Singular Value Decomposition Inversion Research Articles

Study on Inversion Algorithm of NMR T2 Spectrum

Nuclear magnetic resonance (NMR) logging technology is a kind of technology developed gradually in recent decades, which has great significance for the development of logging technology. Nuclear magnetic resonance logging technology compared to traditional logging technology, not only has the ideal signal - hydrogen nuclei, and the exploration of the original signal is not affected by the influence of the rock skeleton, is the only provides information that is unrelated to formation rock porosity, can better provide the formation permeability, movable fluid, oil and gas content, such as information, It is a kind of logging method suitable for low porosity and low permeability and complex special lithology, and overcomes the defects of traditional logging methods .With the development of NMR theory, NMR has been applied to the analysis of well logging data. The original NMR logging data are the spin-echo string observed by the spin echo string, and its amplitude decays with time. The T2 spectrum can be inversely calculated from the measured logging data, and then the T2 spectrum can be further solved to obtain key data such as permeability and porosity. In this paper, a simulated T2 spectrum is constructed by forward modeling, and the inversion is carried out under different SNR. On this basis, the specific relationship between the optimal number of singular values retained and the change of SNR is summarized, and an improved singular value decomposition algorithm is proposed. Inverse problem can be converted to nonlinear fitting optimization problems with nonnegative constraints, linear truncation algorithm on calculation precision has increased significantly, compared with the traditional singular value decomposition inversion algorithm, a new algorithm of signal-to-noise ratio on the solution of smaller and has better stability, and through the improvement of nonnegative constraint processing structure was more rapid inversion iteration algorithm.

The analysis of influencing factors on Tikhonov and truncated singular value decomposition inversion of flowing aerosol particle in dynamic light scattering

The analysis of influencing factors on Tikhonov and truncated singular value decomposition inversion of flowing aerosol particle in dynamic light scattering

基于核矩阵扩展的动态光散射截断奇异值分解反演

基于核矩阵扩展的动态光散射截断奇异值分解反演

Accelerating nonlinear reconstruction in laminar optical tomography by use of recursive SVD inversion.

Image reconstruction in the most model-based biophotonic imaging modalities essentially poses an ill-posed nonlinear inverse problem, which has been effectively tackled in the diffusion-approximation-satisfied scenarios such as diffuse optical tomography. Nevertheless, a nonlinear implementation in high-resolution laminar optical tomography (LOT) is normally computationally-costly due to its strong dependency on a dense source-detector configuration and a physically-rigorous photon-transport model. To circumvent the adversity, we herein propose a practical nonlinear LOT approach to the absorption reconstruction. The scheme takes advantage of the numerical stability of the singular value decomposition (SVD) for the ill-posed linear inversion, and is accelerated by adopting an explicitly recursive strategy for the time-consuming repeated SVD inversion, which is based on a scaled expression of the sensitivity matrix. Experiments demonstrate that the proposed methodology can perform as well as the traditional nonlinear one, while the computation time of the former is merely 26.27% of the later on average.

1-D DC Resistivity Inversion Using Singular Value Decomposition and Levenberg-Marquardt’s Inversion Schemes

Exploration of natural or energy resources requires geophysical survey to determine the subsurface structure, such as DC resistivity method. In this research, field and synthetic data were used using Schlumberger configuration. One-dimensional (1-D) DC resistivity inversion was carried out using Singular Value Decomposition (SVD) and Levenberg-Marquardt (LM) techniques to obtain layered resistivity structure. We have developed software to perform both inversion methods accompanied by a user-friendly interface. Both of the methods were compared one another to determine the number of iteration, robust to noise, elapsed time of computation, and inversion results. SVD inversion generated faster process and better results than LM did. The inversion showed both of these methods were appropriate to interpret subsurface resistivity structure.

Full three-dimensional imaging via ground penetrating radar: assessment in controlled conditions and on field for archaeological prospecting

This paper deals with an advanced microwave tomographic approach capable of providing full 3D images of buried targets from scattered field data gathered by means of Ground Penetrating Radar (GPR) systems. The approach is based on an approximated model of the scattering phenomenon and it is capable of accounting for the vectorial nature of the interactions occurring between electromagnetic waves and probed materials. Moreover, the Truncated Singular Value Decomposition inversion scheme is exploited to solve the involved linear inverse scattering problem in a stable and accurate way. The advantages offered by the full 3D inversion algorithm with respect to a commonly adopted strategy, which produces 3D images by interpolating 2D reconstructions, are assessed against GPR data gathered in laboratory controlled conditions. Moreover, to provide an example of the full 3D imaging capabilities in on field conditions, we report on a GPR measurement campaign carried out at Grotte dell’Angelo, Pertosa, (SA), Southern Italy, one of the most famous sites of the Cilento and Vallo di Diano geopark.

