Inversion with a grain of salt

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 99-109 ◽  
Author(s):  
Larry R. Lines ◽  
Sven Treitel
Keyword(s):  
Least Squares ◽  
Jacobian Matrix ◽  
Quantitative Measure ◽  
Parameter Estimates ◽  
Inversion Problem ◽  
Seismic Profiles ◽  
Traveltime Inversion ◽  
Vertical Seismic Profiles ◽  
Model Response ◽  
Value Decomposition

Although least‐squares inversion is a useful tool in data analysis, nonuniqueness is an inevitable problem. This problem can be analyzed by considering the sensitivity of a model response to the parameter estimates. Such sensitivity methods produce extremal solutions which barely satisfy some resolution (or stability) criteria. Two closely related methods for producing such solutions are the “Edgehog” and “Most Squares” methods due to Jackson (1973, 1976). These techniques, which evaluate the “degree of nonuniqueness” in a least‐squares inversion, require only the information computed in a singular value decomposition (SVD) solution. While the “edgehog” and “most squares” techniques are mathematically similar, the “most squares” estimate is the simpler to compute. Both methods show that the credibility of an inversion depends on both the specified error criterion as well as on the properties of the Jacobian matrix associated with the least‐squares solution. The similar performance of these two closely related methods is demonstrated with the traveltime inversion of both synthetic and real vertical seismic profiles (VSPs). The sensitivity analysis of this inversion problem provides a quantitative measure of solution reliability.

Traveltime inversion of offset vertical seismic profiles—A feasibility study

Geophysics ◽  
1984 ◽  
Vol 49 (3) ◽  
pp. 250-264 ◽  
Author(s):  
L. R. Lines ◽  
A. Bourgeois ◽  
J. D. Covey
Keyword(s):  
Inverse Method ◽  
Synthetic Data ◽  
Real Data ◽  
Seismic Profile ◽  
Inversion Method ◽  
Earth Model ◽  
Seismic Profiles ◽  
Traveltime Inversion ◽  
Vertical Seismic Profiles ◽  
Inversion Techniques

Traveltimes from an offset vertical seismic profile (VSP) are used to estimate subsurface two‐dimensional dip by applying an iterative least‐squares inverse method. Tests on synthetic data demonstrate that inversion techniques are capable of estimating dips in the vicinity of a wellbore by using the traveltimes of the direct arrivals and the primary reflections. The inversion method involves a “layer stripping” approach in which the dips of the shallow layers are estimated before proceeding to estimate deeper dips. Examples demonstrate that the primary reflections become essential whenever the ratio of source offset to layer depth becomes small. Traveltime inversion also requires careful estimation of layer velocities and proper statics corrections. Aside from these difficulties and the ubiquitous nonuniqueness problem, the VSP traveltime inversion was able to produce a valid earth model for tests on a real data case.

Curved‐ray traveltime inversion of shallow vertical seismic profiles

2002 ◽  
Author(s):  
Geoff Moret ◽  
William P. Clement ◽  
Michael D. Knoll
Keyword(s):  
Seismic Profiles ◽  
Traveltime Inversion ◽  
Vertical Seismic Profiles

Multichannel Wiener deconvolution of vertical seismic profiles

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1500-1511 ◽  
Author(s):  
Jakob B. U. Haldorsen ◽  
Douglas E. Miller ◽  
John J. Walsh
Keyword(s):  
Least Squares ◽  
Amplitude Spectrum ◽  
Seismic Source ◽  
Seismic Profile ◽  
Signal Spectrum ◽  
Seismic Profiles ◽  
Vertical Seismic Profiles ◽  
Vsp Data ◽  
Zero Offset ◽  
Noise Signature

We describe a technique for performing optimal, least‐squares deconvolution of vertical seismic profile (VSP) data. The method is a two‐step process that involves (1) estimating the source signature and (2) applying a least‐squares optimum deconvolution operator that minimizes the noise not coherent with the source signature estimate. The optimum inverse problem, formulated in the frequency domain, gives as a solution an operator that can be interpreted as a simple inverse to the estimated aligned signature multiplied by semblance across the array. An application to a zero‐offset VSP acquired with a dynamite source shows the effectiveness of the operator in attaining the two conflicting goals of adaptively spiking the effective source signature and minimizing the noise. Signature design for seismic surveys could benefit from observing that the optimum deconvolution operator gives a flat signal spectrum if and only if the seismic source has the same amplitude spectrum as the noise.

