new approach treats the electrical resistivity, electric potential, moisture content fields, and spatially varying parameters of Archie's law as spatially random fields. It then assimilates sparse point measurements of the moisture content, electrical resistivity, and electric potential as well as parameters of the resistivity-moisture content relation according to prior information about the geologic and moisture content structures in a given geologic medium and principle of electric field to directly estimate three-dimensional moisture content distributions, instead of changes in moisture content in the vadose zone. Numerical experiments were conducted to investigate the effect of uncertainties in the prior information on the estimate. The effects of spatial variability in the constitutive relation were then examined on the interpretation of the change in moisture content, based on the change in electrical resistivity from the ERT survey. Finally, the ability of the integrative approach was tested by directly estimating moisture distributions in three-dimensional, heterogeneous vadose zones. Results show that the integrative approach can produce accurate estimates of the moisture content distributions, and that incorporating some measurements of the moisture content is essential to improve the estimate. Finally, a systematic approach to process and analyze ERT field data using the stochastic estimator is illustrated.
http://www.hwr.arizona.edu/yeh
NS23A-07
Funnel function approach to determine uncertainty: Some advances
Given a finite number of noisy data it is difficult (perhaps impossible) to obtain unique average of the model value in any region of the model (Backus & Gilbert, 1970; Oldenburg, 1983). This difficulty motivated Backus and Gilbert to construct the averaging kernels that is in some sense close to delta function. Averaging kernels describe how the true model is averaged over the entire domain to generate the model value in the region of interest. An unique average value is difficult to obtain theoretically. However we can compute the bounds on the average value and this allows us to obtain a measure of uncertainty. This idea was proposed by Oldenburg (1983). As the region of interest increases the uncertainty decreases associated with the average value giving a funnel like shape. Mathematically this is equivalent to solving minimization and maximization problem of average value (Oldenburg, 1983). In this work I developed a nonlinear interior point method to solve this min-max problem and construct the bounds. The bounds determined in this manner honors all types of available information: (a) geophysical data with errors (b) deterministic or statistical prior information and (c ) complementary information from other data sets at different scales (such as hydrology or other geophysical data) if they are formulated in a joint inversion framework.
http://cgiss.boisestate.edu/~routh
NS23A-08
Plausible Solution Space Sampling to Quantify Resolution and Uncertainty
In the geosciences, parameter estimation problems are typically non-unique due to a lack of sufficient data. There are many different models of the subsurface that can explain a particular set of data. Parameter non-uniqueness arises when the available data are unable to resolve, or are insensitive to, the subsurface at the scale of the parameter (e.g. the discretized cell). These small-scale insensitive features are in the null space of the data, and therefore the corresponding parameter may assume any value without affecting the data fit. It is this null space that leads to uncertainty in parameter estimates, and requires practitioners to apply some type of apriori constraint to the inverse problem in order to obtain a plausible solution. In this work, we present a method of quantifying uncertainty by sampling the space of models that both fit the data and honor a specified spatial covariance structure as given by one or more semivariograms. Sampling is accomplished by formulating the inverse problem with the joint objective of fitting the data and honoring the semivariogram(s). The constrained solution space includes all plausible models where plausibility is defined by the joint data and semivariogram constraints. By sampling from this space, we build an ensemble of solutions that can be used to generate statistical measures of uncertainty, such as ensemble mean and variance. Those smaller-scale features in the null space of the data are included (or sampled) in order to honor the semivariogram constraints at smaller-scale lag distances. These 'null' features are not preserved in the ensemble mean, and therefore the ensemble mean describes the resolution of the joint data and semivariogram constraints. The corresponding ensemble variance of each parameter describes the uncertainty in the solutions given the joint constraints. In order to accurately describe the solution space, the model samples in the ensemble must be unbiased and equally probable. We explore this aspect of the problem in detail and present evidence that such is the case. In addition, the method is efficient and can be applied to existing deterministic inversion codes, providing a powerful method of exploring the plausible solution space to quantify resolution and uncertainty.