Inference of emission rates from multiple sources using Bayesian probability theory

The determination of atmospheric emission rates from multiple sources using inversion (regularized least-squares or best-fit technique) is known to be very susceptible to measurement and model errors in the problem, rendering the solution unusable. In this paper, a new perspective is offered for this problem: namely, it is argued that the problem should be addressed as one of inference rather than inversion. Towards this objective, Bayesian probability theory is used to estimate the emission rates from multiple sources. The posterior probability distribution for the emission rates is derived, accounting fully for the measurement errors in the concentration data and the model errors in the dispersion model used to interpret the data. The Bayesian inferential methodology for emission rate recovery is validated against real dispersion data, obtained from a field experiment involving various source-sensor geometries (scenarios) consisting of four synthetic area sources and eight concentration sensors. The recovery of discrete emission rates from three different scenarios obtained using Bayesian inference and singular value decomposition inversion are compared and contrasted.

A mixin algorithm for geoacoustic inversion

Shear wave velocities in the sediment can be inverted by using the dispersion curves of interface wave. It can be done by use of an algorithm based on singular value decomposition (SVD). This method can only invert the shear wave velocities and requires knowledge of the thicknesses and densities of the sediment layers. This paper presents a new algorithm for inversion of geoacoustic parameters based on dispersion curves that we have coined "mixin inversion". Mixin inversion combines SVD and global search using genetic algorithm (GA). The GA searches the depths and densities by doing a shear wave velocity inversion with SVD for each of its parameter values. This algorithm can be applied on cases where the depths, densities and shear wave velocities of the sediments are unknown. The advantage is that it is faster than pure GA since the search space is much smaller and it can be applied on cases where pure SVD inversion fails because necessary information about depths and densities is not known.

A Multiarray Tomographic Approach for Through-Wall Imaging

A through-wall imaging problem for a 2-D scalar geometry is addressed. It is cast as an inverse scattering problem and tackled under the linear Born model by means of the truncated singular value decomposition inversion scheme. A multiarray-based inversion strategy is considered. In particular, first the data collected by each single array are processed to obtain different tomographic images of the same scene under test. Then, the different images are suitably combined to obtain the overall image. The inversion scheme is tested for the challenging case of objects located within a complex environment resembling a room in a building.

Through-Wall Imaging via a Linear Inverse Scattering Algorithm

A through-wall imaging problem for a 2-D scalar geometry is addressed. It is cast as an inverse scattering problem and tackled under the linear model of the electromagnetic scattering that is provided by the Born approximation. A truncated singular value decomposition inversion scheme is exploited, and the performances that are achievable by such an inversion scheme are assessed by exploiting synthetic data. The cases of weakly and strongly scattering objects are both considered. Finally, an example of reconstruction that is obtained by exploiting experimental data is presented.

Application of the singular value decomposition method for inversion of interferometer measurements in fusion plasmas

We present applications of a recently developed method, the singular value decomposition inversion (SVD-I), for inverting line integrated multichannel interferometer measurements in fusion devices. The method expands the local electron density profiles into a limited series of orthogonal basis functions formed from the singular value decomposition of Thomson scattering measurements, which are obtained at much lower frequencies than the interferometer data. Because these basis functions are formed from a local measurement of the electron densities in the same plasmas, they are well adapted to the class of density profiles to be reconstructed. Using this expansion, the inversion of the interferometer data are reduced to solving an overdetermined set of equations. SVD-I is significantly faster than regularization methods and more accurate for reconstructing hollow profiles. This method, which combines the high bandwidth of interferometer systems with the spatial accuracy of Thomson scattering, is applied to invert data from a 14-chord interferometer on the TCV tokamak.

Frequency-Dependent Lg Q within the Continental United States

Frequency-dependent crustal attenuation (1/ Q ) is determined for seven distinct physiographic/tectonic regions of the continental United States using high-quality Lg waveforms recorded on broadband stations in the frequency band 0.5 to 16 Hz. Lg attenuation is determined from time-domain amplitude measurements in one-octave frequency bands centered on the frequencies 0.75, 1.0, 3.0, 6.0, and 12.0 Hz. Modeling errors are determined using a delete-j jackknife resampling technique. The frequency-dependent quality factor is modeled in the form of Q = Q 0 f η. Regions were initially selected based on tectonic provinces but were eventually limited and adjusted to maximize ray path coverage in each area. Earthquake data was recorded on several different networks and constrained to events occurring within the crust ( m b 3.5 in size. A singular value decomposition inversion technique was applied to the data to simultaneously solve for source and receiver terms along with Q for each region at specific frequencies. The lowest crustal Q was observed in northern and southern California where Q is described by the functions Q = 152(±37) f 0.72(±0.16) and Q = 105(±26) f 0.67(±0.16) , respectively. The Basin and Range Province, Pacific Northwest, and Rocky Mountain states also display lower Q and a strong frequency dependence characterized by the functions Q = 200(±40) f 0.68(±0.12) , Q = 152(±49) f 0.76(±0.18) , and Q = 166(±37) f 0.61(±0.14) , respectively. In contrast, in the central and northeast United States Q functions are Q = 640(±225) f 0.344(±0.22) and Q = 650(±143) f 0.36(±0.14) , respectively, show a high crustal Q and a weaker frequency dependence. These results improve upon previous Lg modeling by subdividing the United States into smaller, distinct tectonic regions and using significantly more data that provide improved constraints on frequency-dependent attenuation and errors. A detailed attenuation map of the continental United States can provide significant input into hazard map mitigation. Both scattering and intrinsic attenuation mechanisms are likely to play a comparable role in the frequency range considered in the study.

