Kirchho Simulation, Migration, and Inversion using Finite
Dierence Traveltimes and Amplitudes
William W. Symes, Roelof Versteeg, Alain Sei, and Quang Huy Tran
July 19, 1994
Abstract
High frequency asymptotic approximation of the acoustic Green's function leads to
ecient modeling and migration methods for primaries only reection seismograms.
A volume (as opposed to interface) oriented description of reectivity allows symmetry between modeling and migration. With proper selection of amplitudes and discretization, the migration and modeling operations are mathematically adjoint, hence
suitable for use in iterative inversion algorithms. Further simplications are possible
under the additional assumptions that incident and reected rays carry signicant
high-frequency energy propagate near the vertical and arc rst arrivals. Then the
necessary traveltime and amplitude tables may be computed by ecient high order
nite dierence solution of the eikonal and transport equations. Careful ordering of
the operations results in an accurate algorithm far more ecient than direct nite
dierence solution of the primaries-only (linearized) wave equation.
1 Introduction
Imaging methods based on high frequency asymptotic representation of the seismic waveeld play an important role in the data processing repertoire, and are still the subject of
research (e.g. [13]). In common with many other authors, and somewhat inaccurately, we
will call such methods Kirchho.
Kirchho imaging methods enjoy several advantages which perhaps explain their persistent popularity. Amongst these are their ability to produce images only in a portion
of the subsurface, using only a fraction of the available data. Also, they provide natural
imaging methods for data binning schemes corresponding to infeasible physical experiments, notably the common oset sort. Finally, Kirchho methods enjoy signicant cost
advantages over their chief rivals, imaging techniques based on nite dierence solution
of paraxial (one-way) or full (two-way) acoustic wave equations, even when used to image
the full subsurface volume using the full data volume (e.g. [13]; see also comparison below). For some of the same reasons, the high frequency asymptotic representation of the
The Rice Inversion Project, Department of Computational and Applied Mathematics, Rice University,
Houston TX 77251-1892
1
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Symes, Versteeg, Sei, Tran
acoustic Green's function has been studied as the basis for modeling of seismic primary
reections (for example [6], [5]) and (noniterative) inversion or true-amplitude migration
([2], [3], [4]).
Kirchho methodology also suers from several drawbacks, as one might expect from
its basis in an approximation with well-understood and limited domain of validity. For
example, the modeling methods mentioned above rely on an interface-based description of
reection: reectivity is modeled as a jump in values of otherwise smoothly varying elastic
parameters (velocities, density, : : : ) across a smooth interface, whereas the inversion (and
migration) methods produce volumetric reectivity, i.e. amplitudes distributed throughout
the subsurface. This mismatch of modeling input and inversion output makes the success
of the inversion methods dicult to quantify.
More seriously, the bulk of Kirchho modeling, imaging and migration methods assume that energy arrives at the target along a single wavefront. White ([21]) showed that
moderately heterogeneous velocity distributions produce multiple wavefronts and the concommitant transmissin caustics at moderate distance from the energy source with high
probability. Geoltrain and Brac (1993) criticize the use of diraction sum migration methods based on unique (rst arrival) wavefronts. Although this topic stills seems controversial
([16], [13]) there is no doubt that, from the point of view of inversion, all signicant energy
paths must be taken into account.
Finally the use of the high frequency asymptotic representation of the acoustic Green's
function implies a description of reection either as interfacial (as in [6], [5]), or as volumetric, and in the latter case the reectivity must be separated in scale from the velocity structure used in propagation. That is, for volumetric reection modeling by high-frequency
asymptotics, the velocity structure must be smooth on the wavelength scale, whereas all
material heterogeneity at the wavelength scale is treated via rst order perturbation theory. In this way only relatively weak primary precritical reections are modeled correctly.
Given the predominantly horizontal orientation of reectors in the earth, this means that
incident and reected energy must travel near the vertical to be modeled correctly by such
schemes.
The purpose of the present article is to collect in one place descriptions of all components
of an accurate and ecient Kirchho algorithm for modeling and migration of marine
reection seismograms.
We accept the last two limitations mentioned above; we will model (and migrate)
only energy traveling near the vertical in smoothly varying velocity structures, describe
volumetric reectivity via rst order perturbation theory (hence model only precritical primary (and weak!) reections from near-horizontal reectors), and guarantee the results
only when unique smooth wavefronts impinge on the reector, i.e. when no transmission
caustics occur near reecting points. These limitations restrict the domain of our approach to velocity and reectivity structures with only slow horizontal variation. However
sedimentary basins contain many such nearly at regions, so that our approach is perhaps
not without utility. Moreover we suspect that it is in reality no more limited in this way
than most other Kirchho implementations.
Our algorithm is distinguished from others mentioned above in several ways. The de2
Kirchho Simulation
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scription of reectivity is volumetric for both modeling and migration, i.e. the reectivities
are depth sections, just as are (depth{) migrated images. The modeling and migration
algorithms are mathematically adjoint, even after discretization, hence may be used as the
basis of iterative (linearized) inversion algorithms to minimize various measures of data
t error. While we give an explicit treatment only of constant density acoustics, general
acoustics and linear elasticity may be treated in an exactly analogous fashion. Iterative
inversion algorithms based on these multiparameter theories can readily incorporate constraints, which the direct inversion formulae of Kirchho type (e.g. [2]) do not. This class
of algorithm may provide a robust and exible tool for AVO interpretation. Finally, we
compute both traveltimes and amplitudes using ecient nite dierence schemes, rather
than ray tracing. These schemes provide the necessary inputs for the Kirchho formulae
directly on the image grid, without interpolation. While several authors have followed
this general approach to computing diraction trajectories (e.g. [13] and references cited
therein), our method for computation of traveltimes and (especially) amplitudes appears
to be new. The nite dierence schemes used here also have limited domain of functionality, but the limits are the same as those already accepted for the methodology in general
| smooth velocities, energy propagates near vertical, only rst arrival traveltimes and
corresponding amplitudes computed.
