A penalty method for PDE-constrained optimization in inverse problems

Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand-sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the paramaters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the non-linearity of the problem and is less sensitive the initial iterate

Quantitative tools for seismic stratigraphy and lithology characterization

Seismological images represent maps of the earth's structure. Apparent bandwidth limitation of seismic data prevents successful estimation of transition sharpness by the multiscale wavelet transform. We discuss the application of two recently developed techniques for (non-linear) singularity analysis designed for bandwidth limited data, such as imaged seismic reflectivity. The first method is a generalization of Mallat's modulus maxima approach to a method capable of estimating coarse-grained local scaling/sharpness/Hölder regularity of edges/transitions from data residing at essentially one single scale. The method is based on a non-linear criterion predicting the (dis)appearance of local maxima as a function of the data's fractional integrations/differentiations. The second method is an extension of an atomic decomposition technique based on the greedy Matching Pursuit Algorithm. Instead of the ordinary Spline Wavelet Packet Basis, our method uses multiple Fractional Spline Wavelet Packet Bases, especially designed for seismic reflectivity data. The first method excels in pinpointing the location of the singularities (the stratigraphy). The second method improves the singularity characterization by providing information on the transition's location, magnitude, scale, order and direction (anti-/causal/symmetric). Moreover, the atomic decomposition entails data compression, denoising and deconvolution. The output of both methods produces a map of the earth's singularity structure. These maps can be overlayed with seismic data, thus providing us with a means to more precisely characterize the seismic reflectivity's litho-stratigraphical information content.Massachusetts Institute of Technology. Industry Consorti

Scaling And Seismic Reflectivity: Implications Of Scaling On Avo

AVO analysis of seismic data is based on the assumption that transitions in the earth consist of jump discontinuities only. The generalization of this type of transition to a more realistic class of transitions shows a drastic change in observed AVO behavior, especially for the large angles currently attained by increasing cable lengths. We propose a simple approach that accounts for this anomalous behavior by renormalizing the observed AVO. This renormalization allows for a separation of the observed AVO effects in terms of a conventional Zoeppritz contribution and a scaling contribution in those cases where the transitions can no longer be considered as isolated jump discontinuities. After renormalization, the inverted fluctuations regain their relative magnitudes which, due to the scaling, may have been significantly distorted. An example of these distortions are tuning effects, often erroneously interpreted as bright spots.Massachusetts Institute of Technology. Borehole Acoustics and Logging ConsortiumMassachusetts Institute of Technology. Earth Resources Laboratory. Reservoir Delineation Consortiu

Seismic Facies Characterization By Scale Analysis

Over the years, there has been an ongoing struggle to relate well-log and seismic data due to the inherent bandwidth limitation of seismic data, the problem of seismic amplitudes, and the apparent inability to delineate and characterize the transitions that can be linked to and held responsible for major reflection events and their signatures. By shifting focus to a scale invariant sharpness characterization for the reflectors, we develop a method that can capture, categorize, and reconstruct the main features of the reflectors, without being sensitive to the amplitudes. In this approach, sharpness is defined as the fractional degree of differentiability, which refers to the order of the singularity of the transitions. This sharpness determines mainly the signature/waveform of the reflection and can be estimated with the proposed monoscale analysis technique. Contrary to multiscale wavelet analysis the monoscale method is able to find the location and sharpness of the transitions at the fixed scale of the seismic wavelet. The method also captures the local orders of magnitude of the amplitude variations by scale exponents. These scale exponents express the local scale-invariance and texture. Consequently, the exponents contain local information on the type of depositional environment to which the reflector pertains. By applying the monoscale method to both migrated seismic sections and welllog data, we create an image of the earth's local singularity structure. This singularity map facilitates interpretation, facies characterization, and integration of well and seismic data on the level of local texture.Massachusetts Institute of Technology. Borehole Acoustics and Logging ConsortiumMassachusetts Institute of Technology. Earth Resources Laboratory. Reservoir Delineation Consortiu