Doug Oldenburg

On optimization techniques for solving nonlinear inverse problems

This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilizing a Gauss-Newton method in which the forward model constraints are implicitly incorporated. Variants of Newtons method which use second-derivative information are rarely employed because their perceived disadvantage in computational cost per step offsets their potential benefits of faster convergence. In this paper we show that, by formulating the inversion as a constrained or unconstrained optimization problem, and by employing sparse matrix techniques, we can carry out variants of sequential quadratic programming and the full Newton iteration with only a modest additional cost. By working with the differential equation explicitly we are able to relate the constrained and the unconstrained formulations and discuss the advantages of each. To make the comparisons meaningful we adopt the same global optimization strategy for all inversions. As an illustration, we focus upon a 1D electromagnetic (EM) example simulating a magnetotelluric survey. This problem is sufficiently rich that it illuminates most of the computational complexities that are prevalent in multi-source inverse problems and we therefore describe its solution process in detail. The numerical results illustrate that variants of Newtons method which utilize second-derivative information can produce a solution in fewer iterations and, in some cases where the data contain significant noise, requiring fewer floating point operations than Gauss-Newton techniques. Although further research is required, we believe that the variants proposed here will have a significant impact on developing practical solutions to large-scale 3D EM inverse problems.

An introduction to linear inverse theory

Linear inverse theory provides a formalism by which many questions fundamental to signal processing may be entertained. Questions pertaining to the resolving power of the data, the types of models that will reproduce the observations, the importance of additional data, the determination of the optimum sampling rate, and the effects of observational inaccuracies can all be meaningfully attacked through inverse theory. This paper presents an elementary overview of some of the approaches used by geophysicists to extract information about a model from the observations. The paper is not intended to be a review of the numerous ways in which a solution of the linear inverse problem has been sought, but rather it concentrates upon the methods and approaches of Backus and Gilbert. The three essential aspects of inverse theory — model construction, appraisal, and inference — are outlined and applied to a single numerical example. It is my hope that, the applications of the techniques presented in this paper to specific problems in signal analysis will be apparent.

Inversion of band-limited reflection seismograms: Theory and practice

This paper examines the problem of recovering the acoustic impedance from band-limited normal incidence reflection seismograms. Recognizing the inherent nonuniqueness in the inversion, we proceed by constructing an impedance model which satisfies the processed seismogram, has a minimum of structural variation, honors any point impedance constraints that are provided, and incorporates information from stacking velocities. The constrained inversion is carried out in a single operation using linear programming methods. The constructed impedance is consistent with available geological and geophysical information and therefore constitutes a well-constrained estimate of the true earth impedance. A basic assumption in our inversion is that each seismic trace is a band-limited representation of the true reflectivity function. When seismic data do not conform with this assumption, pre-inversion processing of the data is required; this involves a series of data checks and possible corrections. A complete processing sequence incorporating all steps of the practical inversion is presented and illustrated with field data examples.

Cost effectiveness of geophysical inversions in mineral exploration Applications at San Nicolas

Effective mineral exploration programs maximize the benefit of appropriate technologies in order to increase cost effectiveness by optimizing the use of drilling, reducing risks, and increasing the speed of discovery. In other words, correct use of available tools can allow exploration programs to find more ore, faster, with less expense.

Root‐mean‐square velocities and recovery of the acoustic impedance

The loss of low‐frequency information in reflection seismograms causes serious difficulties when attempting to generate a full‐band impedance profile. Information about the low‐frequency velocity structure is available from rms (stacking velocities). We show how rms velocities can be inverted with additional point velocity constraints (if they are available) to construct either smooth or blocky velocity structures. Backus‐Gilbert averages of the constructed velocity are then autoregressive solutions for recovering a full band reflectivity from band‐limited seismograms. Our final result is therefore a full‐band acoustic impedance which is consistent with the seismic data section, stacking velocities, and available point constraints.

