## Calculation of Multivalued Ray Theory Travel Times in 3-D Structures.

**Petr Bulant**
### Introduction

The thesis is composed of three author's papers
written during the three-year period of his PhD studies.
The research was devoted mostly to solving the
problems of the ray-theory based computation of seismic wavefields
propagating in complex three-dimensional laterally varying
block structures.
The author continued in the development of the
sophisticated two-point ray tracing algorithm, suggested in
his master thesis.
The robust algorithm was designed to provide all the two-point rays
of given wave for given source-receiver configuration, including multiple
arrivals.
The main attention was paid to the
reliability of the algorithm, and also to its proper and efficient coding and
to the numerical tests of its computer implementation. Resulting
code was incorporated into
already existing Fortran 77 subroutine package ` CRT`.
Considerable attention has then been paid to the development of
a new method for computation of multivalued arrivals at nodes of 3-D grids
in complex 3-D structures.
The method is in principle equivalent to the wavefront tracing method.
It is based on construction of ray tubes,
composed of ray cells, through
the model volume, followed by interpolation inside individual ray cells.
The discussion of the results is classified according to these two topics.
#### 1 Two-point ray tracing

Recent ray-theory based numerical methods enable rays of various
seismic waves to
be computed in 3-D laterally varying inhomogeneous media with a block
structure. Given the initial conditions for the ray, they can be
used to compute the ray trajectory, the time
of propagation, and the slowness vector along any ray. They also
allow the ray-theory amplitudes to be computed and synthetic
seismograms to be constructed. Thus, the problem of
* initial-value ray tracing* is, to some extents, solved.
On the other hand, we are usually more interested in computation
of travel times and other quantities at given receivers
(the points of given coordinates). The task of
computation of the ray connecting a source with the receiver,
* two-point ray tracing,* is much more complex. There are
basically two approaches to the solution of the
two-point ray tracing problem: bending and shooting. The bending method
is dependent on the approximate ray tracer estimating the initial rays
to be bent. In a case of multiples it thus need not provide us with
complete solution (i.e. with all the arrivals). Thus it is more reliable to
use any of shooting methods
for solving the
two-point ray tracing problem.
In the shooting method, it is necessary to find
the boundaries between individual regions of
the same history of rays of the elementary wave under consideration.
The ray history is an
integer-valued function, which assigns the rays to various
groups according to the
structural blocks and interfaces through which the ray has
propagated, as well as according to the position of its endpoint (at the
bottom of the model, at the sides of the model, at the surface,
overcritical incidence at an interface, ...), and the caustics
encountered.
The procedure of identification of individual regions
must be done very carefully in order
to avoid omitting any of the regions,
which may contain two-point rays.
Following these ideas, the two-point ray tracing program being described
is based on the procedures to demarcate the boundaries
between individual regions in the ray-parameter domain, and on the proper
triangularization of these regions.
The boundaries are demarcated by pairs of so called boundary rays,
each of them being situated
at the opposite side of the boundary. The mutual distance of these two rays
is kept under a priori prescribed (very small) limit,
because all the two-point rays
situated inside the demarcation belts will be lost. The distance between
individual pairs of boundary rays along the boundary is also controlled and kept
under prescribed limit, depending on the curvature of the boundary being
demarcated. This very accurate demarcation of the boundaries should
ensure that any of the regions, which is greater than prescribed quantities
(i.e. distances of boundary rays and of pairs of rays), will not be omitted.
After demarcation of the boundaries, individual regions are triangularized
by approximately equilateral triangles with the maximum length of the sides given.
Two-point rays are then iteratively identified within
individual triangles using paraxial ray approximation.
The described two-point ray tracing method was suggested in [1], [2]
and [4] and described in detail in [5], the importance of careful
demarcation of the boundaries between individual regions was discussed
in [3], differences between various triangularizations of
individual regions were shown in [6] and [11].
#### 2 Calculation of multivalued traveltimes at gridpoints of 3-D grids