Since we work in the time domain, a rather ecient natural method for numerical
evaluation of the Kirchho integral is available. We present a simple pair of formulas for
modeling and migration, which are adjoint to each other in the precise sense necessary to
allow their use in least-squares inversion algorithms and of second order accuracy in the
space and time steps. These formulas involve indirect addressing, hence do not vectorize,
but yield an algorithm suitable for use on superscalar workstations and other high performance scalar processors. Parallelization also appears eminently feasible. We remark that
traveltime and amplitude computations are completely vectorizable.
Given accurate components, the primary reection seismogram computed via the Kirchho method is remarkably accurate, for models within its domain of validity. We compare
Kirchho seismograms with those produced from the same model by nite dierence solution of the (full, two-way) linearized acoustic wave equation. We use a suite of models
of increasing complexity, with dimensions characteristic of exploration seismology. Until
multiple arrivals begin to be important in the data, the Kirchho and nite dierence
results are remakable close. In fact, the Kirchho result, being based on quite accurate
traveltimes, does not suer the grid dispersion of the nite dierence scheme, with the remarkable consequence that the nite dierence results appear to converge to the Kirchho
results as the grid is rened!
We also compare the computational cost of Kirchho and (full, two-way) nite dierence simulation. In all these comparisons, we have used a nite dierence scheme of order
two in time and four in space. Somewhat more ecient schemes are available (e.g. [9],
[15]) and would change the qualitative outcome of the comparisons somewhat. However,
we believe that the qualitative conclusion would remain the same. For simulation of a
single shot, the costs are comparable. However a large part of the cost of the Kirchho
computation lies in the generation and access of traveltime and amplitude tables. With
minor geometrical restrictions on acquisition and careful ordering of operations, these can
3
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Symes, Versteeg, Sei, Tran
be amortized over many shots, leading to a dramatic reduction in cost for multishot simulation. For simulation of lines of realistic size, the cost of Kirchho simulation in our
experiments was roughly 5% of the cost of the comparable nite dierence calculation.
Cost and accuracy comparison with paraxial or F-X modeling and migration would also
seem natural We did not made such a comparison, partly because current paraxial codes
perform migration only, or do not control properly the amplitudes of primary reection
modeling (e.g. [13], [1], [10], [8]). Paraxial approximations to the acoustic Green's function
model energy propagation near the vertical, as does the Kirchho approach taken in
this paper. They also construct multiple wavefronts, which rst-arrival-based Kirchho
methods do not. Thus the paraxial methods should be more successful as kinematic
migration tools in complex media, as pointed out by [12]. However at present no assurance
seems to be available that the amplitudes of these paraxial Green's functions approximate
those of the true acoustic Green's function, so that paraxial methodology does not appear
well adapted to inversion (although this situation may change). Since our principal interest
lies in inversion, we did not make a comparison with any paraxial algorithm.
We begin the account of our method with a brief reprise of the Kirchho formulae for
2D and 3D primaries only constant density acoustic modeling and migration. We dene
this migration operator to be the mathematical adjoint to the modeling operator, and
ensure that this property survives the discretization process, and that the discretization
is accurate of second order. The next two sections present our method for computing
traveltimes and amplitudes. The traveltime computation is similar to others reported in
the literature since [20]. However our specic choice of scheme appears to be new. We
use the so called ENO schemes introduced by Osher and colleagues ([17]). We show that
apertures may be controlled and 2D and 2.5D amplitudes computed as well with this set
of tools. Having assembled the necessary elements, we illustrate the method in several
examples, compare with nite dierence calculations, and assess the computational cost.
2 Kirchho Modeling and Migration
2.1 Derivation
The weak primaries only or linearized acoustic constant density model of the seismic
reection response separates the velocity eld into slowly varying, or smooth, and rapidly
varying, or oscillatory, components. In this paper we shall treat mostly the 2D case,
with space coordinates (x; z ). The rapidly varying part of the velocity eld (r(x; z ) the reectivity eld) is treated as a rst order perturbation of the slowly varying part
(v (x; z ) the velocity eld). Mathematically, vtotal = v + v , and r = v=v . That is, r is
the oscillatory relative perturbation of the smooth v , and is dimensionless.
The precise division of a velocity eld into smooth and rough parts is problematic
([7]) unless a clear dichotomy exists between spatial frequency components much longer,
respectively comparable to, than a typical wave length. Nonetheless this perturbational
or single scattering description of the reection process is the basis for most seismic data
processing, in particular for migration. The scale dichotomy between velocity and reectivity is reected in the common conceptual division: velocity is responsible for kinematics,
4
Kirchho Simulation
Trip 94
reectivity for dynamics.
The acoustic eld p of an isotropic point radiator with source f (t) located at (xs ; zs)
satises
!
1 @ 2 r2 p(x; z; t; x ; z ) = f (t) (x x ; z z )
s s
s
s
v (x; z)2 @t2
p 0; t << 0
The rst order perturbation of p resulting from the perturbation v of v satises
!
1 @ 2 r2 p(x; z; t; x ; z ) = 2r(x; z ) 1 @ 2 p(x; z; t; x ; z )
s s
s s
v (x; z)2 @t2
v (x; z)2 @t2
p 0; t << 0
Our aim is to evaluate the perturbational or primaries only eld p at receiver positions
(xr ; zr ). For present purposes we ignore the presence of a boundary at the surface of
the earth, and assume that the velocity v is constant and the reectivity r vanishes for
z < max(zs ; zr).
The Kirchho representation begins, like so many good things, with integration by
parts. The causal Green's function G(x; z; t; xs; zs ) is the solution of the unperturbed
problem above for f (t) = (t); in fact p = f G and p = f G. Using Green's identity
and the wave equation for p we nd that
!
Z Z Z
2
@
1
dxdzdt 2r(x; z) v (x; z)2 @t2 G(x; z; t; xs; zs)
p(xr ; zr ; tr ) = f t
G(x; z; tr t; xr; zr )
where we have used the anticausal Green's function G(; ; tr t; ; ) to keep the domain
of integration bounded.
The next step is to replace the Green's function by the rst term in its progressing wave
expansion. This is a high frequency approximation, as the terms dropped are smoother
than the one kept. In two space dimensions the leading singularity is proportional to the
generalized function
t+1=2 = t 1=2 H (t)
H (t) being the Heaviside function. The progressing wave expansion is
r
G(x; z; t; xs; zs) = a(x; z; xs; zs)t+1=2 (t (x; z; xs; zs)) + :::
where the elided terms are smoother, therefore smaller after convolution with a high frequency source, than the leading term. The traveltime eld solves the eikonal equation
jr j = 1
The initial condition for is
v
lim d v = 0 as d ! 0
5
Symes, Versteeg, Sei, Tran
Trip 94
p
with d = (x xs )2 + (z zs )2 the Euclidean distance. The amplitude eld a solves the
transport equation
2r ra + ar2 = 0
with initial condition
r 1
1
lim d a 2 2vd = 0 as d ! 0
Since we will consider also the traveltime and amplitude from the receiver position
(xr ; zr ), we will make the notation more compact by writing s (x; z ) (x; z ; xs; zs ), etc.
Replacing the Green's functions by the rst terms in the progressing wave expansions,
we get
p(xr ; zr ; tr ; xs; zs ) '
@ 2 Z Z dxdza (x; z)a (x; z) 2r(x; z)t 1=2 t 1=2 (t (x; z) (x; z))
f t @t
r
r
s
r
s
+
2r
v (x; z)2 +
r
[11], p. 116, formula (3') gives
t+1=2 t+1=2 =
2
1
2
H (t)
= H (t)
whence
p(xr; zr ; tr; xs; zs) '
Z Z
dxdzA(x; z; xs; zs; xr; zr)r(x; z) (tr r (x; z) s(x; z))
f t @t@
r
A(x; z; xs; zs; xr; zr) = as (vx;(x;z)az)r(2x; z)
r
This is the Kirchho representation of the primaries only acoustic reection eld.
We are going to view the correspondence between the reectivity r and the sampled
primaries only eld p as a linear operator L : r 7! p. We will also idealize reality by
assuming that the receiver positions form a continuum indexed by xr , with zr xed. We
concentrate on the computation of this map and its adjoint for a single shot, and drop
(xs ; zs ) from the notation. Thus L is the operator which takes as input a reectivity eld
r(x; z) and produces a shot (common source) gather Lr(xr; tr )as output.
The L2 adjoint of L is a prestack migration operator (see eg. [14], [18]). It is dened
by
Z Z
Z Z
dxr dtr (Lr)(xr; tr )g (xr ; tr ) =
dxdzr(x; z)(Lg (x; z))
with g (x; t) being the seismic record. Integration by parts shows that
Lg (x; z) =
Z Z
dxrdtr A(x; z; xs; z; xr; zr ) f t @t@ g (xr; tr )
r
6
Kirchho Simulation
=
Z
Trip 94
(t s(x; z) r(x; z))
@
dxrA(x; z; xs; zs; xr; zr) f t @t g (xr ; s(x; z) + r(x; zr ))
r
Here
f(t) = f ( t)
2.2 Discretization
Because the integral over t can be eliminated, it turns out to be easier to rst discretize the
adjoint operator L (the Kirchho migration operator), and then compute the discretized
Kirchho modeling operator L as its adjoint.
We introduce the total traveltime eld
(xr ; x; z) (x; z; xr; zr ) + (x; z; xs; zs )
(recall that xs ; zs and zr are xed for the moment, hence we drop them from the notation).
g is assumed to vanish near the edges of the xr ; tr grid, and likewise r near the edges
of the x; z grid. In practice this should be ensured by including a mute or cuto function
in the denitions of L and L, on both sides.