Application of singular value decomposition to vertical seismic profiling

Geophysics ◽  
1988 ◽  
Vol 53 (6) ◽  
pp. 778-785 ◽  
Author(s):  
Sergio L. M. Freire ◽  
Tad J. Ulrych
Keyword(s):  
Singular Value Decomposition ◽  
Singular Value ◽  
Seismic Profiles ◽  
New Approach ◽  
Vertical Seismic Profiles ◽  
Additional Information ◽  
Noise Rejection ◽  
Regular Sampling ◽  
Important Additional Information ◽  
Value Decomposition

An essential part of the interpretation of vertical seismic profiles (VSP) is the separation of the upgoing and downgoing waves. This paper presents a new approach which is based on the decomposition of time‐shifted VSP sections into eigenimages, using singular value decomposition (SVD). The first few eigenimages of the time‐shifted VSP section contain the contributions of the horizontally aligned downgoing waves. The last few eigenimages contain the contribution of uncorrelated noise components. The separated upgoing waves are recovered as a partial sum of the eigenimages. Important aspects of this approach are that regular sampling of the recording levels is not required, that the first‐break times need not be measured with extreme accuracy, that noise rejection may be automatically included in the processing, and that eigenimages or sums of eigenimages which may be computed as part of the approach can provide important additional information.

Structure Parameter Identification of Articulated Arm Coordinate Measuring Machines Based on the Least Squares Method

2013 ◽  
Vol 303-306 ◽  
pp. 638-641
Author(s):  
Jian Lu ◽  
Guan Bin Gao ◽  
Hui Ping Yang
Keyword(s):  
Least Squares ◽  
Jacobian Matrix ◽  
Least Squares Method ◽  
Structural Parameters ◽  
Structure Parameters ◽  
Measuring Accuracy ◽  
Total Differential ◽  
Cartesian System ◽  
Main Factors ◽  
Value Decomposition

The AACMM is a kind of new coordinate measuring instrument based on non-Cartesian system. The errors of structure parameters are the main factors which affect the measuring accuracy, and the structural parameters identification can effectively improve the measuring accuracy of AACMM. The measuring model of an AACMM was established using the D-H method. Then the error model of the AACMM was established with total differential transforming method, from which the Jacobian matrix was obtained. The structure parameters’ errors which need to be identified through singular value decomposition of Jacobian matrix and the decomposition of orthogonal matrix elementary row transform can be determined. Finally, the least squares method was used in identifying the structure parameters’ errors of the AACMM, the results shows that the method proposed in this paper can identify the structure parameters efficiently.

Separation of signal and noise applied to vertical seismic profiles

Geophysics ◽  
1988 ◽  
Vol 53 (7) ◽  
pp. 932-946 ◽  
Author(s):  
William S. Harlan
Keyword(s):  
Least Squares ◽  
Earth Model ◽  
Seismic Profiles ◽  
One Dimensional ◽  
Vertical Seismic Profiles ◽  
Probability Densities ◽  
Vsp Data ◽  
Band Limited ◽  
Tube Wave ◽  
Impedance Functions

Inversion of the band‐limited one‐dimensional VSP response is nonunique because impedance functions with very different statistics produce equivalent responses. Least‐squares methods of inversion linearly transform noise and tend to produce impedance functions with a Gaussian distribution of amplitudes. I modify a least‐squares inversion procedure to exclude nonzero impedance derivatives that are significantly influenced by noise. The resulting earth model shows homogeneous intervals unless the data have reliable information to the contrary. The data are modeled with a one‐dimensional wave equation and three invertible functions: acoustic impedance, a source wavelet, and the traces’ amplification. First, a linearized least‐squares inverse perturbs the source function to model the downgoing wave. A relinearized inverse finds perturbations of all three modeling functions to account for first‐order reflections. Further iterations explain higher order reflections. To estimate the reliability of impedance perturbations, each linearized inversion is repeated for pure noise that equals or exceeds the noise in the data. Amplitude histograms are used to estimate probability density functions for the amplitudes of the signal and of the noise in the perturbations. Nonzero impedance derivatives are accepted as reliable if, according to the probability functions, the perturbations contain, with a high probability, only a small amount of noise. For a set of VSP data provided by L’Institut Francais du Petrole, four iterations allowed only a few nonzero impedance derivatives and modeled a recorded VSP as well as did a least‐squares inversion that accepted all proposed perturbations. Estimated probability densities for the remaining signal and noise were used to extract a tube wave that contained little signal.

