Raghu K. Chunduru

Hybrid optimization methods for geophysical inversion

Local and global optimization algorithms are used commonly in geophysical data inversion. Each type of algorithm has unique advantages and disadvantages. Here we propose several methods of combining the two algorithms such that we can overcome their drawbacks and make use of the salient features of the two methods. In particular, we combined a local conjugate gradient (CG) method with a global very fast simulated annealing (VFSA) approach to solve problems of geophysical interests. We conducted a systematic study to find an efficient strategy to combine CG and VFSA optimization schemes and recommend a couple of ways for future implementations. Seven different hybrid algorithms were first tested on a set of field 1-D Schlumberger resistivity sounding data and their performances were compared with stand‐alone genetic algorithm (GA), simulated annealing, and local search algorithms. Almost all of the proposed hybrid algorithms were found to be computationally more efficient than the conventional global optim...

2-D resistivity inversion using spline parameterization and simulated annealing

Successful inversion of geophysical data depends on prior information, proper choice of inversion scheme, and on effective parameterization of the model space such that the model representation is appropriate and efficient. Inversion of resistivity data has long been recognized as a nonlinear or quasi-linear problem. Traditionally, 2-D resistivity inversion has been performed by trial and error methods and with linear and iterative linear methods. The linear and iterative linear methods are limited because of the requirement of good prior knowledge of the subsurface. Unlike linear and iterative linear methods, most nonlinear inversion schemes do not depend strongly on the starting solution, but prior information helps to reduce the computational cost and to obtain geologically meaningful results. In the present study, we have applied a nonlinear optimization scheme called very fast simulated annealing (VFSA) in the inversion of 2-D dipole-dipole resistivity data to image the subsurface. Unlike Metropolis simulated annealing (SA) in which each new model is drawn from a uniform distribution, VFSA draws a model from a Cauchy-like distribution, which is also a function of a control parameter called temperature. The advantage of using such a scheme is that at high temperatures, the algorithm allows for searches far beyond the current position, while at low temperatures, it looks for improvement in the close vicinity of the current model. We have used the mean square error between the synthetics and original data as the error function to be minimized. The synthetic response for 2-D models was obtained by finite-difference modeling, and cubic splines were used to parameterize the model space to get smooth images of the subsurface and to reduce computational cost. VFSA was used to estimate the conductivity at each spline node location. The inversion was applied to various synthetic data to study the influence of the starting solution and the location of the spline nodes. Finally, we applied it to real data collected over a disseminated sulfide zone at Safford, Arizona, and compared the results with those obtained from a linearized inversion and from a model based on geologic and well-log data. The VFSA results are in good agreement with the previously published results.

Simultaneous inversion of formation shear‐wave anisotropy parameters from cross‐dipole acoustic‐array waveform data

This study presents an effective technique for obtaining formation azimuthal shear‐wave anisotropy parameters from four‐component dipole acoustic array waveform data. The proposed technique utilizes the splitting of fast and slow principal flexural waves in an anisotropic formation. First, the principal waves are computed from the four‐component data using the dipole source orientation with respect to the fast shear‐wave polarization azimuth. Then, the fast and slow principal waves are compared for all possible receiver combinations in the receiver array to suppress noise effects. This constructs an objective function to invert the waveform data for anisotropy estimates. Finally, the anisotropy and the fast shear azimuth are simultaneously determined by finding the global minimum of the objective function. The waveform inversion procedure provides a reliable and robust method for obtaining formation anisotropy from four‐component dipole acoustic logging. Field data examples are used to demonstrate the app...

A combined genetic and linear inversion algorithm for seismic waveform inversion

We combine a genetic algorithm (GA) with a linearized inversion (LI) scheme to develop a new approach to seismic waveform inversion. By incorporating the LI method into GA we intend (i) to overcome the limitations of the knowledge of a good starting model in LI and (ii) to reduce the computational cost of GA. The new method takes advantage of the convergence properties and local search approach of the linear method while the global search is carried out using GA. The two methods working together improve the directivity of the model ensemble increasing the fitness and accelerating the convergence to near the global optimum. To illustrate the procedure, we derive estimates of vp, and density for a 1-D elastic earth structure by modeling plane wave decomposed seismic data.

Determining bed boundaries from inversion of EM logging data using general measures of model structure and data misfit

In this paper we propose a new method to locate bed boundaries by carrying out a 1-D nonlinear inversion of electromagnetic (EM) logging data. We first solve for 1-D resistivity structure in which the earth is modeled using layers of constant thickness. This thickness is determined based upon the tool resolution and the desired resolution from the user. We use general measures of data misfit and model structure in the inversion to construct piecewise‐constant models through the iteratively reweighted least‐squares (IRLS) procedure, and we minimize a generic global model objective function subject to data constraints. The general measure includes the traditional lp norm as a special case. The recovered piecewise‐constant models simulate traditional models comprised of a few uniform layers and hence permit easy determination of bed boundaries. The generic model objective function allows the incorporation of prior geological information and provides measures of the closeness to the reference model and the am...

Geophysical applications of global optimization methods

Model based inversion methods as applied to geophysical problems involve finding the global minimum of an objective function (given by a suitably chosen norm) that corresponds to an earth model which best explains the observed geophysical data. It has long been realized that geophysical inverse problems are nonlinear and the objective functions are multimodal. Traditionally such problems have been solved using iterative linear methods which require a good starting solution.

