NG33B-01 INVITED 13:45h
Uncertainty Estimation in Inverse Problems; an Uncertain Proposition
The focus of inverse problems often is on the construction of a model. The user of inverse problems is, however, rarely interested in a model. Inverse problems are in practice used for making decisions that associated with academic issues, business decisions, or risk analysis. The uncertainty in the models is often extremely difficult to quantify. For linear inverse problems the size of the inverse problems is a real hurdle. For nonlinear inverse problems there is no theory to quantify the uncertainty in models. Stochastic methods can be used for this purpose, but their applicability is limited to relatively small-scale inverse problems. In this presentation I will sketch a number of open research questions that are crucial for using inverse problems effectively in decision making.
NG33B-02 INVITED 14:05h
Extended Models in Nonlinear Seismic Waveform Inversion
Most progress in seismic inverse scattering has come through use of linearization (``Born approximation''), in which the reference model is also regarded as unknown - so the problem is still nonlinear, albeit less so. The data-fitting formulations familiar from other work on inverse problems have had remarkably little influence on the development of practical solution methods for the seismic problem. Instead, industrial seismologists use an apparently ad-hoc collection of techniques dubbed {\em velocity analysis}, the effectiveness of which depends on the redundancy of seismic reflection surveys. This talk will explain how to view velocity analysis as the solution of an extended Born inverse problem. The various special techniques correspond to various extensions; Analysis shows that some of these are suitable for inversion in the presence of strong refraction, while others are not. Open problems abound, not the least of which is to get rid of the Born approximation. A natural generalization of an extended inverse formulation due originally to Claerbout combines the methodology of industrial velocity analysis with full nonlinear wavefield modeling. This presentation will describe some preliminary numerical results based on acoustic modeling, as well as a more general theory of nonlinear elastic waveform inversion.
NG33B-03 INVITED 14:25h
Invariance, Non-uniqueness, and Symmetry Groups
Non-uniqueness is an important aspect of most geophysical inverse problems. Non-uniqueness is equivalent to the invariance of the constraints with respect to changes in the Earth model. In this talk I will examine approaches to the analysis of invariance and non-uniqueness which are based upon Lie group methods. A key aspect of these methods involves the determination of symmetry groups associated with the constraint equations. In a general sense, a symmetry group of a system of equations is a group which transforms solutions of the system to other solutions. Therefore, the symmetry groups associated with an inverse problem may be used to explore the range of possible solutions. Lie group methods provide an alternative to conventional perturbation approaches. Mathematical methods from the theory of continuous groups may also be used to determine if a nonlinear inverse problem can be transformed into a linear problem. From the analytical form of the equations representing the inverse problem a set of linear partial differential equations, the defining equations, may be derived. The solutions to the defining equations provide the generators from which a transformation group may be constructed.
NG33B-04 INVITED 14:40h
Amplitude-variation-with-offset and nonlinear prestack waveform inversion
Use of prestack data for quantitative analysis of hydrocarbon reservoirs has gained popularity over the past decades. Most of these quantitative studies rely on the seismic reflection amplitude variation with offset or AVO analysis. AVO makes a simple assumption that every reflection event on the prestack seismic data is a primary P-wave reflection, and there is no contamination of these reflections from other wave modes. Using synthetic data for a finely layered elastic model, we demonstrate that such a simple assumption is incorrect, and there is a lot of interference of P-wave reflection amplitudes with other wave modes, especially at high offsets/angles. To properly model such interference, it is necessary to use prestack waveform inversion (PSWI), where these effects are correctly accounted for. Over the recent years, PSWI has been successfully used not only in a variety of reservoir applications but also in geo-hazard analyses such as pre-drill pore-pressure prediction and shallow water flow analysis. The problem of PSWI is nonlinear in nature. In this paper, we use a nonlinear optimization methodology, based on genetic algorithm (GA) for inversion. Such an inversion is computer intensive, since it must compute many forward synthetic models to obtain an estimate of the optimum earth model at a given common midpoint (CMP) location. To effectively apply inversion over large areas, we therefore combine PSWI with poststack inversion methodologies in a hybrid inversion scheme. Using a variety of synthetic and real data examples, application of PSWI and hybrid inversion methodologies will be discussed during the presentation.
