Iterative Inversion Technique Research Articles

An iterative system inversion technique

An inversion iterative algorithm for discrete-time non-linear systems is presented. The algorithm is tuned depending on system characteristics. Under the assumption that the system inverse exists, the iterative algorithm constructs the system inverse by attempting to achieve on-line the perfect tracking. The inversion process is based on the observation of the input–output pairs. The resulting control scheme requires a very low a priori knowledge of the system dynamics. Moreover, low-engineering efforts are needed to apply the control scheme to a particular control problem or to modify it in order to accommodate changes in the physical system. Copyright © 2001 John Wiley & Sons, Ltd.

Acoustic bubble counting technique using sound speed extracted from sound attenuation

Sound attenuation has been solely used to estimate bubble size distributions of bubbly water in the conventional acoustic bubble sizing methods. These conventional methods are useful for the void fraction around 10/sup -6/ or lower. However, the change of compressibility in the bubbly water also should be considered in bubble sizing for the void fraction around 10/sup -5/ or higher. Recently the sound speed as well as sound attenuation was considered for acoustic bubble size estimation in bubbly water. In this paper, the sound speed estimated from sound attenuation in bubbly water by an iterative method is used for a bubble counting. This new iterative inverse bubble sizing technique is numerically tested for bubble distributions of single-size Gaussian, and power-law functions. The numerical simulation results are in agreement with the given bubble distributions even for the high void fractions of 10/sup -4/-10/sup -3/. It suggests that the iterative inverse technique can be a very powerful tool for practical use in acoustic bubble counting in the ocean.

Non-linear inversion for the hydrostatic structure of the solar interior

We present the results of a non-linear inverse analysis for the hydrostatic, spherically symmetric component of the solar internal structure using the observed p-mode frequencies. The iterative non-linear inversion technique used here is based on the succesive Born approximation description of solar p-modes developed by Roxburgh & Vorontsov. This description can give a high resolution of regions of rapid variation of seismic parameters with depth (e.g., the base of the convection zone), and accounts accurately for the strong influence of gravity perturbations on low-degree modes which penetrate deep into the solar core. The inversion procedure is non-linear; the eigenfrequency equation obtained from the Born approximation is solved by iteration. The particular target of our inverse analysis is to achieve the highest possible resolution of the region near the base of the solar convection zone, searching for possible signatures of penetrative convection, element diffusion and/or strong magnetic fields. The results of the global inversion obtained with solar p-mode frequencies provided by the recent high-quality observational data (GONG, SOI/MDI, GOLF) are presented and discussed.

Satellite retrieval of the absorption coefficient of phytoplankton phycoerythrin pigment: theory and feasibility status

Oceanic radiance model inversion methods are used to develop a comprehensive algorithm for retrieval of the absorption coefficients of phycourobilin (PUB) pigment, type I phycoerythrobilin (PEB) pigment rich in PUB, and type II PEB deficient in PUB pigment (together with the usual "big three" inherent optical properties: the total backscattering coefficient and the absorption coefficients of chromophoric dissolved organic matter (CDOM)-detritus and phytoplankton). This fully modeled inversion algorithm is then simplified to yield a hybrid modeled-unmodeled inversion algorithm in which the phycoerythrin (PE) absorption coefficient is retrieved as unmodeled 488-nm absorption (which exceeds the modeled phytoplankton and the CDOM-detritus absorption coefficients). Each algorithm was applied to water-leaving radiances, but only hybrid modeled-unmodeled inversions yielded viable retrievals of the PE absorption coefficient. Validation of the PE absorption coefficient retrieval was achieved by relative comparison with airborne laser-induced PEB fluorescence. The modeled-unmodeled retrieval of four inherent optical properties by direct matrix inversion is rapid and well conditioned, but the accuracy is strongly limited by the accuracy of the three principal inherent optical property models across all four spectral bands. Several research areas are identified to enhance the radiance-model-based retrievals: (a) improved PEB and PUB absorption coefficient models, (b) PE spectral shifts induced by PUB chromophore substitution at chromophore binding sites, (c) specific absorption-sensitive phytoplankton absorption modeling, (d) total constituent backscattering modeling, (e) unmodeled carotinoid and phycocyanin absorption that are not now accounted for in the chlorophyll-dominated phytoplankton absorption coefficient model, and (f) iterative inversion techniques to solve for six constituents with only five radiances. Although considerable progress has been made toward the satellite recovery of PE absorption, the maturity of the retrieval is presently insufficient for routine global application. Instead it must currently be used on a regional basis where localized ship and aircraft validation can be made available. The algorithm was developed for the MODIS (Moderate-Resolution Imaging Spectroradiometer) sensor but is applicable to any sensor having comparable band locations.

