Inverse Theory

• Type of physical science: Mathematical methods
• Field of study: Signal processing

To determine the interior geometric and material properties of an object, from analysis of a limited number of surface or interior measurements, is termed a solution to the inverse problem. Inverse problems not only arise in physical sciences employing remote sensing (such as seismology, acoustics, and electromagnetic sounding) but also are frequently the basis of much precision signal and image processing in nuclear physics and astrophysics, as well as engineering.

Overview

Broadly considered, inverse theory is a loosely organized set of mathematical techniques and concepts for reducing data to extract specific information about particular properties of a medium, object, or system, subject to only limited direct- and remote-sensed measurements. The specific media properties are often called model parameters, in that it is assumed that one or more models can be found fitting model parameters to actually measured data. Typically, a model stands for a physical-mathematical theory of a system whose unknown (interior, unobservable) processes are under examination and measurement. Depending on where the border is drawn between a system and its surrounding environment, it is frequently possible to characterize most key system parameters by the ratio of its output to input, or transfer function. The system of unknown process(es) may be an amplifier box comprising unknown internal electronic components as probed by voltage measurements on its external terminal. A model is basically something that simulates or reproduces observed measurable system behaviors. Although models can, and frequently do, begin as conceptual and empirical relations, in most inverse theory the tendency is to formalize the latter into a system of mathematical equations, whose unknown coefficients can be solved for.

In most cases, there is also at least some a priori knowledge, from physical insight or from previous or analogous studies, into the process(es) under examination, such that the general class of models and the range of expected data behaviors can be delimited. From this perspective, inverse theory can be seen as comprising several stages or components. The first stage is that of selecting the best appropriate model(s) for the unknown process(es), and stage two is that of estimating the values of the parameters defining the model, so that simulations using the model yield results agreeing sufficiently well with measured data. Every inverse problem thus necessarily contains what is called a forward problem. Forward modeling is the process of predicting the results of measurements on the basis of some assumed model and a set of model parameters. Inverse theory or inversion, by contrast, begins with measured data and only a general model and general model parameter guess, and seeks to determine the underlying model parameters.

Some of the earliest and conceptually most straightforward examples of practical inversion arise in analogue or digital signal filtering via deconvolution, as widely used in astronomy, satellite imaging, seismology, and radar engineering. The goal of deconvolution is to reverse or remove the detrimental effects suffered by a signal when it has passed through a distorting medium or noisy channel, and/or caused by limitations of the measuring sensors themselves. When the principle of linearity of the total sensor-signal-system can be assumed, the process of signal or image distortion can be represented by the convolution product g = hf, of the true signal f with some channel, system, or sensor response h. By the convolution product for Fourier transforms, if g = hf, G = HF, where G, H, and F are the Fourier transforms of g, h, and f. Thus, in principle, the true signal can be restored by computing F = G/H, and then inverse Fourier transform (FT). This process was first set forth by Norbert Wiener in 1942 and designated "inverse filtering."

The relationships between deconvolution and inverse filtering, and more general inverse problems, can be seen when considering that inverse filtering problems generally seek to remove the effects of the system interacting with the desired signal. By contrast, inversion (while frequently incorporating inverse filtering as an initial data-processing stage) seeks explicitly to determine the true model and parameters for the system itself, where the input signal is known.

Perhaps the most widely used generalization of inverse filtering is the so-called generalized linear inverse (GLI) method of Gilbert Backus and Freeman Gilbert (1966), related to the independent method of generalized eigenvector analysis of Cornelius Lanczos (1961). In GLI, inversion is a general method to estimate the full information content of the system measured by a given data set, which former is said to be defined by the system kernel, source, or Green's function. This goal is achieved by recasting insofar as possible all stages of inversion into the framework of linear algebra, because linear algebra has well-established conditions for matrices, arising from solving systems of equations for model-fit parameters, to have a unique or stable solution.

To linearize the general inverse problem (of finding the system kernel) in GLI means to seek a sufficiently good linear approximation between small changes in model values and assumed small effects on measurable properties calculated by the model. The matrix to solve for the kernel which algebraically inverts the linear system of equations is called the "general linear inverse" and is related to, but distinct from, ordinary algebraic matrix inversion. A related matrix arising from the solution process is the data-resolution matrix, which quantitatively describes how well model predictions match the data. Numerical values in the rows of the data resolution matrix indicate how well adjacent data values can be independently predicted or resolved. Yet another associated matrix, the covariance matrix, characterizes the degree of error growth and noise amplification in a given data set. Many computationally intensive forms of GLI are named after the particular type of matrix inversion algorithms used (steepest-descent, gradient, or singular-value-decomposition methods). The conditions under which these matrix systems can be accurately solved significantly influence the efficiency and practical utility of inversion results.

An inverse problem, for example, is said to be "well posed" when a unique solution exists that is stable to changes in the data values. Numerical problems in computation (such as division by very small numbers) frequently arise in inverse computations when the solution is very sensitive to small perturbations in data or noise values (that is, when the solution is not very linear).