Assume that all grids are uniform. Write
uk;l ' u(xmin + (k 1)x; zmin + (l 1)z)
for functions u(x; z ) of the space coordinates, and
gi;j ' g (xr;min + (i 1)xr; tmin + (j 1)tr )
k;l;i ' (xr;min + (i 1)xr; xmin + (k 1)x; zmin + (l 1)z)
A~k;l;i ' A~(xr;min + (i 1)xr; xmin + (k 1)x; zmin + (l 1)z)
Since the integrand vanishes at the boundaries of the grid, the trapezoidal rule applied to
the formula above dening L is equivalent to the left hand rule:
(Lg )k;l = xr
X
i
A~k;l;i @ (f@tt g ) (xr;min + (i 1) xr; k;l;i ) + O(x2r )
r
r
We approximate the evaluation in tr by piecewise linear interpolation, then the tr -derivative
by the central dierence formula, in both cases committing a second order error: set
k;l;i
Jk;l;i = int t
r
Tk;l;i = k;l;i tr Jk;l;i
Dt0 Uj = Uj+12tUj 1
r
r
7
Symes, Versteeg, Sei, Tran
Trip 94
for any sampled function U of tr . Then
@ (f t g ) (x
r;min + (i 1) xr ; k;l;i )
r
@tr
= (1 Tk;l;i)Dt0 (f t g )i;J
r
r
So nally we obtain
k;l;i
+ Tk;l;i Dt0 (f t g )i;J
(Lg )k;l = xr
Tk;l;i)Dt0 (f t
r
r
X
i
0
+ Tk;l;iDt (f t
k;l;i
+1
+ O(t2r )
A~k;l;i [(1
g )i;J
g )i;J +1 ] + O(t2r ) + O(x2r )
Throwing away the discretization error we take this expression as the denition of the
(second-order) discretization of L.
The correct inner products to use for evaluation of the adjoint of this adjoint are
r
r
k;l;i
r
r
< g1; g2 >x ;t = xr tr
r
r
< u1; u2 >x;z = xz
k;l;i
X
i;j
X
k;l
Thus L will be dened by
g1;i;j g2;i;j
u1;k;lu2;k;l
< Lr; g >x ;t =< r; Lg >x;z
r
= xz
X
k;l
r
r;k;lxr
X
i
A~k;l;i
[(1 Tk;l;i)Dt0 (f t g )i;J + Tk;l;i Dt0 (f t g )i;J
X
X
= xz r;k;lxr tr A~k;l;i r
r
k;l;i
r
r
k;l;i
+1 ]
i;j
Tk;l;i)Dt0 (f t g )i;j j J =tr + Tk;l;iDt0 (f t g )i;j j J 1 =tr ]
X
= xr tr Dt0 (f t g )i;j i;j
X
xz r;k;l A~k;l;i [(1 Tk;l;i )j J =tr + Tk;l;i j J 1 =tr ]
k;l
k;l
[(1
r
r
k;l;i
r
r
k;l;i
r
r
k;l;i
k;l;i
The convolution with the source time series f is adjoint to convolution with f on the discrete level as well, and the central dierence operator Dt0 is skew adjoint for gridfunctions
vanishing near the grid boundaries, as has been assumed for all of the major functions
used here. Thus the above is equal to
r
= xr tr
8
<
z X r A~ [(1
Dt0 f t : x
tr k;l ;k;l k;l;i
r
X
i;j
gi;j Tk;l;i)j
r
8
J
k;l;i
+ Tk;l;ij J
k;l;i
9
=
1 ];
Kirchho Simulation
Trip 94
from which we can read o the second-order approximation to L, adjoint at the discrete
level to that derived above for L .
Concerning the level of discretization necessary to produce a given level of accuracy,
we oer only a few preliminary observations; a more thorough analysis is in progress. The
integration rule used here is the (compound) trapezoidal rule. The maximum error E of
this rule is given by the formula
2
t f 00(t)
E t max
12
With f (t) = sin 2!t, the error is bounded roughly by 4! 2 t2 = 4=G2, where G is the
number of gridpoints per wavelength. Thus to achieve an error of about 5% in evaluation of
the Kirchho integral, roughly 10 gridpoints per wavelength are necessary. This estimate
is of course somewhat pessimistic - since the Kirchho representation is a path integral,
the frequency content of the integrand along the path is actually at the determinant of
accuracy. For small angle reection and near-horizontal reectors (or their reections, in
case of migration) the variation of the integrand along the Kirchho integration paths
is considerably lower in frequency than the actual maximum spatial frequency. So one
would expect reasonable results from somewhat coarser grid spacing than is indicated by
the basic error analysis.
2.3 Algorithms
Next we state explicitly discrete algorithms for evaluation of the Kirchho simulation and
migration formulae. First we recall some notation.
Indices:
i is the xr index
j is the tr index
k is the x index
l is the z index
Gridfunctions:
k;l;i = s;k;l + r;k;l;i is the total travel time
h
i
Jk;l;i = int t is the integer part of the number of time steps
Tk;l;i = k;l;i tr Jk;l;i is the fractional part of the number of time steps
k;l;i
r
ar;k;l;i
Ak;l;i = as;k;l
2
v
k;l
is the geometrical spreading factor
9
Symes, Versteeg, Sei, Tran
Trip 94
rk;l is the input reectivity eld for modeling
gi;j is the input shot record for migration
A second-order 2D Kirchho simulation (forward modeling) formula is given by
(Lr)i;j = Dt0 f
r
8
<
X
t : xtz rk;lAk;l;i[(1
r k;l
Tk;l;i)j
r
J
k;l;i
+ Tk;l;ij J
Our modeling algorithm is given by:
1. Precompute the scaling eld
xz
Bk;l r;k;l
2
v t
k;l
2. For each i (receiver index):
(a) initialize output trace to zero
(b) for each k; l (depth point):
i. compute
r
Pk;l;i = Bk;las;k;l ar;k;l;i
ii. calculate Jk;l;i ; Tk;l;i
iii. compute Q = Pk;l;i Tk;l;i
iv. add to output sample Jk;l;i the quantity
P Q = rk;lAk;l;i (1 Tk;l;i)
v. add to output sample Jk;l;i + 1 the quantity
Q = rk;lAk;l;iTk;l;i
(c) apply centered time dierence operator Dt0 to output trace
(d) convolve output trace with source wavelet f
r
A second order Kirchho adjoint (migration) formula is given by
(Lg )k;l = xr
X
i
Ak;l;i (1 Tk;l;i)Dt0 (f t g )i;J + Tk;l;i Dt0 (f t g )i;J
One possible arrangement of the algorithm for migration:
r
1. Precompute the scaling eld
r
k;l;i
Bk;l v2xr
k;l
10
r
r
k;l;i
+1
k;l;i
9
=
1 ];
Kirchho Simulation
Trip 94
2. initialize output depth section to zero
3. for each i (receiver index):
(a) apply centered time dierence operator Dt0 to input trace, convolve result with
time-reversed source wavelet f (compute h f t (Dt0 g ))
(b) for each k,l (depth point):
i. compute
Pk;l;i = Bk;las;k;l ar;k;l;i
ii. calculate Jk;l;i ; Tk;l;i
iii. add to k; l sample of output depth section
r
r
Pk;l;i [(1 Tk;l;i)hi;J
k;l;i
r
+ Tk;l;ihi;J
k;l;i
+1 ]
3 Traveltime and Amplitude Calculation
3.1 Geometrical Considerations
Recall the limitations on the Kirchho algorithm accepted in the Introduction: rays carrying signicant reected energy propagate \near the vertical," and reections occur only
at points on simple, single-arrival wavefronts. The rst of these restrictions simply means
that we foresake the possibility of imaging via turning rays: incident rays are entirely
downgoing, and reected rays are entirely upcoming. In terms of the traveltime eld
(x; z; xs; zs),
@ > 0
@z
at least in the zone below the source (z > zs ) where reections occur. A nondimensionalized and safeguarded version of this criterion is j sin j sin max < 1, where
sin = v @ = 1 @
@x
s @x
is the sine of the angle made by the ray with the vertical.
Evidently point sources actually produce rays traveling in all directions near the source
point. Rays traveling horizontally in the water column are irrelevant to the simulation
of reections. More generally, rays incident at high angles tend to turn before reaching
deeper reectors, in the typical velocity structure increasing generally with depth. To
eliminate this class of rays from the calculation, consistent with the assumptions made
in the Introduction, we shall modify both the point source initial condition for the travel
time and amplitude elds, and the eikonal equation itself.
We rst pick a convenient datum depth zd > zs , which might be the water bottom in
an idealized model, but need not correspond to any actual physical transition | it must
merely lie entirely in the water layer, i.e. in a homogeneous layer containing the source.
11
Symes, Versteeg, Sei, Tran
Trip 94
[Even that restriction might be eliminated at the price of algorithmic complexity.] On the
surface fz = zd g we replace the traveltime data of a point source:
q
pt(x; zd; xszs) = v1 (x xs)2 + (zd zs)2
(IPT)
0
by a condition which limits the sines of incidence angles to sin max:
8 1p
2
2
>
if
>
v0 (x xs ) + (zd zs )
>
>
>
>
>
>
jx xsj tan maxjzs zd j ;
>
>
>
>
<
j
mod (x; zd; xs; zs) = > v0jzcos zmax
>
>
>
>
>
+ sinv0max (jx xs j tan maxjzs zd j)
>
>
>
>
>
>
: else
s
d
(IMOD)
This modication enforces the maximum angle condition at the initial datum depth
z = zd . To enforce it at depth, we modify the eikonal equation, written as an evolution
equation in depth:
s
@ = 1 @ 2
(EPT)
@z
v2
@x
with the aim of limiting the angles of wavefront normals to max at most. A simple
modication which accomplishes this objective is
@
@z
v
(
u
u
= tmax
1
v2
!
@ 2 ; cos2 max
@x
v2
)
This modied eikonal equation is satisfactory for computation of the traveltime eld, but
not for the computation of the associated amplitude: as the transport equation involves
the Laplacian of the traveltime, the discontinuity in derivative caused by the use of the max
function produces spurious oscillations in the amplitude eld at the edge of the aperture.
Instead we use a tapered transition to constant under the square root:
v
u
@ = u
tbmax
@z
(
1
v2
!
@ 2 ; cos2 true
@x
v2
)
(EMOD)
Here bmax is the piecewise quadratic function described by the following conditions: for
x; y 0
bmax(x; y ) = x if x y
= 21 y if x 0
2!
x
1
= 2 y + y if 0 < x < y
12
Kirchho Simulation
Trip 94
For the sake of vectorization, bmax may also be written
!
2!
1
x
bmax(x; y ) = min 2 y + y ; max(x; y )
Thus the modied eikonal equation (EMOD) is identical to the original whereever the
angle subtended with the vertical by the ray velocity vector is less that true. The modied
equation is compatible with the modied initial conditions (IMOD) when
cos2 (true) = 2cos2(max)
which denes the relation between true and max.
With these two changes, we have ensured that @=@z > 0 always, i.e. that \rays will
always be downgoing." However we have certainly not computed the traveltime eld of a
point source, and it is important to identify exactly what we have changed.
It is possible to show that the time mod (x; z ) computed by solving the modied eikonal
equation (EMOD) as an evolution equation in depth, with initial condition (IMOD), is
the same as the rst arrival time from the source point (xs ; zs ) to (x; z ) provided that the
associated ray velocity vector never makes an angle with the vertical greater than true.
Therefore we can use these times to compute the Kirchho integral, provided that all
reection points are connected to both source and receiver by unique rays which obey the
angle condition | as assumed in the Introduction. While this condition does not hold in
general, even for layered media, it does hold for muted reection data. We will assume
that a suitable mute has been applied to the data, and that true has been chosen to be
large enough so that the angle condition holds for all events in the muted data.
The possibility of devising an automatic scheme to verify the angle condition, relating
the choice of aperture true and the mute pattern, is probably worth some thought.
Remark. It is possible to weaken the condition of unique source-reection point-receiver
rays sightly, by modifying the velocity outside of the cone bounded by the rays taking o
at angles rmtrue to be layered. Since the horizontal slowness is constant in a layered
medium, rays beginning outside the cone are guaranteed never to change direction. Since
the rays associated to the modied initial condition (IMOD) are leftgoing to the left of
the cone, and rightgoing to the right of the cone, in fz = zd g the same remains true
throughout the depth range. These modied rays form a parallel ray family, hence never
penetrate the above-mentioned cone. In this case the arrival times computed within the
cone are necessarily associated with rays having takeo angles < true . Our code computes
the rst arrival from amongst the times associated with this restricted family of rays. This
may not necessarily be the (global) rst arrival in the point source time eld, thus the
slight weakening noted above.
This aperture-driven velocity modication is an option in our algorithm. For nearly layered models velocity modication appears to be unnecessary to avoid computing spurious
rst arrivals, as one might expect.
Of course times along other rays not belonging to the point source eld are also computed. To ensure that these rays do not contribute inadvertently to the computed Kirchho integral, we diminish the amplitude assigned to them. The transport equation for the
13
Symes, Versteeg, Sei, Tran
Trip 94
amplitudes may be written as
!
@ + @ 1 @ @ log a = 1 @ 1 r2
@z @z @x @x
2 @z
(TPT)
This form is advantageous in computation: the amplitude eld changes by perhaps several
orders of magnitude over a typical computational domain, whereas its logarithm changes
much less | thus it is considerably easier to maintain uniform accuracy for the logarithm
in the presence of discretization and round-o error.
Initial conditions at fz zd g are
v
u
v0
A(x; zd) = 21 u
t q
2 (x xj )2 + (zd zs )2
(TIPT)
As a by-product of the solution of the modied eikonal equation (IMOD), we also compute
a tapered cuto function g , dened by
g = bcut
1
v2
!
@ 2 ; cos2 (max ) ; cos2 (true )
@x
v2
v2
!
where bcut(x; a; b) is the piecewise cubic twice continuously dierentiable function which
is 0 for x < a and 1 for x > b. Specically, set
r = min(b a; max(0; x a)):
Then
bcut(x; a; b) = 2r3 + 3r2
We use this cuto function to modify the right hand side of the transport equation:
!
@ + @ 1 @ @ log a =
@z @z @x @x
"
#
1 g @ 1 r2 1 (1 g )max @ 1 r2
(TMOD)
x @z
2 @z
2
That is, we force the amplitudes outside of the aperture dened by true to decay at least
as quickly as those inside the aperture. Since these amplitudes are decaying rather quickly
for geometrical reasons also, the upshot is to suppress the out of aperture amplitudes.
As for the traveltimes, the amplitudes produced by solving the modied system (TIPT),
(TMOD) are identical to those of the original point source system (TIPT), (TPT) so
long as the rays carrying signicant incident and reected energy make an angle with
the vertical of at most true during their entire transit from source to reecting point to
receiver. We assume that this condition holds for all reection events surviving the mute.
14
Kirchho Simulation
Trip 94
3.2 Traveltime Computation
From amongst the large number of competing nite dierence and related schemes for
solution of the eikonal equation suggested in the recent literature, we have selected the essentially nonoscillatory (\ENO") schemes of [17] as particularly appropriate. This class of
schemes is attractive for three essential reasons: (1) stable schemes of arbitrarily high order of accuracy exist; (2) versions exist in any dimension | in particular three dimensional
analogues of our algorithm are available; (3) the ENO schemes are easy to program.
Introduce basic nite dierence operators as follows. At x = j x, z = kz the value of
the discrete traveltime eld will be written jk similarly with skj = v (x; z ) 1. The forward
(+) and backward ( ) divided dierence operators in x are
h
Dx jk =
jk1 jk
i
x
Similar divided dierence operators are dened in z .
The basic rst order upwind scheme is:
bmax
(skj )2
2
max(Dx jk ; 0)
min
1
2 2
k
2
2
+
k
Dx j ; 0 ; (sj ) cos true
jk+1 = jk + z
The essential principle in this scheme is that the update of is computed from directions
in which the rays are owing (\upwind"). Ray tracing would do this naturally; the decision
tree within the right hand side of the above formula enforces the upwind choice of dierence
operator. Since the choice is made via min and max operators, the scheme vectorizes.
A necessary detail in a practical code is enforcement of stability. Like most other explicit
dierence schemes, the scheme explained above is only conditionally stable. The CourantFriedrichs-Levy (\CFL") condition necessary (and sucient in this case) for stability is
z Dxupwind < 1
y Dz+ Here Dxupwind denotes the upwind choice of Dx made in the above formula. This condition
must be satised; most likely, many of the reported failures of nite dierence schemes
for the eikonal equation are due to failure to enforce it. Enforcement is easy: each full
step, e.g. in the down cycles from kz to (k + 1)z , is subdivided into partial steps of
length zlocal. The local step length zlocal is chosen so that the above CFL condition is
satised with z replaced by zlocal. Of course it is also necessary to ensure that zlocal
is not too small relative to z , to avoid arbitrarily long internal loops.
Osher and Sethian ([17]) describe families of upwind schemes of arbitrarily high order,
based on the concept of essentially nonoscillatory (ENO) higher order dierences. For
example, the second order ENO correction of Dx+ is:
Dx+;2 = Dx+ 21 x(Dx+ Dx+ ; Dx Dx+ )
15
Symes, Versteeg, Sei, Tran
Trip 94
where
(u; v ) = u if juj jv j and uv > 0
= v if jv j < juj and uv > 0
= 0 if uv 0
That is, the rst order divided dierence is corrected by a second derivative approximation given by a one sided dierence formula if that is smaller, else by the centered
dierence. For example at a discontinuity of the rst derivative, the one sided dierence
would be chosen, if the two possibilities agree in sign. Thus the upwind character of the
rst order scheme is preserved at discontinuities of the traveltime gradient. If the signs
of the two dierence approximations disagree, no correction is made, on the presumption
that the rate of change of the second derivative is too high for higher dierences to be
accurate. Similarly,
Dx ;2 = Dx + 21 x(Dx Dx ; Dx Dx+ )
To obtain the second-order ENO scheme used here from the rst order upwind scheme
above,
1. replace Dx with Dx;2 in 1st order upwind formula
2. use second-order Runge-Kutta formula to step in z direction.
Explicitly, set
G[ ]kj =
q
Then
bmax((skj )2 (max(Dx2; jk ; 0))2 (min(Dx2;+ jk ; 0))2; (skj )2cos2 (true))
1
jk+ 2 = jk + 12 zG[ ]kj
1
jk+1 = jk+ 2 + 12 zG[ k+ 12 ]j
Remark. Since we have assumed (in the Introduction) that the portion of the traveltime
eld responsible for reections is smooth and single-valued, conceivably some other choice
of scheme, e.g. an analogue of the leapfrog or Lax-Wendro schemes, would function
as well. However, the ENO schemes and other similar schemes have the advantage of
nonlinear stability: if a discontinuity of r (signaling the formation of a caustic) developes
in the eective aperture, i.e. along rays maintaining an angle of less than true with the
vertical, the ENO scheme continues to compute the rst arrival time, whereas other nonupwind schemes may fail altogether or propagate oscillations away from the discontinuity.
We prefer to have a successful computation, even if the single-arrival hypothesis is violated.
Note that success or failure of this hypothesis cannot be determined a priori, in general.
16
Kirchho Simulation
Trip 94
As noted earlier, the transport equation for the amplitude eld involves the traveltime
Laplacian. To make possible a two level scheme for depth matching compatible with the
traveltime scheme, it is useful to write the traveltime Laplacian in a form involving only
rst z -derivatives. Dierentiation of the eikonal equation with respect to s and z , followed
by some algebra, yields
r2
= S2
@ 2 @ 2 + 1 @ @z @x2 2 @z
1
@ (s2)
@z
@ 1 @ @ (s2)
@z @x @x
!
Second-order approximations to the rst and second traveltime x derivatives and the
traveltime z derivative are by-products of the step computation for the eikonal equation:
@ k D [ ]k = max(D2; k ; 0) + min(D2;+ k ; 0)
x j
x j
x j
@x j
@ 2
@x2
!k
j
1x2 jk+1 + jk 1 2jk
@ k G[ ]k
j
@z j
Substitutions of these approximations in the above expression for the Laplacian gives
the second order approximation
r2 L[ ]kj
An upwind choice of derivatives of u = log a guarantees that discretization error from
discontinuities in @=@x (if such occur) does not pollute the solution, as such discontinuities only occur downwind of neighboring points. Also upwind dierences for log a simplify
the computation at the boundary, just as in the eikonal scheme.
This reasoning suggests use of the Beam-Warming scheme, an upwind variant of the
well-known Lax-Wendro scheme (see eg. [19], pp. 330 .), based on one-sided second
order approximations to the rst and second derivatives:
Dpuj = 1x 32 uj + 2uj+1 21 uj+2
Dm uj = 1x 32 uj 2uj 1 + 12 uj 2
Dppuj = 1x2 (uj + uj+2 2uj+1 )
Dmm uj = 1x2 (uj + uj 2 2uj 1)
Set
h
k ; uk
i
j
(
1 max(D [ ]k ; 0) Dm uk
=
x j
j
G[ ]kj
17
z Dx[ ]kj Dmm uk
j
2 G[ ]kj
!
Symes, Versteeg, Sei, Tran
Trip 94
+ min(Dx[ ]kj ; 0) Dp ukj
Then the Beam-Warming scheme is
z Dx[ ]kj Dppuk
j
2 G[ ]kj
!)
1 L[ ]kj
2 G[ ]kj
ukj +1 = ukj + z[ k ; uk ]j
The transport equation is a scalar hyperbolic system, and the Beam-Warming scheme is
accurate of order 2 and dissipative of order 4, hence stable by a theorem of Parlett ([19],
p. 119), so long as its coecients remain smooth. We have seen no evidence of unstable behaviour resulting from nonsmoothness in the coecients (traveltime derivatives),
although the stability theory of dierence schemes appears to have nothing directly to say
about this matter. The stability condition appears to be somewhat more strict than that
required by the ENO scheme for the eikonal equation. Experimentally we have found that
zDx[ ]kj
< 0:5
xG[ ]kj
provides conditional stability.
4 Numerical Examples
4.1 Accuracy of Eikonal, Transport Approximations
To give some idea of the accuracy obtainable with the dierence schemes outlined in the
preceding section, we have carried out several tests using constant velocity, for which
comparison with the analytic quantities is straightforward.