Study of interdiffusion in thin Fe film deposited on Si(111) by x-ray reflectivity and secondary ion mass spectrometry

We report on the effect of annealing on a thin Fe film deposited on a Si(111) substrate, using x-ray reflectivity and secondary ion mass spectrometry (SIMS) techniques. Using Fourier transform of the x-ray reflectivity data, we have estimated the layer thickness of the film. From the estimated thickness and critical value of the scattering vector qc obtained from the reflectivity data, an initial guess model of the electron density profile of the film is made. Using an iterative inversion technique, based on the Born approximation, with the obtained initial guess model, we have extracted the actual electron density profile of the film as a function of depth from the specular x-ray reflectivity data. On annealing, we observe interdiffusion of Fe and Si resulting in an increase in the thickness of the film. We have also carried out a SIMS measurement on the annealed sample to support the result of the annealing effect observed from the analysis of x-ray reflectivity data. The SIMS analysis indicates that the top of the film is rich in Si which has diffused from the substrate to the surface of the Fe film on annealing.

A program to create permeability fields that honor single-phase flow rate and pressure data

Accurate prediction (or simulation) of reservoir performance or contaminant transport in groundwater requires a realistic geological model representative of the reservoir/aquifer heterogeneity. Geostatistics provides tools for constructing such complex geological models constrained by different types of available (hard and soft) data and providing an assessment of related uncertainty. Permeability and flow data are nonlinearly related through the flow equations. Derivation of permeability models that honor flow response data is typically an inverse problem. This paper presents a FORTRAN program for generating permeability fields conditional to multiple-well single-phase flow rate and pressure data through an iterative inverse technique, called the sequential self-calibration (SSC) method. The SSC method is geostatistically-based, that is, it generates multiple equiprobable realizations that honor the input geostatistics of permeability and match pressure data for the given flow rate, under the given boundary conditions. The unique aspects of SSC are: (1) the master point concept that reduces the amount of computation, (2) a propagation mechanism based on kriging that accounts for spatial correlations of perturbations and (3) a fast method for computing all sensitivity coefficients within a single flow simulation run. Results from running the SSC code using an example data set are presented.

Lateral variations of Pn wave velocity in northwestern Italy

The Pn arrival times recorded from seismic networks operating throughout northwestern Italy and surrounding regions were inverted to map the structural variations of the uppermost mantle over the area and to estimate the crustal static delays at each station. By means of careful data selection a quality data set was obtained removing statistical outliers, poorly recorded events, and scantily sampled stations. Moreover, synthetic data were used to evaluate the resolution power of the available data set and the adopted iterative inversion technique. The agreement between synthetic and calculated models is more satisfactory where the path coverage of the rays is quite complete. A low‐velocity zone was found beneath the western and northwestern side of the Alps; it may be related to an increase in depth of the Moho, as supported by other geophysical data. The high Pn velocities found in the eastern side of the western Alps indicate the presence of high velocity and density, lower crust rocks of the so‐called “Ivrea Body.” The high Pn velocity underlying the Ligurian Sea could be related to a high‐velocity structure existing in the upper mantle. The shape of the velocity anomalies matches not only the major tectonic features, but also the Bouguer gravity anomalies of the area.