Reconstruction of resistivity structure from 2-D inversions of resistivity logging data using equality and inequality constraints

The inversion technique is a powerful tool in estimating residual and movable hydrocarbon and has been widely used in the oil industry with great su ccess. However, geophysical inverse problems are ill-posed b ecause of incomplete measurements in the survey, noise in the data, various approximations in the physical models, and numerical errors in the computations. The solutions from these ill-posed problems are nonunique and unstable. If the problem is nonlinear, then the inversion is further complicated by local minima. To overcome these difficulties, it is necessary to regularize the inverse problems and apply appropriate constraints on the solutions. In this paper, we develop a model-norm-based inverse algorithm to estimate 2-D resistivity distribution around the borehole using resistivity logging data. This algorithm features a model objective function that contains a priori information about the geological structure and an outside penalty function that augments the unconstrained inversion with equality and inequality constraints on the solutions. This penalty function is introduced through a weighting parameter that is increased from a starting value to a theoretically infinite value. The linear system of equations to be solved at each iteration in the inversion is preconditioned to avoid unstable and slow convergence caused by the large weighting on the penalty term. In this paper, we show the usefulness of two particular types of inequality constraints, the range and the relative constraints. The range constraint defines the feasible region for the model parameters, and the relative constraint dictates the relative strength among the model parameters. We apply our algorithm to both synthetic and field data.

Detection of Layer Boundaries From Array Induction Tool Responses Using Neural Networks

Summary Delineation of bed boundaries is one of the important aspects of petrophysical interpretation. Proper location of bed boundaries plays a key role in log data inversion, especially in resistivity inversion, where the subsurface formation parameters are commonly parameterized into layers of varying thickness and resistivity. Traditionally, the location of bed boundaries is chosen based on the inflection points, maximum change in slope, etc. These algorithms work well for focussed responses and reasonably well for unfocussed responses, but the algorithms fail in presence of noise and in thin-bed regions. In order to overcome these difficulties, we propose an algorithm using Artificial Neural Networks (ANN) for detection of bed boundaries. We demonstrate the applicability of the algorithm on array induction data for a synthetic Oklahoma formation model and a field data from the Gulf of Mexico. The results obtained from the proposed scheme are compared with existing bed boundary

Petrophysical Inversion of Resistivity Logging Data

Inversion is a powerful tool for interpreting resistivity-logging data in complex situations, such as deep invasion, high conductive shoulders, thin beds, anisotropic formations, and highly deviated and horizontal wells. Inversion also provides a general means to interpret data from earlier, unfocused resistivity tools as well as the new generation of array-type instruments, allowing for accurate delineation of the resistivity structure. Although, in most cases, inversion produces solutions that are consistent with petrophysics and geology, there are occasions where consistency is not achieved. In order to address these inconsistencies, we have developed a petrophysical inversion algorithm that involves gamma ray, neutron, density, acoustic, and resistivity data in a single, unified interpretation process. The steps in the interpretation process consist of estimating bed boundary positions from all available measurements, including gamma ray, neutron, density, and resistivity data, using a weighted inflection point method. Bed boundary positions are then adjusted using the tool response functions associated with the various instruments involved and integrated to produce a consistent set of bed boundaries, which best represents the subsurface geology and lithology. Next, upper and lower bounds for formation resistivity and flushed zone resistivity are estimated using an appropriate water saturation equation. Input for resistivity bound estimation includes shale volume, porosity, as well as the possible range of variation for water saturation, formation water resistivity, and shale resistivity. The resistivity bounds are incorporated into the inversion algorithm via an outside penalty function added to the original objective function of the optimization. The proposed inversion algorithm utilizes a generic objective function including a reference model and data weighting based on uncertainties. Furthermore, a first-order spatial-finite difference operator has been built into the objective function to eliminate unrealistic oscillations in the inversion results. A field case example shows that the proposed inversion process can effectively handle systematic noise in the data caused by borehole washouts and inappropriate bed boundary positioning, and generates petrophysically meaningful inversion results.

Integrated Interpretation of 3D Seismic and Wireline Data to Delineate Thin Oil- Producing Sands in San Jorge Basin, Argentina

Hydrocarbon-bearing sand units in San Jorge Basin, Argentina exhibit thicknesses between 0.5 and 15m, with a mean value of 1.5m. These sand units are associated with ephemereal and laterally heterogeneous fluvial architectures embedded within much thicker shale units of lacustrine and flood-plain origin. Tuffaceous laminations originated from pulses of volcanic activity are also present in the sedimentary column. In view of the wide range of spatial variability and size of porous sands, a typical well is planned to intersect vertically as many sand units as possible. The best producing sands have average porosities of 10% and permeabilities of a few millidarcies. There are several problems faced by the petrophysicist in assessing whether or not a given sand unit should be perforated, namely, (a) discrimination between oil- and waterbearing sands is not trivial because of very low salinity water, (b) there exist substantial vertical variations of effective porosity within an individual sand because of shale laminations, and (c) it is often impossible to assess lateral extent away from the well. We have successfully addressed most of these difficulties using an interpretation procedure centered about the full 2D inversion of wireline array induction data. Full 2D inversion of array induction data is necessary to accurately estimate shallow and deep resistivities, as well as invasion radii in light of significant shoulder bed effects. This procedure has been complemented with the use of borehole NMR data to provide estimates of effective porosity within individual sand units. Finally, we have made use of geostatistical inversion of 3D post-stack seismic data to estimate the lateral extent of hydrocarbon-bearing sands laterally away from the well. We present several application examples that yield results consistent with borehole testing and production data.