New objective functions for waveform inversion

Geophysics ◽  
1998 ◽  
Vol 63 (1) ◽  
pp. 213-222 ◽  
Author(s):  
L. Neil Frazer ◽  
Xinhua Sun
Keyword(s):  
Green’S Function ◽  
Objective Function ◽  
Impulse Response ◽  
Green's Function ◽  
Waveform Inversion ◽  
Data Sets ◽  
Minimum Phase ◽  
Objective Functions ◽  
Data Set ◽  
Parameter Values

Inversion is an organized search for parameter values that maximize or minimize an objective function, referred to here as a processor. This note derives three new seismic processors that require neither prior deconvolution nor knowledge of the source‐receiver wavelet. The most powerful of these is the fourwise processor, as it is applicable to data sets from multiple shots and receivers even when each shot has a different unknown signature and each receiver has a different unknown impulse response. Somewhat less powerful than the fourwise processor is the pairwise processor, which is applicable to a data set consisting of two or more traces with the same unknown wavelet but possibly different gains. When only one seismogram exists the partition processor can be used. The partition processor is also applicable when there is only one shot (receiver) and each receiver (shot) has a different signature. In fourwise and pairwise inversions the unknown wavelets may be arbitrarily long in time and need not be minimum phase. In partition inversion the wavelet is assumed to be shorter in time than the data trace itself but is not otherwise restricted. None of the methods requires assumptions about the Green’s function.

The Search of Diffusive Properties in Ambient Seismic Noise

2021 ◽  
Author(s):  
José Piña-Flores ◽  
Martín Cárdenas-Soto ◽  
Antonio García-Jerez ◽  
Michel Campillo ◽  
Francisco J. Sánchez-Sesma
Keyword(s):  
Green’S Function ◽  
Surface Waves ◽  
Green's Function ◽  
Seismic Noise ◽  
Elastic Field ◽  
Spectral Ratio ◽  
Ambient Seismic Noise ◽  
Data Sets ◽  
Original Experiment ◽  
Imaging Theory

ABSTRACT Ambient seismic noise (ASN) is becoming of interest for geophysical exploration and engineering seismology, because it is possible to exploit its potential for imaging. Theory asserts that the Green’s function can be retrieved from correlations within a diffuse field. Surface waves are the most conspicuous part of Green’s function in layered media. Thus, the velocities of surface waves can be obtained from ASN if the wavefield is diffuse. There is widespread interest in the conditions of emergence and properties of diffuse fields. In the applications, useful approximations of the Green’s function can be obtained from cross correlations of recorded motions of ASN. An elastic field is diffuse if the background illumination is azimuthally uniform and equipartitioned. It happens with the coda waves in earthquakes and has been verified in carefully planned experiments. For one of these data sets, the 1999 Chilpancingo (Mexico) experiment, there are some records of earthquake pre-events that undoubtedly are composed of ASN, so that the processing for coda can be tested on them. We decompose the ASN energies and study their equilibration. The scheme is inspired by the original experiment and uses the ASN recorded in an L-shaped array that allows the computation of spatial derivatives. It requires care in establishing the appropriate ranges for measuring parameters. In this search for robust indicators of diffusivity, we are led to establish that under certain circumstances, the S and P energy equilibration is a process that anticipates the diffusion regime (not necessarily isotropy), which justifies the use of horizontal-to-vertical spectral ratio in the context of diffuse-field theory.

Fuzzy C-Means in High Dimensional Spaces

2013 ◽  
pp. 1-16
Author(s):  
Roland Winkler ◽  
Frank Klawonn ◽  
Rudolf Kruse
Keyword(s):  
Objective Function ◽  
Local Minimum ◽  
High Dimensional ◽  
Controlled Environment ◽  
Data Sets ◽  
Local Minima ◽  
High Dimensions ◽  
Data Set ◽  
Fcm Algorithm ◽  
Centre Of Gravity

High dimensions have a devastating effect on the FCM algorithm and similar algorithms. One effect is that the prototypes run into the centre of gravity of the entire data set. The objective function must have a local minimum in the centre of gravity that causes FCM’s behaviour. In this paper, examine this problem. This paper answers the following questions: How many dimensions are necessary to cause an ill behaviour of FCM? How does the number of prototypes influence the behaviour? Why has the objective function a local minimum in the centre of gravity? How must FCM be initialised to avoid the local minima in the centre of gravity? To understand the behaviour of the FCM algorithm and answer the above questions, the authors examine the values of the objective function and develop three test environments that consist of artificially generated data sets to provide a controlled environment. The paper concludes that FCM can only be applied successfully in high dimensions if the prototypes are initialized very close to the cluster centres.