Inversion technique for recovering the elastic constants of transversely isotropic materials

A least‐squares iterative inversion technique has been developed for the determination of the elastic parameter δ* of any transversely isotropic modeling material in the laboratory. For most applications in petroleum geophysics, the elastic parameter δ* is very important and is the crucial anisotropic parameter for near‐vertical P‐wave propagation. Despite the potential importance of δ* in seismic exploration and for resolution in an anisotropic medium, the conventional procedures adopted in estimating its value unfortunately are faced with many ambiguities and the reliability of its measurement is doubtful prior to the development of this technique. The anisotropic inverse modeling technique finds the best fitting solution. To optimize the accuracy of the results presented in this paper, analytical rather than numerical differentiations were implemented and the modeling procedures allow for controlled iterative adjustments in resolving the parameter δ*. Inversion of the first‐arrival traveltimes obtained for vertical P‐waves through an anisotropic material known as phenolite yield estimates of the elastic parameter δ* as well as the vertical P‐wave velocity [Formula: see text] of the material. Accurate picking of the first‐arrival traveltimes is essential since δ* is found to be very sensitive to small differences between vertical and oblique traveltime picks. The inversion results have been found to be stable and convergent, and they also highlight the need for good angular coverage to determine the anisotropy parameters in materials suspected of being anisotropic.

This issue of GEOPHYSICS

Okoye et al. develop a least-squares iterative inversion technique determining of the elastic parameters δ* and vertical P-wave velocity (α0) of any transversely isotropic modeling material in the laboratory. The anisotropic inverse modeling technique finds the best fitting solution and implements analytical rather than numerical differentiations to optimize the accuracy of the results.

Nonlinear teleseismic tomography at Long Valley Caldera, using three‐dimensional minimum travel time ray tracing

We explore the impact of three‐dimensional minimum travel time ray tracing on nonlinear teleseismic inversion. This problem has particular significance when trying to image strongly contrasting low‐velocity bodies, such as magma chambers, because strongly refracted and/or diffracted rays may precede the direct P wave arrival traditionally used in straight‐ray seismic tomography. We use a simplex‐based ray tracer to compute the three‐dimensional, minimum travel time ray paths and employ an iterative inversion technique to cope with nonlinearity. Results from synthetic data show that our algorithm results in better model reconstructions compared with traditional straight‐ray inversions. We reexamine the teleseismic data collected at Long Valley caldera by the U.S. Geological Survey. The most prominent feature of our result is a 25–30% low‐velocity zone centered at 11.5 km depth beneath the northwestern quadrant of the caldera. Beneath this at a depth of 24.5 km is a more diffuse 15% low‐velocity zone. In general, the low velocities tend to deepen to the south and east. We interpret the shallow feature to be the residual Long Valley caldera magma chamber, while the deeper feature may represent basaltic magmas ponded in the midcrust. The deeper position of the prominent low‐velocity region in comparison to earlier tomographic images is a result of using three‐dimensional rays rather than straight rays in our ray tracing. The magnitude of our low‐velocity anomaly is a factor of ∼3 times larger than earlier models from linear arrival time inversions and is consistent with models based on observations of ray bending at sites within the caldera. Our results imply the presence of anywhere from 7 to 100% partial melt beneath the caldera.

Tomography of the transition zone from the inversion of higher-mode surface waves

Abstract Higher-mode surface waves have a uniform depth resolution to about 1500 km depth in the period range 60–250 s. Therefore, these modes can be used for retrieving the S-wave velocity structure in the mantle and, more specifically, in the transition zone. The procedure used for computing a tomographic model of the mantle is divided into two steps. First, the iterative inverse technique developed by Stutzmann and Montagner (Geophys. J. Int., 113: 669–683, 1993) is used for determining higher-mode phase velocities and a spherically symmetric velocity model associated with each path. A ‘path’ is defined to be the minor or major arc (along a great circle) connecting a small epicentral area, in which several earthquakes are selected, and a station. The use of several earthquakes improves the depth resolution. Simultaneously for each datum, the difference between the spectrum of the observed seismogram and the sum of the synthetic spectra corresponding to the different modes is inverted to retrieve (1) the phase velocity dispersion curves of the fundamental mode and the three first higher modes and (2) the spherically symmetric S-wave velocity model corresponding to this path. In a second step, the higher-mode phase velocity dispersion curves and the S-wave velocity model corresponding to each path are regionalized without a priori constraint to obtain their lateral variations. This method has been applied to long-period GEOSCOPE data. We present preliminary results obtained with about 300 seismograms corresponding to 86 paths for the R1 or R2 trains. The three-dimensional S velocity model has been then expanded in spherical harmonics, to compare it with other models and to determine characteristics of the transition zone.

