387 research outputs found
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
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
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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
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