Another important mathematical result is the non-uniqueness of most solutions to inverse problems. Non-uniqueness means that there is more than one (and frequently many) combinations of model parameters that can yield theoretical results sufficiently close to observed data. Considered physically, an underground acoustics impedance value, defined in seismology and acoustics as the product of density ρ and velocity v (for example, measuring 6 from seismic or sonar records), can be obtained from a geological layer having a density of 2 and a velocity of 3 units, or by a density of 3 and a velocity of 2. Prior information about the geometry and material properties of the specific problem at hand can often be used to eliminate nonphysical solutions to inverse problems (for example, almost never are rocks found having high densities but low velocities).

Non-uniqueness in data version is generally made worse by underdetermination (inadequate quantity) or insufficiency (inadequate quality) of measured data. In most cases of generalized linear inversion, merely adding unexamined data points typically weakens rather than improves the quality of parameter fit and inverse problem solution. Improvement frequently requires collecting the data again under new conditions. For example, large variances in measured data can be reduced by increasing the effective sample size by widening the total spatial area or volume measured. In so doing, howeverbecause the sample space is now larger for the same number of pointssmaller features within this area will no longer be detected with the same resolution. Thus, all inverse filtering and inversion is a compromise, trading off some accuracy and precision in the resulting signal and parameter estimates for gains in computational speed and stability.

The selection and implementation of inversion methods depend very strongly on type, quantity, and quality of measured data. Frequently inversion of data from a very large and complex medium or system is facilitated by decomposing the total problem (having a large number of layers) into several simpler (fewer-layered) submodels to be independently solved.

This aspect of parameterization requires careful division of the total spatial structure in which data values are measured into a uniform layer or grid, typically defined by the number, shape, and depth position of discontinuities or changes in the key variables (such as temperature, acoustic impedance, and electrical resistivity). Usually uniform layer/grid structures are assumed as part of the basic underlying model, since true layer positions are generally unknown. Another simplification in practical inversion problems is assuming certain approximate (for example, smooth) functions for variations of key parameters in and between layer or grid points.

Applications

Over the decades since about 1950, needs for characterizing smaller and more complex phenomena in many disciplines have led to increasingly sophisticated procedures of data acquisition, processing, and interpretation. These latter include new methods for recognizing and filtering out unwanted data and noise, to recapture the original form of the input measurement or sounding signal, or the system or transfer function with which the signal interacts. The precise kind of inversion method(s) employed generally depends on the type of input data measuring the system/medium; the number and spatial sampling of data points; the degree and type of noise in the data and/or system; the number and kind of a priori physical constraints; and the accuracy, precision, and time requirements of a given application.

Many of the earliest inverse problem solutions were determined through operational calculus, where exact inverse transforms of the Fourier, Bessel, Hankel, Laplace, Hilbert, and Radon types are often available for a given equation. The geophysicists I. Stefanescu and P. Schlichter in 1930 and 1932 introduced a kernel function method into geoelectrical subsurface prospecting for groundwater and mineral deposits. Here, the electrical potential measured by sensor arrays at the ground's surface over an assumed horizontally layered subsurface used Bessel's integral transform of the true source potential of the buried deposits. The radar-sonar specialist C. Pekeris in 1940 developed a general relation to estimate the system source function starting from knowledge of the first subsurface layer successively down to the deepest (so-called strip-off inversion). When such integral transforms are written in terms of Fourier or Hankel transform representations, they can be rapidly computed by the fast Fourier transforms (FFTs).

The seismologists Backus and Gilbert originally introduced the GLI method to compute the mass density and compressional and shear wave velocities in the interior of a spherically stratified Earth from seismic measurements made on the Earth's surface. Since then, GLI has been widely applied to other linearizable inverse problems, involving electrical, gravimetric, magnetic, thermal, seismic, ultrasonic, X-ray, neutrino, and many other field measurements in Earth, atmospheric, space, physical, and medical sciences. Since the 1970s in particular, in addition to statistical inversion techniques such as GLI, physically more exact inversion methods such as those of M. Born and I. Rytov, and other integral transform varieties, such as the Gelfand-Levitan, Marchenko, and Gopinath-Sondhi-Sarwar inversions, have found increasing applications as adjuncts to imaging, microscopy, and tomography, in such diverse areas as audio and optical systems engineering, radar and spread-spectrum communications, acoustic holography, nondestructive evaluation, earthquake and exploration seismology, and meso-scale ocean current circulation.

Context

There are several general viewpoints from which inverse theory has been considered.

One is that of engineering uses of probability, statistics, and optimal systems theory. Another perspective is that of the particular subdisciplines of physicssuch as quantum mechanics, seismology, and acousticswhich gave rise to certain inverse methods. A third view of inversion is that of pure and applied mathematics.

Historically, inverse theory was developed more or less independently in several disciplines, seen at the time as part of other methods of physics or mathematics. Gustav Herglotz and Emil Wiechert in seismology, circa 1907, derived the first working equations for solving the inverse problem of estimating the compressional-wave velocity function versus depth in the Earth's crust and mantle, from analysis of seismic ray travel-time curves. Max Born, Emil Wolf, and Pieter van Cittert, among others in physical optics, sought various techniques of optical image restoration and enhancement, the (Fourier) inverse problem of which is that of determining the true (near-field) source characteristics from remote (far-field) diffracted measurements. Analogous efforts were developed during and following World War II in radar and underwater acoustics, where it is necessary to determine the presence, location, characteristics, and motions of passively reradiating and actively emitting radiation sources, from what are generally very sparse and error-mixed measurements using limited sensor arrays. Between 1942 and 1949, Wiener first formulated and applied what later became the principles of inverse filtering, which since the early 1960s has been widely applied in seismic prospecting and forensic seismic monitoring of underground nuclear tests.

From the pure mathematics view, Edwin Lighthill and others developed the "operational calculus" of Laplace and Fourier transforms, a standard tool in electrical and mechanical engineering systems analysis. The key to all such mathematically exact inverse methods is being able to find, and then compute, an inverse integral transformation. Another mathematical interest in inversion theory is the development and application of general computational strategies, such as quasi-linearization, dynamic programming, and invariant embedding, to simplify complicated (non)linear partial differential equations from boundary value problems to easier linear initial value problems. Mathematical inverse theory has the further hope of finding "limiting theorems" as a natural way to classify solutions of inverse problems, to enable an optimal choice and parameterization strategy in any given case. The work of applied mathematicians such as Robert Bracewell in the early 1950s formulated the basis of the mathematical principles of aperture synthesis and image formation using exact and approximate inverse transforms.

The viewpoint of Claude Shannon's information theory complements and extends that of traditional probability and statistics, as well as Wiener's approach. Here the data values to be estimated are considered as "messages" (signals) sent from an information source (system) received by sensors, where the components of transmitter, channel, system, and receiver can all be quantified. As noted, another view of inversion as an identification problem, from probability theory, assumes one or more probability distribution functions for a process model and then employs special statistical estimation methods, such as Bayesian or maximum likelihood methods, to confirm the specific model class and to specify it fully. Almost all the above approaches to inverse theory were notably facilitated by the development of digital computers and the FFT algorithm in the mid-1960s.

Further work is needed to determine which of the many approximate and exact inverse techniques offers the best accuracy and resolution, alone and in combination with other inversion methods. It is increasingly important to reduce and quantify all possible errors and uncertainties affecting estimates of system/media properties provided by inverse methods. Expert systems and artificial intelligence techniques for pattern recognition have been recognized for their important potential contributions to the initial input and final interpretation stages of the total inverse problem. The wider use of generalized linear and other inversion techniques by chemical, biological, economic, and social sciences is foreseen. Many of the requisite computational advances will be closely tied to the development and availability of faster and cheaper computing hardware and entire programs for deconvolution, generalized linear and other inversion, and wave propagation modeling on IC chips.

Principal terms

APPROXIMATE (INVERSE) SOLUTIONS: solutions providing only limited precision and accuracy, as a result of simplifying assumptions concerning the physical process or mathematical technique representing the process

CONVERGENCE: the condition, in a progressive or iterative process, when successive calculated values approach closer to observed data values; mathematically, the condition whereby computed values approach definite limits as the number of computational terms used increases

CROSS-CORRELATION: a mathematical measure of the similarity, or correspondence, between matching members of two or more data series

DECONVOLUTION: a process to restore a data series or signal's true form, by removing the filtering effects imposed by the medium or channel and sensors through which the data are recorded

EXACT (INVERSE) SOLUTIONS: an integral transform operator, which exactly yields the unknown system function f, in the sense that the inverse Fourier transform F^-1 applied to the Fourier transform of a function, F[f(x)], yields the function f(x); that is, F^-1 {F[f(x)]} = f(x)

NOISE: any unwanted signal; white noise is random energy containing any frequencies in equal proportion, and red (blue) time-series noise spectra are those having most of their statistical variance concentrated in the long (short) wavelength band

NONLINEARITY: a property of a measuring device, filter, or medium, whereby the output result does not increase proportionally to the input; in many inversion applications, nonlinearity designates parameters of the channel or system to be estimated whose measured values depend on the amplitude of the radiations used in measurement

PARAMETER ESTIMATION: any of a number of techniques that seek to minimize certain predefined error criteria to fit an assumed process model to observed data

PARAMETERIZATION: the representation, in a mathematical model, of generally complicated physical effects in terms of simpler equivalent conditions

UNIQUENESS: conditions under which there is only one definite solution to a given inverse problema condition only rarely encountered in practice; the inverse problem is inherently non-unique since, in principle, there are a very large number of different models, and model parameter values, that can give rise to results matching the observed data values

Bibliography

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