Image Interpretation Using Appraisal Analysis

In geophysical inversion, a significant effort is invested to obtain images of the Earth from finite data. The first step is to obtain an image i.e. solve the inverse problem. This step alone provides significant challenges that are not addressed inthis paper. The next step is to interpret the image in terms of specific questions. For example, what can we say about the average value of a physical property within a certain region of the model? What scale information can we resolve from the data? These questions are problem dependent and may require that inversion be carried out several times to arrive at a satisfactory answer. Therefore the solution to an inverse problem is only a step towards answering these questions. Appraisal analysis of the solution takes the next step by providing a set of tools to judge and select from the possibly infinite suite of images that adequately fit our observations. We discuss the use of point spread functions and averaging kernels in the interpretation of images. We use a controlled source electromagnetic example to demonstrate the methodology

Wavelet Deconvolution in a Periodic Setting Using Cross-Validation

The wavelet deconvolution method WaveD using band-limited wavelets offers both theoretical and computational advantages over traditional compactly supported wavelets. The translation-invariant WaveD with a fast algorithm improves further. The twofold cross-validation method for choosing the threshold parameter and the finest resolution level in WaveD is introduced. The algorithm’s performance is compared with the fixed constant tuning and the default tuning in WaveD

Wavelet Reconstruction of Nonuniformly Sampled Signals

For the reconstruction of a nonuniformly sampled signal based on its noisy observations, we propose a level dependent l1 penalized wavelet reconstruction method. The LARS/Lasso algorithm is applied to solve the Lasso problem. The data adaptive choice of the regularization parameters is based on the AIC and the degrees of freedom is estimated by the number of nonzero elements in the Lasso solution. Simulation results conducted on some commonly used 1_D test signals illustrate that the proposed method possesses good empirical properties

Inversion for Non-Smooth Models with Physical Bounds

Geological processes produce structures at multiple scales. A discontinuity in the subsurface can occur due to layering, tectonic activities such as faulting, folding and fractures. Traditional approaches to invert geophysical data employ smoothness constraints. Such methods produce smooth models and thefore sharp contrasts in the medium such as lithological boundaries are not easily discernible. The methods that are able to produce non-smooth models, can help interpret the geological discontinuity. In this paper we examine various approaches to obtain non-smooth models from a finite set of noisy data. Broadly they can be categorized into approaches: (1) imposing non-smooth regularization in the inverse problem and (2) solve the inverse problem in a domain that provides multi-scale resolution, such as wavelet domain. In addition to applying non-smooth constraints, we further constrain the inverse problem to obtain models within prescribed physical bounds. The optimization with non-smooth regularization and physical bounds is solved using an interior point method. We demonstrate the applicability and usefulness of these methods with realistic synthetic examples and provide a field example from crosswell radar data

A Field Comparison of Fresnel Zone and Ray-Based GPR Attenuation-Difference Tomography for Time-Lapse Imaging of Electrically Anomalous Tracer or Contaminant Plumes

Ground-penetrating radar (GPR) attenuation-difference tomography is a useful tool for imaging the migration of electrically anomalous tracer or contaminant plumes. Attenuation-difference tomography uses the difference in the trace amplitudes of tomographic data sets collected at different times to image the distribution of bulk-conductivity changes within the medium. The most common approach for computing the tomographic sensitivities uses ray theory, which is well understood and leads to efficient computations. However, ray theory requires the assumption that waves propagate at infinite frequency, and thus sensitivities are distributed along a line between the source and receiver. The infinite-frequency assumption in ray theory leads to a significant loss of resolution (both spatially and in terms of amplitude) of the recovered image. We use scattering theory to approximate the sensitivity of electromagnetic (EM) wave amplitude to changes in bulk conductivity within the medium. These sensitivities occupy the first Fresnel zone, account for the finite frequency nature of propagating EM waves, and are valid when velocity variations within the medium do not cause significant ray bending. We evaluate the scattering theory sensitivities by imaging a bromide tracer plume as it migrates through a coarse alluvial aquifer over two successive days. The scattering theory tomograms display a significant improvement in resolution over the ray-based counterparts, as shown by a direct comparison of the tomograms and also by a comparison of the vertical fluid conductivity distribution measured in a monitoring well, located within the tomographic plane. By improving resolution, the scattering theory sensitivities increase the utility of GPR attenuation- difference tomography for monitoring the movement of electrically anomalous plumes. In addition, the improved accuracy of information gathered through attenuation-difference tomography using scattering theory is a positive step toward future developments in using GPR data to help characterize the distribution of hydrogeologic propertie

Application of Time-Lapse ERT Imaging to Watershed Characterization

Time-lapse electrical resistivity tomography (ERT) has many practical applications to the study of subsurface properties and processes. When inverting time-lapse ERT data, it is useful to proceed beyond straightforward inversion of data differences and take advantage of the time-lapse nature of the data. We assess various approaches for inverting and interpreting time-lapse ERT data and determine that two approaches work well. The first approach is model subtraction after separate inversion of the data from two time periods, and the second approach is to use the inverted model from a base data set as the reference model or prior information for subsequent time periods. We prefer this second approach. Data inversion methodology should be consideredwhen designing data acquisition; i.e., to utilize the second approach, it is important to collect one or more data sets for which the bulk of the subsurface is in a background or relatively unperturbed state. A third and commonly used approach to time-lapse inversion, inverting the difference between two data sets, localizes the regions of the model in which change has occurred; however, varying noise levels between the two data sets can be problematic. To further assess the various time-lapse inversion approaches, we acquired field data from a catchment within the Dry Creek Experimental Watershed near Boise, Idaho, U.S.A. We combined the complimentary information from individual static ERT inversions, time-lapse ERT images, and available hydrologic data in a robust interpretation scheme to aid in quantifying seasonal variations in subsurface moisture content

Incorporating Geostatistical Constraints in Nonlinear Inverse Problems

In this paper we present a method of incorporating semivariogram constraints into nonlinear inversion problems. That is, we describe a method of sampling the space of inverse solutions that honor a specified semivariogram or set of semivariograms and also explain a set of state data. The approach can be considered a method of conditional simulation where model conditioning is based upon state data (as opposed to parameter data). The difference between this approach and other simulation approaches is that the simulation is posed as an optimization problem with the joint objective of matching the semivariograms and honoring the state data. This approach requires computing the sensitivities of the semivariograms with respect to the distributed parameter. We derive these sensitivities and find that they are efficient to compute and store, making the method tenable for large models. We demonstrate the method with one synthetic and one field example using radar velocity tomography, where radar velocity is related through a petrophysical transform to saturated porosity. We address biasing issues and demonstrate ensemble generation and the resulting resolution and uncertainty analysis using ensemble statistics. We also demonstrate how the method can be applied to existing deterministic inversion codes with the field example

Electromagnetic coupling in frequency domain induced polarisation data

Frequency domain induced polarization (IP) surveys are commonly carried out to provide information about the chargeability structure of the earth. The goals might be as diverse as trying to delineate a mineralized and/or alteration zone for mineral exploration, or to find a region of contaminants for an environmental problem. Unfortunately, the measured responses can have contributions from inductive and galvanic effects of the ground. The inductive components are called EM coupling effects. They are considered to be "noise" and much of this thesis is devoted towards either removing these effects, or reformulating the inverse problem so that inductive effects are part of the "signal". If the forward modeling is based on galvanic responses only, then the inductive responses must first be removed from the data. The motivation for attacking the problem in this manner is that it is easier to solve D.C. resistivity equation than the full Maxwell's equation. The separation of the inductive response from the total response is derived by expressing the total electric field as a product of an IP response function, and an electric field which depends on EM coupling response. This enables me to generate formulae to obtain IP amplitude (PFE) and phase response from the raw data. The data can then be inverted, using a galvanic forward modeling. I illustrate this with 1D and 3D synthetic examples. To handle field data sets, I have developed an approximate method for estimating the EM coupling effects based upon the assumption that the earth is locally 1D. The 1D conductivity is obtained from a 2D inversion of the low frequency DC resistivity data. Application of this method to a field data set has shown encouraging results. I also examine the EM coupling problem in terms of complex conductivity. I show that if the forward modeling is carried out with full Maxwell's equation, then there is no need to remove EM coupling. I illustrate this with 1D synthetic example. In summary, I have investigated the EM coupling problem in IP and developed a practical removal methodology that can be applied to data sets from 1D, 2D and 3D earth structures.Science, Faculty ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat

The Point-Spread Function Measure of Resolution for the 3-D Electrical Resistivity Experiment

The solution appraisal component of the inverse problem involves investigation of the relationship between our estimated model and the actual model. However, full appraisal is difficult for large 3-D problems such as electrical resistivity tomography (ERT). We tackle the appraisal problem for 3-D ERT via the point-spread functions (PSFs) of the linearized resolution matrix. The PSFs represent the impulse response of the inverse solution and quantify our parameter-specific resolving capability. We implement an iterative least-squares solution of the PSF for the ERT experiment, using on-the-fly calculation of the sensitivity via an adjoint integral equation with stored Green\u27s functions and subgrid reduction. For a synthetic example, analysis of individual PSFs demonstrates the truly 3-D character of the resolution. The PSFs for the ERT experiment are Gaussian-like in shape, with directional asymmetry and significant off-diagonal features. Computation of attributes representative of the blurring and localization of the PSF reveal significant spatial dependence of the resolution with some correlation to the electrode infrastructure. Application to a time-lapse ground-water monitoring experiment demonstrates the utility of the PSF for assessing feature discrimination, predicting artefacts and identifying model dependence of resolution. For a judicious selection of model parameters, we analyse the PSFs and their attributes to quantify the case-specific localized resolving capability and its variability over regions of interest. We observe approximate interborehole resolving capability of less than 1–1.5 m in the vertical direction and less than 1–2.5 m in the horizontal direction. Resolving capability deteriorates significantly outside the electrode infrastructure