Intro The aim of prestack seismic imaging is to obtain an estimate of the reflectivity of the subsurface from a seismic survey conducted on the surface. A basic requirement for this process is the ability to propagate a seismic wavefield through the subsurface. In order to propagate the wavefield we need to know the subsurface velocity structure. This appears to create an impasse, we need to know the subsurface structure in order to image the subsurface structure. Fortunately this problem is not insurmountable, the imaging and the velocity estimation can be performed as a coupled process. As the estimate of the velocity structure improves, the image improves, and vice versa. A well known testbed for the issues involved in prestack imaging and velocity estimation is the ``Marmousi'' dataset. The dataset was originally created by the Institute Fran c ais du P e trole as a test of velocity inversion methods. It was distributed as a ``blind ...
Modeling in polar coordinates using one-way wave equations
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Polgreens In this chapter I show how to calculate approximate Green's functions using one-way wave equations in polar coordinates. I use one-way extrapolators to downward or outward continue an initial condition until a solution is obtained in the whole region of interest. The solutions obtained using one-way equations are approximations to the the full Green's functions. However, they contain the information that is usually of greatest interest, the amplitude and phase of the primary arrivals. Two-way wave equations ...
Modeling, adjoint modeling, and migration in V(z) media,
using mono-frequency Green's functions
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VZmodmig Modeling and migration can be simply expressed in terms of the Green's functions of the wave equation. In the previous chapter I discussed calculating mono-frequency Green's functions using one way wave equations in polar coordinates. The complete outgoing Green's function is the superposition of all the mono-frequency Green's functions. Once Green's functions at all frequencies have been estimated they can be used for both modeling and migration. However, most practical implementations do not use the Green's function explicitly. The reason for this is that, using currently available computers, it is too expensive to routinely calculate and store these functions for all frequencies and all surface locations. In this chapter I exploit the fact that when the velocity model is independent of the horizontal coordinate, only one set of Green's functions needs to be calculated. They can be used for prestack and poststack modeling and migration. Because my Green's functions are ...
Band-limited Green's functions
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TTparam A parametric model of the Green's function When Green's functions are calculated at a few distinct frequencies they need to be interpolated to all frequencies to be useful in seismic imaging. The interpolation requires a model of the amplitude and phase behavior of the Green's functions. The most common model is one that parameterizes the Green's function in terms of ``arrival time'' and ``amplitude''. This parameterization has the advantage that these quantities can be used directly in time domain Kirchhoff migration or modeling schemes. The Green's function can be characterized by one or more ``events'' that arrive at each location. Parameterization by a traveltime implies that the phase is a linear function of frequency for each event. This simple model can also be extended to incorporate frequency dispersion and attenuation in the Green's function. In a weakly dispersive model the phase becomes a smooth function of frequency rather than a linear function. ...
Band-limited Green's functions in the Marmousi model
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MarmTT In this chapter I study different ways of calculating Green's functions in the Marmousi model. I compare traveltimes and amplitudes computed using the band-limited traveltime method, with full one-way and two-way wavefields, and with traveltimes calculated by finite difference solutions of the eikonal equation and paraxial ray tracing. Figure marmslow is a plot of the slowness field for the model. It has a complicated faulted structure that impedes imaging of the anticlinal trap at the bottom of the model. marmslow height=7.5in,width=5.5in . The slowness model for the Marmousi dataset. The slowness model for the Marmousi dataset. Comparison of band-limited and first-arriving traveltimes Figure marm2-tt shows contour plots of three sets of traveltimes superimposed on the Marmousi velocity field. At the top are the first arriving traveltimes, calculated by a finite difference solution to the Eikonal equation. In the middle are maximum amplitude traveltimes from paraxial ray tracing. At the bottom are band-limited traveltimes, calculated using my method. The initial impression is that the first ...
Kirchhoff migration with band-limited traveltimes
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MigMarm In this chapter I compare the images created by Kirchhoff migration using Green's functions computed by three different methods. First arrival traveltimes from a finite difference solution to the eikonal equation. Maximum amplitude arrival traveltimes, amplitudes and phases from paraxial raytracing. Maximum energy traveltimes amplitudes and phases computed by my band-limited Green's function method. itemize Kirchhoff migration in the time domain The modeling equation for a scalar scattering function was previously stated in the frequency domain. If we wish to use this method in the time domain it must be inverse Fourier transformed over frequency: The approximate, single-event form for the combination of the two Green's functions can be written as: