SUMMARY
Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Frechet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Frechet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint-state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved.

Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

SUMMARY By specifying a discrete matrix formulation for the frequency^space modelling problem for linear partial diierential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss^Newton method and the full Newton method We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI)The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general The FDI methods only require that the original partial diierential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem) Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the mis¢t function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a ¢nal computation to compute a step length) This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss^Newton method, and the full Hessian matrix used in the full Newton method In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss^Newton method, can be e⁄ciently computed using a backpropagation approach similar to that used to compute the gradient vector The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of ¢rst-order multiples (such as free-surface multiples in seismic data) Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear eiects In a numerical test, the Gauss^Newton and full Newton methods prove eiective in helping to solve the di⁄cult non-linear problem of extracting a smooth background velocity model from surface seismic-re£ection data

Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion

Plane‐wave destruction filters originate from a local plane‐wave model for characterizing seismic data. These filters can be thought of as a time–distance (T‐X) analog of frequency‐distance (F‐X) prediction‐error filters and as an alternative to T‐X prediction‐error filters. The filters are constructed with the help of an implicit finite‐difference scheme for the local plane‐wave equation. Several synthetic and real data examples show that finite‐difference plane‐wave destruction filters perform well in applications such as fault detection, data interpolation, and noise attenuation.

/pdf/applications-of-plane-wave-destruction-filters-3dl6kh42zm.pdf

Applications of plane-wave destruction filters

Least-squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long-scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single-scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey-sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey-sinking (depth-oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning-based (surface-oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth-oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton-like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model-based data fitting.

http://www.trip.caam.rice.edu/reports/2007/geoprosp0207.pdf

Migration velocity analysis and waveform inversion

Marine controlled-source electromagnetic CSEM surveying has been in commercial use for predrill reservoir appraisalandhydrocarbonexplorationfor10 years.Althougha recent decrease has occurred in the number of surveys and publications associated with this technique, the method has become firmly established as an important geophysical tool intheoffshoreenvironment.Thisisaconsequenceoftwoimportant aspects associated with the physics of the method: First, it is sensitive to high electrical resistivity, which, although not an unambiguous indicator of hydrocarbons, is an important property of economically viable reservoirs. Second, although the method lacks the resolution of seismic wave propagation, it has a much better intrinsic resolution than potential-field methods such as gravity and magnetic surveying, which until now have been the primary nonseismicdatasetsusedinoffshoreexploration.Althoughbymany measures marine CSEM is still in its infancy, the reliability and noise floors of the instrument systems have improved significantly over the last decade, and interpretation methodology has progressed from simple anomaly detection to 3D anisotropic inversion of multicomponent data using some of theworld’sfastestsupercomputers.Researchdirectionspresently include tackling the airwave problem in shallow water by applying time-domain methodology, continuous profiling tools, and the use of CSEM for reservoir monitoring during production.

Ten years of marine CSEM for hydrocarbon exploration

Differential semblance optimization (DSO) is an approach to inversion of reflection seismograms which avoids the severe convergence difficulties associated with nonlinear least‐squares inversion. The method exploits both moveout and amplitude characteristics of reflections. We have implemented a version appropriate to plane‐wave (p‐tau) seismograms and layered constant‐density acoustic earth models. Theoretical and numerical analyses of this version of DSO indicate that stable and reasonably accurate estimates of both velocity trend and reflectivity can be derived. To test DSO further, we applied it to a marine data set from the Gulf of Mexico, where the method produced results which compare favorably to well‐log information. The method can be extended to incorporate laterally heterogeneous velocity models.

Velocity inversion by differential semblance optimization

A method for deriving reservoir lithology and fluid content for a target location (400) from pre-stack seismic reflection data (110). The method uses inversion (200) of pre-stack seismic reflection data for both the target location and a calibration location (110) having known subsurface lithology and fluid content to derive the subsurface lithology and fluid content at the target location (400). The inversion process (200) is preferably a viscoelastic inversion to account for the effects of friction on seismic wave propagation. The results of the inversion process are a set of subsurface elastic parameters for both the target and calibration locations. Relative magnitudes of these subsurface elastic parameters are compared (300), together with the known subsurface lithology and fluid content at the calibration location (146), to derive the subsurface lithology and fluid content at the target location (400).

Method for deriving reservoir lithology and fluid content from pre-stack inversion of seismic data

Formation anisotropy should be incorporated into the analysis of controlled-source electromagnetic CSEM data becausefailuretodosocanproduceseriousartifactsintheresulting resistivity images for certain data configurations of interest.Thisfindingisdemonstratedinmodelandcasestudies. Sensitivity to horizontal resistivity will be strongest in the broadside electric field data where detectors are offset from the tow line. Sensitivity to vertical resistivity is strongestforoverflightdatawherethetransmittingantennapasses directlyoverthedetectingantenna.Consequently,consistent treatment of overflight and broadside electric field measurements requires an anisotropic modeling assumption. To produce a consistent resistivity model for such data, we develop and use a 3D CSEM imaging algorithm that treats transverse anisotropy. The algorithm is based on nonlinear conjugate gradients and full wave-equation modeling. It exploits parallel computing systems to effectively treat 3D imaging problems and CSEM data volumes of industrial size. We use it to demonstrate the anisotropic imaging process on model and field data sets from the North Sea and offshore Brazil. We alsoverifythatisotropicimagingofoverflightdataaloneproduces an image generally consistent with vertical resistivity. However,superiordatafitsareobtainedwhenthesameoverflight data are analyzed assuming an anisotropic resistivity model.

/pdf/imaging-csem-data-in-the-presence-of-electrical-anisotropy-4zkbkqvx3o.pdf

Imaging CSEM data in the presence of electrical anisotropy

Three Dimensional Imaging of Marine CSEM Data

A method for removing the air wave effect from offshore frequency domain controlled source electromagnetic survey data. The region of interest is modeled with (111) and without (113) a top air layer. The electromagnetic field due to the source is computed at the survey receiver positions for each source position for both models. The difference between the fields computed (116) from the two models is the air wave effect, which is then subtracted (117) from the field data for the corresponding source-receiver geometry.

Carazzone James J

Papers

Velocity inversion by differential semblance optimization

Method for deriving reservoir lithology and fluid content from pre-stack inversion of seismic data

Imaging CSEM data in the presence of electrical anisotropy

Three Dimensional Imaging of Marine CSEM Data

Method for removing air wave effect from offshore frequency domain controlled-source electromagnetic data