Fuzzy C-Means in High Dimensional Spaces

2011 ◽  
Vol 1 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Roland Winkler ◽  
Frank Klawonn ◽  
Rudolf Kruse
Keyword(s):  
Objective Function ◽  
Local Minimum ◽  
High Dimensional ◽  
Controlled Environment ◽  
Data Sets ◽  
Local Minima ◽  
High Dimensions ◽  
Data Set ◽  
Fcm Algorithm ◽  
Centre Of Gravity

High dimensions have a devastating effect on the FCM algorithm and similar algorithms. One effect is that the prototypes run into the centre of gravity of the entire data set. The objective function must have a local minimum in the centre of gravity that causes FCM’s behaviour. In this paper, examine this problem. This paper answers the following questions: How many dimensions are necessary to cause an ill behaviour of FCM? How does the number of prototypes influence the behaviour? Why has the objective function a local minimum in the centre of gravity? How must FCM be initialised to avoid the local minima in the centre of gravity? To understand the behaviour of the FCM algorithm and answer the above questions, the authors examine the values of the objective function and develop three test environments that consist of artificially generated data sets to provide a controlled environment. The paper concludes that FCM can only be applied successfully in high dimensions if the prototypes are initialized very close to the cluster centres.

A Bayesian approach to modeling 2D gravity data using polygons

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. G1-G21 ◽  
Author(s):  
William J. Titus ◽  
Sarah J. Titus ◽  
Joshua R. Davis
Keyword(s):  
Gravity Data ◽  
Synthetic Data ◽  
Gravity Inversion ◽  
Limiting Factor ◽  
Data Sets ◽  
Data Set ◽  
Local Optima ◽  
Occupancy Probability ◽  
Compute Model ◽  
Parameter Values

We apply a Bayesian Markov chain Monte Carlo formalism to the gravity inversion of a single localized 2D subsurface object. The object is modeled as a polygon described by five parameters: the number of vertices, a density contrast, a shape-limiting factor, and the width and depth of an encompassing container. We first constrain these parameters with an interactive forward model and explicit geologic information. Then, we generate an approximate probability distribution of polygons for a given set of parameter values. From these, we determine statistical distributions such as the variance between the observed and model fields, the area, the center of area, and the occupancy probability (the probability that a spatial point lies within the subsurface object). We introduce replica exchange to mitigate trapping in local optima and to compute model probabilities and their uncertainties. We apply our techniques to synthetic data sets and a natural data set collected across the Rio Grande Gorge Bridge in New Mexico. On the basis of our examples, we find that the occupancy probability is useful in visualizing the results, giving a “hazy” cross section of the object. We also find that the role of the container is important in making predictions about the subsurface object.

Tackling cycle skipping in full-waveform inversion with intermediate data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R411-R427 ◽  
Author(s):  
Gang Yao ◽  
Nuno V. da Silva ◽  
Michael Warner ◽  
Di Wu ◽  
Chenhao Yang
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Intermediate Data ◽  
Full Waveform ◽  
L2 Norm ◽  
Recorded Data

Full-waveform inversion (FWI) is a promising technique for recovering the earth models for exploration geophysics and global seismology. FWI is generally formulated as the minimization of an objective function, defined as the L2-norm of the data residuals. The nonconvex nature of this objective function is one of the main obstacles for the successful application of FWI. A key manifestation of this nonconvexity is cycle skipping, which happens if the predicted data are more than half a cycle away from the recorded data. We have developed the concept of intermediate data for tackling cycle skipping. This intermediate data set is created to sit between predicted and recorded data, and it is less than half a cycle away from the predicted data. Inverting the intermediate data rather than the cycle-skipped recorded data can then circumvent cycle skipping. We applied this concept to invert cycle-skipped first arrivals. First, we picked up the first breaks of the predicted data and the recorded data. Second, we linearly scaled down the time difference between the two first breaks of each shot into a series of time shifts, the maximum of which was less than half a cycle, for each trace in this shot. Third, we moved the predicted data with the corresponding time shifts to create the intermediate data. Finally, we inverted the intermediate data rather than the recorded data. Because the intermediate data are not cycle-skipped and contain the traveltime information of the recorded data, FWI with intermediate data updates the background velocity model in the correct direction. Thus, it produces a background velocity model accurate enough for carrying out conventional FWI to rebuild the intermediate- and short-wavelength components of the velocity model. Our numerical examples using synthetic data validate the intermediate-data concept for tackling cycle skipping and demonstrate its effectiveness for the application to first arrivals.