As was discussed above, the described two-point ray tracing algorithm
is based on demarcation of the boundaries
between individual ray-parameter regions of
the same history of the elementary wave under consideration,
and on the triangularization of these regions.
Once the individual regions of ray take-off parameters are triangularized,
we can use the triangles to construct ray tubes
through the model volume. Each ray tube is then bounded by three rays,
the vertices of the corresponding triangle in the ray-parameter domain.
Each ray tube is composed of ray cells.
Each ray cell corresponds to the space between two consecutive wavefronts
and/or structural interfaces.
Travel times
and other quantities may be then interpolated inside individual ray cells,
like in the wavefront tracing method.
The described method
is a useful extension of two-point ray tracing programs
based on triangularization. It provides interpolated values of
ray-theory travel times and other quantities at arbitrary 3-D grids located
in the seismic model, including multiple arrivals.
Inside blocks a ray cell is determined by 6 points
located on the corresponding three rays, and it is determined
by 4, 5 or 6 points in the vicinity of structural interfaces.
The described interpolation method
then reduces into proper organization of the loop over the ray cells
and in proper decision, which gridpoints of the target grid
are located in the ray cell being examined. The interpolation formula
for individual gridpoint is connected with the system of equations
for decision of whether the gridpoint is located in the selected ray cell.
Thus the interpolation of the travel time and other quantities
is very straightforward once
the gridpoint is identified as being located in the ray cell.
Described method for computation of multivalued arrivals was suggested in [6] and [11],
and the first numerical experiments were performed ([9],[10]). Recently the method
is being coded in Fortran 77 as a post-processing program to the
` CRT` program [8].
### List of author's publications

[1]
Bulant, P.,
1994,
Two-point Ray Tracing in 3-D.
Diploma Thesis,
Dep. Geophys., Charles Univ., Prague.

[2]
Bulant, P.,
1995,
Two-point ray tracing in 3-D.
In: Seismic Waves in Complex 3-D Structures, Report 3, pp. 37-64,
Dep. Geophys., Charles Univ., Prague.

[3]
Bulant, P.,
1995,
Numerical examples of two-point ray tracing in 3-D.
In: Seismic Waves in Complex 3-D Structures, Report 3, pp. 65-92,
Dep. Geophys., Charles Univ., Prague.

[4]
Bulant, P., & Klimes, L.,
1996,
Two-point ray tracing in 3D heterogeneous structures.
Extended Abstracts, **C045**, EAGE 58th Conference and Technical Exhibition, Amsterdam.

[5]
Bulant, P.,
1996,
Two-point ray tracing in 3-D.
Pure and Applied Geophysics **148**, 421-446.

[6]
Bulant, P.,
1996,
Two-point ray tracing and controlled initial-value ray
tracing in 3-D heterogeneous block structures.
In: Seismic Waves in Complex 3-D Structures, Report 4, pp. 61-76,
Dep. Geophys., Charles Univ., Prague.

[7]
Bulant, P., & Klimes, L.,
1996,
Examples of seismic models. Part 2.
In: Seismic Waves in Complex 3-D Structures, Report 4, pp. 39-52, Dep. Geophys., Charles Univ., Prague.

[8]
Bulant, P., Cerveny, V., Klimes, L. & Psencik, I.,
1996,
MODEL version 5.00 & CRT version 5.00 (2 floppy disks).
In: Seismic Waves in Complex 3-D Structures, Report 4, p. 91, Dep. Geophys., Charles Univ., Prague.

[9]
Bulant, P.,
1997,
Controlled initial-value ray tracing and two-point ray tracing in 3D.
Extended Abstracts, **P025**, EAGE 59th Conference and Technical Exhibition, Geneva.

[10]
Bulant, P.,
1997,
Two-point ray tracing and
controlled initial-value ray tracing in 3-D heterogeneous block structures.
Expanded Abstracts, 67th SEG International Exposition and Annual Meeting,
Dallas, in press.

[11]
Bulant, P.,
1997,
Two-point ray tracing and Controlled initial-value
ray tracing in 3-D heterogeneous block structures.
Geophysics, submitted.

[12]
Bulant, P.,
1997,
Recent development of the shooting algorithm.
In: Seismic Waves in Complex 3-D Structures,
Report 6, submitted.

PhD Thesis, Department of Geophysics,
Charles University, Prague, 1997.

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