NG33B-05 14:55h
Geostatistical Surface Classification
Geostatistical surface classification is aimed at distinguishing objects - surface provinces or surface types - objectively and automatically. The basic idea is to calculate spatial structure functions from surface data and extract parameters from those functions that constitute a feature vector. If feature vectors can be designed to capture characteristic properties of surface types, then a classification of surface provinces is possible. Application in a moving-window operation facilitates segmentation of a given study area into surface provinces. Application to time series of surface data provides a means to study morphogenetic processes and changes in environmental conditions. Geostatistical classification provides a number of mathematical challenges and solutions as well as a wide range of applications. The problem of extraction of parameters in an ill-posed noise-to-resolution situation motivates the introduction of vario functions of higher order. Association of surface classes may be performed using deterministic functions or connectionist association. Applications range from a segmentation of marine-geologic provinces to a study of self-organisational processes in an alpine snow pack.
NG33B-06 15:10h
Eigen-Space Modeling for Anisotropic Multifractality Characterization From 2-D Mineralization-Associated Geochemical Patterns
Two-dimensional map patterns commonly observed in geoscieces such as distribution of geochemical concentration values of elements in various surface media, spatial distribution of mineral deposits in a mineral district, and most remotely sensed images can often be considered as end products of multiplicative cascade processes. Modeling the anisotropic scaling property and heterogeneity of these types of patterns are essential for information retrieval in characterizing the underlying processes. The paper introduces a new method to characterize the multifractality of 2-D map from its Eigenvalues and Eigenvectors derived at multiple scales. First, it calculates all Eigenvalues and Eigenvectors from a given map, treated as an asymmetrical matrix at multiple map resolutions, and then multifractal distributions of these values are explored using number-frequency as well as value-frequency models. The links between these distributions and anisotropic property of the map patterns are discussed. Potential application of this method has been demonstrated using 2-D maps created from As, Cu, Pb, and Ag geochemical values analyzed from lake sediments and U, Th, and K gamma ray spectrometer data from southwestern Nova Scotia, Canada.
http://www.yorku.ca/gisweb
NG33B-07 15:25h
Detection of Local Anomalies in High Resolution Hyperspectral Imagery Using Geostatistical Filtering and Local Spatial Statistics
Spatial data are periodically collected and processed to monitor, analyze and interpret developments in our changing environment. Remote sensing is a modern way of data collecting and has seen an enormous growth since launching of modern satellites and development of airborne sensors. In particular, the recent availability of high spatial resolution hyperspectral imagery (spatial resolution of less than 5 meters and including data collected over 64 or more bands of electromagnetic radiation for each pixel offers a great potential to significantly enhance environmental mapping and our ability to model spatial systems. High spatial resolution imagery contains a remarkable quantity of information that could be used to analyze spatial breaks (boundaries), areas of similarity (clusters), and spatial autocorrelation (associations) across the landscape. This paper addresses the specific issue of soil disturbance detection, which could indicate the presence of land mines or recent movements of troop and heavy equipment. A challenge presented by soil detection is to retain the measurement of fine-scale features (i.e. mineral soil changes, organic content changes, vegetation disturbance related changes, aspect changes) while still covering proportionally large spatial areas. An additional difficulty is that no ground data might be available for the calibration of spectral signatures, and little might be known about the size of patches of disturbed soils to be detected. This paper describes a new technique for automatic target detection which capitalizes on both spatial and across spectral bands correlation, does not require any a priori information on the target spectral signature but does not allow discrimination between targets. This approach involves successively a multivariate statistical analysis (principal component analysis) of all spectral bands, a geostatistical filtering of noise and regional background in the first principal components using factorial kriging, and finally the computation of a local indicator of spatial autocorrelation to detect local clusters of high or low reflectance values as well as anomalies. The approach is illustrated using one meter resolution data collected in Yellowstone National Park. Ground validation data demonstrate the ability of the filtering procedure to reduce the proportion of false alarms, and its robustness under low signal to noise ratios. In almost all scenarios, the proposed approach outperforms traditional anomaly detectors (i.e. RXD) and fewer false alarms were obtained when using statistic S2 (average absolute deviation of p-values from 0.5 through all spectral bands) to summarize information across bands. Image degradation through addition of noise or reduction of spectral resolution tends to blur the detection of anomalies, leading to more false alarms, in particular for the identification of the least pure pixels. Results from the tailings site demonstrated that the approach still performs reasonably well for highly complex landscape with multiple targets of various sizes and shapes. By leveraging both spectral and spatial information, the technique requires little or no input from the user, and hence can be readily automated.
http://www-personal.engin.umich.edu/~goovaert/IEEE03_P-27.pdf