Surface heat flux histories from inversion of geothermal data: Energy balance at the Earth's surface

Past changes in the Earth's surface energy balance propagate into the subsurface and appear as perturbations of the subsurface thermal regime. This paper presents a singular value decomposition inversion method used to reconstruct surface heat flux histories (SHFH) from the heat flux anomalies detected in the shallow subsurface. Synthetic tests were used to assess the robustness of the inversion procedure. It was found that data noise can have a significant effect on the stability of the SHFH inferred from inversion. This translates in SHFHs having lower temporal resolution than ground surface temperature histories (GSTHs) obtained from the same data, but the long‐term trends are robust. Results are encouraging for temperature data noise levels typically encountered in field measurements. Synthetic data tests yield results in agreement with analytical expressions derived from GSTHs for the same parameterization. Temperature‐depth profiles from Canada's geothermal database were used to illustrate the inversion procedure. Individual temperature profile inversions are shown as examples. All 112 temperature logs in the database were used to obtain a mean heat flux history for the region. Results indicate that the ground heat flux has increased an average of 24 mW m−2 over the last 200 years in Canada. Application of this method to the existing global geothermal data base should allow for a quantification of the global energy balance at the Earth's surface for the past few centuries and may be useful for land surface models.

Surface heat flux histories from geothermal data: Inferences from inversion

Past changes in the Earth's surface energy balance propagate into the subsurface and appear as perturbations of the subsurface thermal regime. A singular value decomposition inversion (SVD) scheme has been used to reconstruct surface heat flux histories (SHFH) from the heat flux anomalies detected in the subsurface. Synthetic data tests yield results in agreement with analytical expressions. Inversion of a temperature log from central Canada is used to illustrate this procedure. Results show that reconstruction of SHFH is possible with existing data. Application of this method to the existing geothermal data base should permit a quantification of the energy balance at the Earth's surface for the past few centuries.

Source parameters for the 1952 Kern County earthquake, California: A joint inversion of leveling and triangulation observations

Coseismic leveling and triangulation observations are used to determine the faulting geometry and slip distribution of the July 21, 1952, Mw 7.3 Kern County earthquake on the White Wolf fault. A singular value decomposition inversion is used to assess the ability of the geodetic network to resolve slip along a multisegment fault and shows that the network is sufficient to resolve slip along the surface rupture to a depth of 10 km. Below 10 km, the network can only resolve dip slip near the fault ends. The preferred source model is a two‐segment right‐stepping fault with a strike of 51° and a dip of 75° SW. The epicentral patch has deep (6–27 km) left‐lateral oblique slip, while the northeastern patch has shallow (1–12.5 km) reverse slip. There is nearly uniform reverse slip (epicentral, 1.6 m; northeast, 1.9 m), with 3.6 m of left‐lateral strike slip limited to the epicentral patch. The seismic moment is M0 = 9.2 ± 0.5 × 1019 N m (Mw = 7.2). The signal‐to‐noise ratio of the leveling and triangulation data is reduced by 96% and 49%, respectively. The slip distribution from the preferred model matches regional geomorphic features and may provide a driving mechanism for regional shortening across the Comanche thrust and structural continuity with the Scodie seismic lineament to the northeast.

Seismic cycle and plate margin deformation in Costa Rica: GPS observations from 1994 to 1997

Global Positioning System (GPS) observations in Costa Rica from 1994 to 1997 reveal a complex pattern of motion consistent with the superposition of seismic cycle and secular plate margin deformation. In the south, velocity vectors are consistent with motion of the Panama Block plus postseismic deformation following the 1991 Limon earthquake and interseismic strain due to partial locking of the Middle America Trench (MAT) thrust. In the northwest, sites west of the volcanic arc are moving to the NW as a forearc sliver. Superimposed on this sliver motion are vertical and horizontal interseismic deformations from the adjacent Nicoya segment of the MAT. We apply two different inverse methods to understand the source of the seismic strain in NW Costa Rica. We compare fault‐locking models derived using a singular value decomposition inversion with that of a simulated annealing global optimization approach. Both methods yield similar models for partial locking of the thrust interface beneath the Nicoya Peninsula. Our results define an area of nearly fully locked fault beneath the outer coast of the southern portion of the peninsula, with somewhat lower coupling beneath the northern half and with low coupling elsewhere. These initial results show the promise for detailed imaging of the locked portion of a thrust interface responsible for future large subduction zone earthquakes.

The Inversion of NMR Log Data Sets with Different Measurement Errors

We present a composite-data processing method which simultaneously processes two or more data sets with different measurement errors. We examine the role of the noise level of the data in the singular value decomposition inversion process, the criteria for a proper cutoff, and its effect on the uncertainty of the solution. Examples of processed logs using the composite-data processing method are presented and discussed. The possible usefulness of the apparent T1/T2 ratio extracted from the logs is illustrated.

Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition

We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type—such as numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform. We propose to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire scale of Besov spaces. The important case of Besov spaces B σ p, q , p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures— SVD included—cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure.