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

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

Multifractional splines: from seismic singularities to geological transitions

A matching pursuit technique in conjunction with an imaging method is used to obtain quantitative information on geological records from seismic data. The technique is based on a greedy, non-linear search algorithm decomposing data into atoms. These atoms are drawn from a redundant dictionary of seismic waveforms. Fractional splines are used to define this dictionary, whose elements are not only designed to match the observed waveforms but also to span the appropriate family of geological patterns. Consequently, the atom’s parameterization provides localized scale, order and direction information that reveals the stratigraphy and the type of geological transitions. Besides a localized scaling characterization, the atomic decomposition allows for an accurate denoised reconstruction of data with only a small number of atoms. Application of this approach to angles gathers allows us to track geological singularities from seismic data. Our characterization bridges the gap between the analysis of the main features within geologic processes, i.e. the geologic patterns, and the interpretation of their associated seismic response. A case study of Valhall data is presented.Massachusetts Institute of Technology. Earth Resources Laborator