An Experimental Study of Resolved Acceleration Control of Robots at Singularities: Damped Least-Squares Approach

1997 ◽  
Vol 119 (1) ◽  
pp. 97-101 ◽  
Author(s):  
M. Kirc´anski ◽  
N. Kirc´anski ◽  
D. Lekovic´ ◽  
M. Vukobratovic´
Keyword(s):  
Least Squares ◽  
Degrees Of Freedom ◽  
Jacobian Matrix ◽  
Control Level ◽  
Singular Values ◽  
Control Scheme ◽  
Resolved Acceleration Control ◽  
Robot Task ◽  
Damped Least Squares ◽  
Value Decomposition

Most of the robot task space control methods based on inverse Jacobian matrix suffer from instability in singular regions of workspace. Methods based on damped least-squares algorithm (DLS) for matrix inversion have been developed but not experimentally confirmed. The application of DLS method at the kinematic control level has been reported in (Chiaverini et al., 1994). In this article, a modified DLS method combined with the resolved-acceleration control scheme, is experimentally verified on two degrees of freedom of a PUMA-560 robot. In order to decrease the position error introduced by the damping, only small singular values are damped, in contrast to the conventional damping method were all the singular values are damped. The symbolic expressions of the singular value decomposition of the Jacobian matrix were used, to decrease the computational burden.

Generalised DOPs with Consideration of the Influence Function of Signal-in-Space Errors

2011 ◽  
Vol 64 (S1) ◽  
pp. S3-S18 ◽  
Author(s):  
Yuanxi Yang ◽  
Jinlong Li ◽  
Junyi Xu ◽  
Jing Tang
Keyword(s):  
Least Squares ◽  
Stochastic Models ◽  
Influence Function ◽  
A Priori ◽  
Functional Model ◽  
Global Navigation Satellite Systems ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Integrated Navigation ◽  
Satellite Systems

Integrated navigation using multiple Global Navigation Satellite Systems (GNSS) is beneficial to increase the number of observable satellites, alleviate the effects of systematic errors and improve the accuracy of positioning, navigation and timing (PNT). When multiple constellations and multiple frequency measurements are employed, the functional and stochastic models as well as the estimation principle for PNT may be different. Therefore, the commonly used definition of “dilution of precision (DOP)” based on the least squares (LS) estimation and unified functional and stochastic models will be not applicable anymore. In this paper, three types of generalised DOPs are defined. The first type of generalised DOP is based on the error influence function (IF) of pseudo-ranges that reflects the geometry strength of the measurements, error magnitude and the estimation risk criteria. When the least squares estimation is used, the first type of generalised DOP is identical to the one commonly used. In order to define the first type of generalised DOP, an IF of signal–in-space (SIS) errors on the parameter estimates of PNT is derived. The second type of generalised DOP is defined based on the functional model with additional systematic parameters induced by the compatibility and interoperability problems among different GNSS systems. The third type of generalised DOP is defined based on Bayesian estimation in which the a priori information of the model parameters is taken into account. This is suitable for evaluating the precision of kinematic positioning or navigation. Different types of generalised DOPs are suitable for different PNT scenarios and an example for the calculation of these DOPs for multi-GNSS systems including GPS, GLONASS, Compass and Galileo is given. New observation equations of Compass and GLONASS that may contain additional parameters for interoperability are specifically investigated. It shows that if the interoperability of multi-GNSS is not fulfilled, the increased number of satellites will not significantly reduce the generalised DOP value. Furthermore, the outlying measurements will not change the original DOP, but will change the first type of generalised DOP which includes a robust error IF. A priori information of the model parameters will also reduce the DOP.

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