Assessing the Quality of Electromagnetic Data for the Discrimination of UXO Using Figures of Merit

The need for assessing data quality in unexploded ordnance (UXO) remediation problems arises from two sources. In the planning stage it is essential that the data are acquired in sufficient numbers and with sufficient accuracy to answer the detection or discrimination problem of relevance. At the interpretation stage it is critical to objectively assess whether the data are of sufficient quality to warrant subsequent processing, inversion, and classification. Faced with this practical challenge of defining data quality we propose a Figure of Merit (FOM). FOM is a reliability indicator derived from quantities that affect the quality of data, such as anomaly coverage, line spacing, station spacing, instrument noise, survey location errors, etc. The FOM can also include informative features of the inversion, such as the variance of key model parameters, and thus it depends on the inverse model to be applied. Anomalies associated with higher values of FOM should have increased reliability in classification. A...

Robust Inversion of Time-domain Electromagnetic Data: Application to Unexploded Ordnance Discrimination

We invert time-domain electromagnetic data for the purpose of discriminating between buried unexploded ordnance (UXO) and non-hazardous metallic clutter. The observed secondary magnetic field radiated by a conductor is forward modeled as a linear combination of decaying, orthogonal dipoles. We show via a perturbation analysis that errors in the measurement of sensor position propagate to non-normal errors on the observed data. A least squares (L2) inversion assumes normal errors on the data, so non-normal errors have the potential to bias dipole parameter estimates. In contrast, robust norms are designed to downweight the effect of outlying (noisy) data and so can provide useful parameter estimates when there is a non-normal component to the noise. When positional errors are modeled as independent Gaussian perturbations, we find that weighted least squares and robust inversions have comparable performance. Both inversion techniques estimate data uncertainties from observed data, and this has the effect of making the least squares inversion robust to outliers. However, when simulated errors are correlated, robust inversion with a bisquare norm provides a marked improvement over L2 inversion. Application of robust inversion to real data sets from Camp Sibert, Alabama produced an incremental improvement to the initial L2 inversion, identifying outlying ordnance items and improving discrimination performance.

Controlled Source Electromagnetic (CSEM) Technique For Detection And Delineation of Hydrocarbon Reservoirs: an Evaluation

The marine Controlled Source Electromagnetic technique was developed almost three decades ago to study the conductivity structure beneath the seafloor. An exhaustive and still valid treatment of the CSEM techniques can be found in Chave et al. (1991). One advantage of sub-sea measurements is that the highly conductive sea (approximately 3.2 S/m) acts as a low pass filter for fluctuating EM fields generated above it either in the ionosphere or magnetosphere. At frequencies as low as 1 Hz, a few hundred meters of water will practically completely eliminate the effect of above-water EM sources including the man-made ones or those due to cultural noises. As a result weak electromagnetic fields that propagate in the underlying sediments from a sea-bottom artificial source are measurable at large transmitter-receiver separations of the order of kilometers.

Incorporating Uncertainty in Unexploded Ordnance Discrimination

We examine representations of feature vector uncertainty in the context of unexploded ordnance (UXO) discrimination with electromagnetic data. We compare a local uncertainty estimate derived from the curvature of the misfit function with global estimates of the model posterior probability density (PPD) obtained with Markov chain sampling. For well-posed experiments (i.e., with high SNR and adequate spatial coverage), the two methods of uncertainty appraisal agree. However, when the inverse problem is ill posed, we find out that the PPD can be multimodal. To incorporate these uncertainties in discrimination, we first develop an extension of discriminant analysis which integrates over the posterior distribution of the model. When dealing with multimodal PPDs, we show that an effective solution is to input all modes of the PPD-corresponding to all models at local minima of the misfit-into discrimination and, then, to classify on the basis of the model which is most likely a UXO.

New methods for constructing flattest and smoothest models

Standard techniques for constructing flattest and smoothest models require the specification of additional information about the model. This extra information consists of an independent value of the model (in the flattest model case) or values for the model and its first derivative (in the smoothest model solution), and is typically prescribed at an endpoint of the interval of model definition. This conventional procedure is generalized in two important ways: (i) the extra information is supplied at arbitrary points within the interval, and (ii) optimum values for these parameters are calculated directly from the observed data. The latter method yields the absolutely flattest (or smoothest) model satisfying the data, and is particularly useful if an independent value of the model (and/or its slope) is difficult to estimate accurately. Both techniques are illustrated by the problem of constructing a smooth refractor elevation profile from a set of point estimates of the refractor depth.