Mass distribution of binary stars using an iterative inversion technique

Proposes a new method to extract the masses of the components of single-lined spectroscopic binary stars. This approach is based on the Richardson-Lucy iterative inversion technique. First, the authors apply this method to derive the mass-ratio distribution from a mass-function distribution assuming a constant mass of the primary. Then, the authors generalize the method in order to derive the distribution of the companion mass given a certain distribution of the primary mass. The method proves to be easy to implement, mathematically founded, and at the same time very efficient. The authors present applications to three cases of astrophysical interest.

Evaluation of thermal field in buoyancy-induced flows by a Schlieren method

The aim of this work is to perform a quantitative experimental investigation of two-dimensional buoyancy-induced flows using a schlieren method. The schlieren technique uses the refraction of light rays to display a pattern related to the temperature field. The employed schlieren system makes use of an opaque filament to identify regions of fluid deflecting light rays by the same amount. An iterative inverse technique is developed to reconstruct the temperature field from optical data. After a description of the experimental technique and the data processing features, results concerning the temperature profile reconstruction for natural convective flows are presented. Attention is devoted to air buoyancy flows induced by a single heated vertical plate and by an array of four heated vertical plates. Reconstructed temperature data are compared with theoretical predictions and with direct temperature measurements performed by thermocouples.

Evaluation of thermal field in Buoyancy-induced flows by a Schlieren method

Abstract The aim of this work is to perform a quantitative experimental investigation of two-dimensional buoyancy-induced flows using a schlieren method. The schlieren technique uses the refraction of light rays to display a pattern related to the temperature field. The employed schlieren system makes use of an opaque filament to identify regions of fluid deflecting light rays by the same amount. An iterative inverse technique is developed to reconstruct the temperature field from optical data. After a description of the experimental technique and the data processing features, results concerning the temperature profile reconstruction for natural convective flows are presented. Attention is devoted to air buoyancy flows induced by a single heated vertical plate and by an array of four heated vertical plates. Reconstructed temperature data are compared with theoretical predictions and with direct temperature measurements performed by thermocouples.

Singular value decomposition of the velocity‐reflector depth tradeoff, Part 2: High‐resolution analysis of a generic model

The symmetries of a block circulant matrix significantly reduce the computational expense of the singular value decomposition (SVD) of the variable velocity inverse problem for a generic reflection seismology model. As a result, the decomposition does not suffer from edge effects or parameterization artifacts that are associated with small model spaces. Using this approach, we study the eigenvector and eigenvalue characteristics for a generic model of a size as large as is used with a variety of iterative inversion techniques (>100 000 parameters). Singular value decomposition of the raypath inverse problem of a discretized generic seismic model having one reflector indicates that the eigenvalue distribution for the inverse problem is nonuniform, with a large concentration near 0 and a gap near 0.4. All but the long horizontal wavelength reflector‐depth variations cannot be uniquely resolved from velocity variations. Lateral velocity variations serve to significantly reduce the ability of seismic data to resolve reflector depth for most of the horizontal wavelength components shorter than twice the cable length. As a result, automatic velocity analysis methods may not be able to resolve reflector variations when the velocity field is allowed to take on an arbitrary structure.

NEPR: A FORTRAN-77 program for determining thermal conductivity and diffusivity by needle-probe inversion

Abstract NEPR is a FORTRAN program for interpretation of needle-probe data by an iterative least-squares inversion technique. Basically, the response of a needle-probe is assumed to be described accurately by the integral solution associated with the heat conduction problem of an infinite region bounded internally by a perfect conducting circular cylinder. Two inversion procedures are given, one for ordinary interpretations and the other for calibration of the needle-probe. In the first situation, thermal conductivity, thermal diffusivity, and thermal conductance are determined, whereas in the latter situation a heat input correction constant, the effective heat capacity of the probe, and thermal conductance are determined. Theoretical responses, that is temperature rises vs time within the probe and in the external medium, are computable with the program. All computed parameters are determined with standard deviations. Examples of responses and inversion results are presented and the program is listed.