Random objective waveform inversion of surface waves

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. EN49-EN61
Author(s):  
Yudi Pan ◽  
Lingli Gao
Keyword(s):  
Objective Function ◽  
Surface Waves ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Initial Model ◽  
S Wave ◽  
Data Set ◽  
Near Surface ◽  
Synthetic Test ◽  
Ground Penetrating

Full-waveform inversion (FWI) of surface waves is becoming increasingly popular among shallow-seismic methods. Due to a huge amount of data and the high nonlinearity of the objective function, FWI usually requires heavy computational costs and may converge toward a local minimum. To mitigate these problems, we have reformulated FWI under a multiobjective framework and adopted a random objective waveform inversion (ROWI) method for surface-wave characterization. Three different measure functions were used, whereas the combination of one measure function with one shot independently provided one of the [Formula: see text] objective functions ([Formula: see text] is the total number of shots). We have randomly chose and optimized one objective function at each iteration. We performed a synthetic test to compare the performance of the ROWI and conventional FWI approaches, which showed that the convergence of ROWI is faster and more robust compared with conventional FWI approaches. We also applied ROWI to a field data set acquired in Rheinstetten, Germany. ROWI successfully reconstructed the main geologic feature, a refilled trench, in the final result. The comparison between the ROWI result and a migrated ground-penetrating radar profile further proved the effectiveness of ROWI in reconstructing the near-surface S-wave velocity model. We also ran the same field example by using a poor initial model. In this case, conventional FWI failed whereas ROWI still reconstructed the subsurface model to a fairly good level, which highlighted the relatively low dependency of ROWI on the initial model.

scholarly journals Wiener estimation of the Green’s function

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. W31-W44 ◽  
Author(s):  
Anton Ziolkowski
Keyword(s):  
Green’S Function ◽  
Impulse Response ◽  
Green's Function ◽  
Seismic Data ◽  
Wiener Filter ◽  
Electromagnetic Data ◽  
Transient Electromagnetic ◽  
Marine Seismic ◽  
Measured Response

I consider the problem of finding the impulse response, or Green’s function, from a measured response including noise, given an estimate of the source time function. This process is usually known as signature deconvolution. Classical signature deconvolution provides no measure of the quality of the result and does not separate signal from noise. Recovery of the earth impulse response is here formulated as the calculation of a Wiener filter in which the estimated source signature is the input and the measured response is the desired output. Convolution of this filter with the estimated source signature is the part of the measured response that is correlated with the estimated signature. Subtraction of the correlated part from the measured response yields the estimated noise, or the uncorrelated part. The fraction of energy not contained in this uncorrelated component is defined as the quality of the filter. If the estimated source signature contains errors, the estimated earth impulse response is incomplete, and the estimated noise contains signal, recognizable as trace-to-trace correlation. The method can be applied to many types of geophysical data, including earthquake seismic data, exploration seismic data, and controlled source electromagnetic data; it is illustrated here with examples of marine seismic and marine transient electromagnetic data.

On estimating the impulse response between receivers in a controlled ultrasonic experiment

Geophysics ◽  
2006 ◽  
Vol 71 (4) ◽  
pp. SI79-SI84 ◽  
Author(s):  
K. van Wijk
Keyword(s):  
Data Processing ◽  
Laboratory Experiment ◽  
Green’S Function ◽  
Detailed Analysis ◽  
Elastic Medium ◽  
Impulse Response ◽  
Green's Function ◽  
Geophysical Data ◽  
Seismic Interferometry ◽  
Band Limited

A controlled ultrasonic laboratory experiment provides a detailed analysis of retrieving a band-limited estimate of the Green's function between receivers in an elastic medium. Instead of producing a formal derivation, this paper appeals to a series of intuitive operations, common to geophysical data processing, to understand the practicality of seismic interferometry. Whereas the retrieval of the full Green's function is based on the crosscorrelation of receivers in the presence of equipartitioned signal, an estimate of the impulse response is recovered successfully with 40 sources in a line covering six wavelengths at the surface.

Land FWI in the Delaware Basin, west Texas: A case study

2020 ◽  
Vol 39 (5) ◽  
pp. 324-331
Author(s):  
Gary Murphy ◽  
Vanessa Brown ◽  
Denes Vigh
Keyword(s):  
Seismic Data ◽  
Waveform Inversion ◽  
Surface Velocity ◽  
High Signal ◽  
Data Sets ◽  
Data Set ◽  
Low Frequencies ◽  
Near Surface ◽  
Delaware Basin ◽  
West Texas

As part of a wide-reaching full-waveform inversion (FWI) research program, FWI is applied to an onshore seismic data set collected in the Delaware Basin, west Texas. FWI is routinely applied on typical marine data sets with high signal-to-noise ratio (S/N), relatively good low-frequency content, and reasonably long offsets. Land seismic data sets, in comparison, present significant challenges for FWI due to low S/N, a dearth of low frequencies, and limited offsets. Recent advancements in FWI overcome limitations due to poor S/N and low frequencies making land FWI feasible to use to update the shallow velocities. The chosen area has contrasting and variable near-surface conditions providing an excellent test data set on which to demonstrate the workflow and its challenges. An acoustic FWI workflow is used to update the near-surface velocity model in order to improve the deeper image and simultaneously help highlight potential shallow drilling hazards.

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