Coding and Decoding

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How do we understand what one person is saying when others are speaking at the same time (the cocktail-party problem)? On what logical basis could one design a machine (technique) for carrying out such an operation (Cherry, 1953)? These two questions are the essence of the famous cross-discipline problem known as the cocktail-party problem, which was formulated by Colin Cherry and his coworkers more than sixty years ago. Figure 1.1 provides an illustration of this problem, which involves several people speaking simultaneously in a room containing two microphones that represent human ears. In all cases, the output of each microphone is a mixture of several voice signals.

Graduate and postgraduate students, along with engineers and scientists working in the fields of neurobiology, biophysics, communications, signal and image processing, computer science, and, more recently, seismology (Ikelle, 2010) have been wrestling with this problem for many decades. Figures 1.2 and 1.3 depict the seismology version of the problem. Seismic waves are generated from four locations almost simultaneously instead of from a single location at a time, as is currently the case. The source signatures are identical for all four shots, but they have different initial firing times. The firing-time delays have been made quite large in this example to facilitate the analysis of the first example of multishot data in this book.

We call the concept of generating waves simultaneously from several locations simultaneous multishooting, or simply multishooting. The data resulting from multishooting acquisition will be called multishot data, and those resulting from the current acquisition approach, in which waves are generated from one location at a time, will be called single-shot data. So multishot data are the coded data, and the decoding process aims to reconstruct single-shot data.

 

 

The end products of seismic-data acquisition and processing are images of the subsurface. As depicted in Figure 1.4, one solution is to decode multishot seismic data. The decoded data can then be imaged to recover the model of the subsurface using current seismic-imaging technology. Another solution is to image multishot data without decoding them. The benefits of directly imaging multishot data, instead of decoding them before imaging, include a reduction in memory and the CPU time needed for imaging processes and an improvement in the signal-to-noise ratio of the resulting images of the subsurface. In the case in which the nonlinear inversion techniques [also known as a full waveform inversion, from the work of Tarantola (1987)] are used to image seismic data, we can significantly reduce the cost of the forward problem by using multishot data instead of the current single-shot data, hence removing one of the major impediments of the application of the nonlinear inversion to seismic data.

 

 

The simultaneous multishooting concept can also be used to improve the ways in which we acquire data. For instance, it can be used to improve the spacing between shot points, especially the azimuthal distribution of shot points, as illustrated in Figure 1.21, and therefore to collect true 3D data [i.e., the full-azimuth survey, as described in Ikelle and Amundsen (2005, 2017)]. In fact, current 3-D acquisitions—say, marine, with a shooting boat sailing along in one direction and shooting only in that direction—do not allow enough spacing between shot points for full azimuthal coverage of the sea surface or land surface.

 

 

Illustrations of a cross-talk in the pseudo-receiver gathers. The firing times of the sources are the same for all the multishots; that is, the time delays occurring in the firing of single shots are the same at all 160 multishot locations. Note that the cross-talk in this case is a set of coherent events.

 

Illustrations of a cross-talk in the pseudo-offset gathers.

(d) The actual zero-offset gather. (e) The pseudo-offset gather.

(f) The cross-talk associated with the pseudo-offset gather.

Note that the cross-talk in this case is a set of coherent events.

 

 

Illustrations of a cross-talk in a pseudo-offset gather.

The firing times of one of the single shots randomly taken between 50 and 200 ms

( (d) The actual zero-offset gather. (e) The pseudo-offset gather.

(f) The cross-talk associated with the pseudo-offset gather.

Note that the cross-talk in this case is incoherent.

 

 

Illustrations of a cross-talk in the pseudo-receiver gathers.

The firing times of one of the single shots is a sawtooth wave function.

(d) The actual zero-offset gather. (e) The pseudo-offset gather.

(f) The cross-talk associated with the pseudo-offset gather.

Notice that the events of the cross-talk are also coherent in this case but their dips are very different from
those of the events of single-shot gathers.

 

 

(a) A multishot response for an inhomogeneous isotropic nonlinear elastic medium. We have a 2D model consisting of a homogeneous isotropic linear elastic layer sandwiched between two isotropic linearly elastic half-spaces. The multishot is made of two single shots. The distance between these two single-shot points is 25 m. The time delay between the firing times of the two shot points is 25 ms.

(b) The residuals between the multishot response and the sum of the single-shot responses.

(c) A multishot response for an inhomogeneous isotropic nonlinear elastic medium with the same
parameters, except for the time delay between the firing time of the two shot points, which is now 50 ms.

(d) The residuals between the multishot response and the sum of the single-shot responses for the
nonlinear elastic medium.

One can still use a small distance between the single-shot points within a given multishot array as long as the time delay between the firing times of single shots are large enough.

 

 

On the morning of September 19, 1985, a magnitude 8.1 earthquake struck some states of Mexico, including Mexico City, causing the deaths of at least 10,000 people. The earthquake was felt over 825,000 square kilometers—as far away as Los Angeles and Houston in the United States. The earthquake actually occurred in the Pacific Ocean off the coast of the Mexican state of Michoacan—a distance of more than 350 km from Mexico City, which is in the Cocos plate subduction zone. As we can see on the tectonic map (Figure 1.39), the Cocos plate pushes against and slides under the North American plate, primarily along the coasts of the Mexican states of Michoacan and Guerrero.

It was later discovered that this earthquake was a two-event phenomenon (i.e., two epicenters), the second movement occurring 26 seconds after the first (Anderson et al., 1986; Astiz et al., 1987; Murillo, 1995), so there were essentially two nearly simultaneous earthquakes. Because of the interference of these two events, the ground shaking lasted more than five minutes in places along the coast, and parts of Mexico City shook for three minutes.

 

 

Chapters 2 and 3 include a list of mathematical notations, definitions, and results used frequently throughout the book. In addition to providing basic mathematical tools for decoding, Chapter 2 discusses solutions to the system of equations correspond to the decoding of multishot data for the particular case in which the mixtures are instantaneous and the number of sweeps is equal to the number of single shots which compose a multishooting experiment for a given sweep. In Chapter 2 we discuss linear instantaneous mixtures. Chapter 3 builds on the results of Chapter 2 to propose solutions to equation, which deals with convolutive mixtures.

Chapter 4 deals with the decoding methods for underdetermined mixtures. Several solutions are presented which take advantage of the source encoding, acquisition geometries, and classic processing tools to compensate for the fact that the systems of equations are underdetermined. Chapter 5 builds on the results of Chapters 2 and 3 to propose decoding solutions to nonlinear mixtures.

Chapter 6 deals with the processing of multishot data without first decoding them. We address the problem of multiple attenuation of multishot data and the issues of velocity estimation, time imaging, and depth imaging. We also address the issue of using the process of coding and decoding to reduce the costs of numerically generating 3D seismic data with the finite-difference modeling (FDM) technique. We take advantage of the fact that in this case the model of the subsurface is known.