Backward reconstruction in acoustical holography based on iterative inversion techniques

Reconstruction of a source surface motion from acoustic holographic data presents computational difficulties because the problem is ill-posed as is well known in the literature. In order to deal with these difficulties, a method based on a recursive algorithm was developed. In this method, the inverse problem is convened to a well-posed forward propagation problem. An initial guess regarding the source images is required to activate the iterative inversion method. Then, the tentative image is forward propagated to the hologram plane and the residue is determined. Next, a feedback operator is used to process the residue by which the image is updated. Two types of feedback operators were investigated: (1) Wiener type, suboptimal operator, and (2) dynamic, optimal operator (designed subject to minimum mean-square error optimization criteria). Both iteration methods provided satisfactory convergence. In addition, these methods were found to be relatively insensitive to the choice of the initial guess and the noise parameters used in feedback operators. The efficiency of the iteration process is greatly enhanced by 2-D FFT. Numerical and experimental results obtained by using iterative inversion techniques will be presented. The iterative inversion techniques will be compared with conventional inverse filtering methods. In addition, performance of the iterative inversion methods when used in connection with the nearfield acoustic holography technique will be discussed. [Work supported by NSF.]

Inverted Shear Modulus from Wave‐induced Soil Motion

A previously developed technique, used mainly in seismology and geophysics, has been adopted to determine the shear modulus from soil displacements measured during wave tank experiments. This iterative inversion technique allowed the determination of the shear modulus and Coulomb specific loss (damping) with depth within a bentonite clay profile. It also allowed the observation of both the modulus reduction and damping curves for large shear strain amplitudes (up to 7%) for the first time. There was fairly good agreement between the modulus reduction curve determined by the inversion method and values found in the literature. Refinements to the inversion technique are suggested to improve determination of shear modulus. An alternate procedure to provide fairly reliable and quick estimates of the shear modulus is also presented.

Simultaneous inversion for hypocenters and lateral velocity variations: An iterative solution with a layered model

An iterative inversion technique has been developed that uses the direct P and S wave arrival times from local earthquakes to compute simultaneously a three‐dimensional velocity structure and relocated hypocenters. Crustal structure is modeled by subdividing flat layers into rectangular blocks. An interpolation function is used to smoothly vary velocities between blocks, allowing ray trace calculations of travel times in a three‐dimensional medium. Tests using synthetic data from known models show that solutions are reasonably independent of block size and spatial distribution but are sensitive to the choice of layer thicknesses. Application of the technique to observed earthquake data from north‐central Utah shows the following: (1) lateral velocity variations in the crust as large as 7% occur over 30‐km distances, (2) earthquake epicenters computed with the three‐dimensional velocity structure were shifted an average of 3.0 km from locations determined assuming homogeneous flat layered models, and (3) the laterally varying velocity structure correlates with anomalous variations in the local gravity and aeromagnetic fields, suggesting that the new velocity information can be valuable in acquiring a better understanding of crustal structure.

DIRECT INVERSION OF TRANSMISSION SYNTHETIC SEISMOGRAMS

Seismograms are routinely used to extract information about the internal structure of the subsurface. The basic philosophy is to idealize the earth with a simple model characterized by resolvable parameters. Further, we assume that the seismograms obey simple mathematical relations, such as the wave equation, and that these seismograms are uniquely determined by the given parameters. The computation of seismograms for a given model whose parameters are specified gives rise to the “forward problem”. The determination of the parameters of the model from the seismogram generally constitutes the “inverse problem”. While the forward problem is often relatively simple, the inverse problem tends to be much more involved. One of the few cases for which the inverse problem has a simple direct solution is given for a model consisting of a homogeneous one‐dimensional layered medium. The layers overlie a homogeneous half‐space, and are excited by a normally incident plane wave. Such direct solutions to the inverse problem are in contrast to the iterative inversion techniques of the Gilbert‐Backus type. We have chosen to treat the inverse problem for the seismogram escaping into the homogeneous substratum, i.e., the transmission seismogram. We do so because this time series is a simple autoregressive, or all‐pole process. The problem is studied both with and without white noise. The Wiener‐Levinson algorithm has been found to be well suited for the direct inversion of the noise‐free case, while the “maximum entropy” algorithm is generally more appropriate for noisy data. Numerical examples serve to clarify and illustrate these points.