Smoothing operators for waveform tomographic imaging

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1646-1654 ◽  
Author(s):  
Steve Cardimona ◽  
Jan Garmany
Keyword(s):  
Fresnel Zone ◽  
A Priori ◽  
Approximate Solutions ◽  
Velocity Model ◽  
Forward Problem ◽  
Nonlinear Inversion ◽  
Approximation Solution ◽  
Waveform Data ◽  
Sensitivity Functions ◽  
Priori Information

The analytic kernel in the space‐time domain for the Frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the Born approximation solution to the integral equation of waveform scattering. Preconditioning operators in the solution of this forward problem, which may incorporate a priori information and approximate solutions, are smoothing operators in the imaging problem, the first iteration of a nonlinear inversion for the slowness model. Some preconditioning operators are determined for solutions to the parabolic wave equation, and then used to create new sensitivity functions that retain appropriate characteristics of the true Frechet kernel in forward calculations. The new sensitivity functions define near‐source, near‐receiver and far‐field kernels, as well as kernels that exhibit an amplitude decay off the ray yielding ray‐perpendicular sensitivity that scales with the Fresnel zone size. A sample calculation from a synthetic cross‐well imaging experiment shows the usefulness of introducing physically appropriate model smoothing directly into the sensitivity function of the forward problem, helping to obtain a geologically reasonable image of the velocity model when ray coverage is insufficient.

scholarly journals Sensitivity maps for time-reverse imaging: an accuracy study for the Los Humeros Geothermal Field (Mexico)

2020 ◽  
Vol 222 (1) ◽  
pp. 231-246
Author(s):  
C Finger ◽  
E H Saenger
Keyword(s):  
A Priori ◽  
Velocity Model ◽  
Source Location ◽  
Geothermal Field ◽  
Location Accuracy ◽  
Accurate Localization ◽  
Localization Scheme ◽  
Priori Information ◽  
Initial Source ◽  
Sensitivity Maps

SUMMARY The estimation of the source–location accuracy of microseismic events in reservoirs is of significant importance. Time-reverse imaging (TRI) provides a highly accurate localization scheme to locate events by time-reversing the recorded full wavefield and back propagating it through a velocity model. So far, the influence of the station geometry and the velocity model on the source–location accuracy is not well known. Therefore, sensitivity maps are developed using the geothermal site of Los Humeros in Mexico to evaluate the spatial variability of the source–location accuracy. Sensitivity maps are created with an assumed gradient velocity model with a constant vp–vs ratio and with a realistic velocity model for the region of Los Humeros. The positions of 27 stations deployed in Los Humeros from September 2017 to September 2018 are used as surface receivers. An automatic localization scheme is proposed that does not rely on any a priori information about the sources and thus negates any user bias in the source locations. The sensitivity maps are created by simulating numerous uniformly distributed sources simultaneously and locating these sources using TRI. The found source locations are compared to the initial source locations to estimate the achieved accuracy. The resulting sensitivity maps show that the station geometry introduces complex patterns in the spatial variation of accuracy. Furthermore, the influence of the station geometry on the source–location accuracy is larger than the influence of the velocity model. Finally, a microearthquake recorded at the geothermal site of Los Humeros is located to demonstrate the usability of the derived sensitivity maps. This study stresses the importance of optimizing station networks to enhance the accuracy when locating seismic events using TRI.

Reflector imaging using trial reflector and crosscorrelation: Application to fracture imaging for sonic data

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. S433-S446 ◽  
Author(s):  
Nobuyasu Hirabayashi
Keyword(s):  
Fresnel Zone ◽  
A Priori ◽  
Image Point ◽  
Complex Structures ◽  
A Priori Information ◽  
Kirchhoff Migration ◽  
Common Depth Point ◽  
Resolution Imaging ◽  
The Common ◽  
Priori Information

I have developed two high-resolution imaging methods that do not require a priori information on the structural dip. The dip information is often necessary to image complex structures using Kirchhoff migration by selecting only the constructively interfering parts of waveforms, especially for data with limited acquisition geometry. However, such dip information is not generally available. The methods that I evaluated use a trial reflector, which is defined for each image point and source-receiver pair, to search for the true geologic reflector. The coincidence of these reflectors is judged by a coherency analysis of event signals for the trial reflector using the crosscorrelations, and the coherency is converted to a weight. The weight is combined with the stacking methods of waveform samples in migration. In the first method, a waveform sample summed at an image point for a source-receiver pair is obtained by the common-depth-point stack of array data for the trial reflector. In the second method, a waveform sample of a source-receiver pair at the traveltime of the reflected ray for the trial reflector is smeared in the Fresnel zone computed for the trial reflector. My methods were applied to image fractures for sonic data, whose frequency range is centered approximately 8 kHz, and they provide higher resolution images than those given by conventional Kirchhoff migration.

A comparison of performances of linearized and global nonlinear 2-D inversions of VLF and VLF-R electromagnetic data

Geophysics ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 462-475 ◽  
Author(s):  
P. Kaikkonen ◽  
S. P. Sharma
Keyword(s):  
A Priori ◽  
Low Frequency ◽  
Model Parameters ◽  
Nonlinear Inversion ◽  
Stable Model ◽  
Very Fast Simulated Annealing ◽  
Iterative Inversion ◽  
Priori Information ◽  
The One ◽  
Nonlinear Joint

The performances of linearized (local) and global nonlinear joint 2-D inversions of very low frequency (VLF) and VLF resistivity electromagnetic measurements are analyzed. A stable iterative inversion scheme is used in linearized inversion while the very fast simulated annealing approach is used in global nonlinear inversion. Synthetic noise‐free and noisy data due to three different models in complexity and two field examples are considered. Synthetic examples show that linearized inversion reveals the subsurface structure better than global nonlinear inversion provided the model has only a few parameters under inversion. Both linearized and global nonlinear inversions must be performed combining all available data in order to obtain the most reliable estimates of the subsurface parameters. Complex models with a large number of parameters are better to invert using global nonlinear inversion although the CPU time needed is always much longer than the one used in linearized inversion. Contrary to global nonlinear inversion, success in linearized inversion requires the good a priori information of all the model parameters under inversion. Noise in data influences the linearized inversion results more than those provided by global inversion. Linearized inversion using as an initial model the mean model due to a few global inversion runs is also a good approach. Even in this case, if there are a large number of model parameters in inversion, linearized inversion can lead to an unstable solution. To overcome such a problem, one can fix the important and stable model parameters from the first step of linearized inversion and then vary and stabilize unstable parameters in the second step.

scholarly journals Accounting for imperfect forward modeling in geophysical inverse problems — Exemplified for crosshole tomography

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. H1-H21 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Knud Skou Cordua ◽  
Bo Holm Jacobsen ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
A Priori ◽  
Velocity Model ◽  
Forward Modeling ◽  
Forward Problem ◽  
Forward Model ◽  
Model Parameters ◽  
Observation Error ◽  
Modeling Error

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.

Innovative Approach to Information Search by Example of a Patent Analysis of an Important Substitution Plan

2020 ◽  
pp. 143-157
Author(s):  
Maria A. Milkova
Keyword(s):  
Information Search ◽  
Topic Modeling ◽  
Cognitive Biases ◽  
A Priori ◽  
Import Substitution ◽  
Innovative Approach ◽  
Iterative Search ◽  
Comprehensive Picture ◽  
Priori Information ◽  
Selection Of

Nowadays the process of information accumulation is so rapid that the concept of the usual iterative search requires revision. Being in the world of oversaturated information in order to comprehensively cover and analyze the problem under study, it is necessary to make high demands on the search methods. An innovative approach to search should flexibly take into account the large amount of already accumulated knowledge and a priori requirements for results. The results, in turn, should immediately provide a roadmap of the direction being studied with the possibility of as much detail as possible. The approach to search based on topic modeling, the so-called topic search, allows you to take into account all these requirements and thereby streamline the nature of working with information, increase the efficiency of knowledge production, avoid cognitive biases in the perception of information, which is important both on micro and macro level. In order to demonstrate an example of applying topic search, the article considers the task of analyzing an import substitution program based on patent data. The program includes plans for 22 industries and contains more than 1,500 products and technologies for the proposed import substitution. The use of patent search based on topic modeling allows to search immediately by the blocks of a priori information – terms of industrial plans for import substitution and at the output get a selection of relevant documents for each of the industries. This approach allows not only to provide a comprehensive picture of the effectiveness of the program as a whole, but also to visually obtain more detailed information about which groups of products and technologies have been patented.

scholarly journals Earthquake location in tectonic structures of the Alpine Chain: the case of the Constance Lake (Central Europe) seismic sequence

2021 ◽  
Author(s):  
Francesca D’Ajello Caracciolo ◽  
Rodolfo Console
Keyword(s):  
Central Europe ◽  
Velocity Model ◽  
Moho Depth ◽  
Seismic Sequence ◽  
Tectonic Structures ◽  
Study Region ◽  
Waveform Data ◽  
Seismic Stations ◽  
Velocity Models ◽  
Master Event

AbstractA set of four magnitude Ml ≥ 3.0 earthquakes including the magnitude Ml = 3.7 mainshock of the seismic sequence hitting the Lake Constance, Southern Germany, area in July–August 2019 was studied by means of bulletin and waveform data collected from 86 seismic stations of the Central Europe-Alpine region. The first single-event locations obtained using a uniform 1-D velocity model, and both fixed and free depths, showed residuals of the order of up ± 2.0 s, systematically affecting stations located in different areas of the study region. Namely, German stations to the northeast of the epicenters and French stations to the west exhibit negative residuals, while Italian stations located to the southeast are characterized by similarly large positive residuals. As a consequence, the epicentral coordinates were affected by a significant bias of the order of 4–5 km to the NNE. The locations were repeated applying a method that uses different velocity models for three groups of stations situated in different geological environments, obtaining more accurate locations. Moreover, the application of two methods of relative locations and joint hypocentral determination, without improving the absolute location of the master event, has shown that the sources of the four considered events are separated by distances of the order of one km both in horizontal coordinates and in depths. A particular attention has been paid to the geographical positions of the seismic stations used in the locations and their relationship with the known crustal features, such as the Moho depth and velocity anomalies in the studied region. Significant correlations between the observed travel time residuals and the crustal structure were obtained.

scholarly journals Computational Method for Wavefront Sensing Based on Transport-Of-Intensity Equation

Photonics ◽  
2021 ◽  
Vol 8 (6) ◽  
pp. 177
Author(s):  
Iliya Gritsenko ◽  
Michael Kovalev ◽  
George Krasin ◽  
Matvey Konoplyov ◽  
Nikita Stsepuro
Keyword(s):  
A Priori ◽  
Spherical Wave ◽  
Computational Method ◽  
Optical Systems ◽  
Radius Of Curvature ◽  
Imaging Method ◽  
Wavefront Sensing ◽  
Phase Imaging ◽  
Transport Of Intensity Equation ◽  
Priori Information

Recently the transport-of-intensity equation as a phase imaging method turned out as an effective microscopy method that does not require the use of high-resolution optical systems and a priori information about the object. In this paper we propose a mathematical model that adapts the transport-of-intensity equation for the purpose of wavefront sensing of the given light wave. The analysis of the influence of the longitudinal displacement z and the step between intensity distributions measurements on the error in determining the wavefront radius of curvature of a spherical wave is carried out. The proposed method is compared with the traditional Shack–Hartmann method and the method based on computer-generated Fourier holograms. Numerical simulation showed that the proposed method allows measurement of the wavefront radius of curvature with radius of 40 mm and with accuracy of ~200 μm.

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