The velocity used in these experiments is 1.0 m/ms, and the computational domain in
all cases is 4096 m (x) 2048 m (z). The source point is oset 512 m from the left edge
of the model, and is 150 m above the top. We chose an aperture of 55 degrees, which
allowed us to use x = 4z throughout the computation - the automatic step selection
would otherwise have a confusing inuence on the accuracy.
Figures 1 and 2 show the traveltimes and relative errors at the bottom of the domain,
and Figures 3 and 4 do the same for the amplitudes. The second order convergence claimed
for the schemes. The relative error in the amplitudes is quite a bit greater than that for
the traveltimes. This is easy to understand, as the error in the traveltimes is second order
and the transport equation involves the second dierence of the traveltime, which is of has
O(1) error. As noted in the last section, a small amount of smoothing appears to suce
to remove the largest part of the inuence of high frequency error from the traveltime
Laplacian.
4.2 Accuracy of Kirchho Simulation
We performed a number of tests using the \gascloud" model presented in last year's annual
report. The velocity is mostly layered, with a slow anomaly embedded in the center (Figure
18
Kirchho Simulation
Trip 94
5). The reectivity is layered (Figure 6). Shot and receiver spacing were 48m, depth 32m,
far oset 1728m, 34 traces per shot gather. We computed traveltimes and amplitudes
using an aperture of 65 degrees, which we found to be well adapted to the mute. We used
a grid with x = z = 32m. The resulting traveltime and amplitude elds are displayed
in Figures 7 and 8. The adaptive aperture denition is particularly clear in the case of the
amplitude. The grid on which velocity and reectivity elds were dened had spacings
x = z = 16m, however. The Kirchho sums were also performed on the 16m grid. The
capability to decimate the grid for traveltime and amplitude calculations, then interpolate
the resulting elds onto a ner grid for combination with the reectivity, is very important
for eciency reasons, as will be reviewed below.
The energy source used in the rst several experiments is a Ricker wavelet with peak
frequency 10Hz. The resulting Kirchho simulation of a shot at oset 3200m from the
left edge of the model is shown in Figure 9. For comparison, we simulated the same
seismograms using a nite dierence scheme of order 2 in time and 4 in space ([9]). Since
the Kirchho code simulates wave propagation in a domain without boundary, we had
to arrange the nite dierence simulator to avoid surface reections. We did not use
absorbing boundary conditions; instead we used free surface conditions on all sides of an
adaptivly estimated optimal subdomain, the boundaries of which are just far enough from
sources and receivers that no edge reections occur in the data. This was not a large loss
for the sides and bottom of the model, as we did not wish to rule out steeply dipping
reectors a priori. However it did require almost doubling the size of the model in order
to avoid reections from the top. To avoid giving misleadingly negative time comparisons
with the Kirchho simulator, I have somewhat optimistically divided all nite dierence
execution times quoted below by two to simulate the added eciency of an absorbing top
surface.
In order to ensure accurate results we used a grid with x = z = 8m, t = 2ms;
another computation with grid sizes twice as large as this yielded less than 3% change in
the output. This accurate nite dierence seismogram is shown in Figure 10. To exhibit
more clearly the close correspondence between the Kirchho and nite dierence results,
we show trace 30 from both shot gathers in Figure 11.
To investigate the change in accuracy as the upper bandlimit is increased, we repeated
these experiments with 15Hz and 20Hz Ricker source wavelets. The results are shown
in Figures 12 through 17. In these two examples, the nite dierence simulator used
x = z = 8m and t = 2ms, whereas the Kirchho simulator computed traveltime and
amplitude elds on a 32m grid and interpolated these to a 16m grid for summation. Up
to 20Hz, at least, the elds are quite close.
Kirchho simulations remain accurate at far coarser grid spacings than do nite dierence simulations, at least of the type considered here. To illustrate this contention, Figures
18 and 19 show Kirchho and nite dierence simulations of the same shot gather, with a
12.5 Hz Ricker source. The Kirchho grid parameters are as before, 4ms time step, 32m
grids for traveltime and amplitude calculations, 16m grid for Krichho summation. The
nite dierence space steps were 16m, the time step 4ms. The grid dispersion evident in
the nite dierence results is largely absent from the Kirchho simulations.
Somewhat more accurate nite dierence methods are of course available, and would
19
Trip 94
Symes, Versteeg, Sei, Tran
yield good results at coarser grids somewhat more eciently than does our primitive (2,4)
scheme. However we believe that the fundamental conclusion of this section would remain
valid even under comparison with these more sophisticated schemes: for the parameter
ranges and type of model considered here, primaries only simulation via Kirchho summation demands less rened grids than does the same simulation via nite dierences.
4.3 Eciency considerations
A straightforward discretization error analysis shows that the Kirchho algorithm presented here has second order truncation error. Worst case estimates of the error suggest
that roughly 10 gridpoints per wavelength would be necessary to obtain output signals at
5% error level. The relative dips of typical (nearly horizontal reectors) and the summation trajectories near reection points are not far apart, which argues for regarding the
worst case error estimates as unduly pessimistic - as our numerical experiments indicate.
In any case the Kirchho summation itself is quite inexpensive. The major cost of
Kirchho simulation lies in the computation of traveltimes and amplitudes. Accordingly
the Kirchho simulator obtains its considerable gain in eciency over nite dierence
simulation from three sources:
the smoothness of the traveltime and amplitude elds allows the use of much coarser
grids for their computation than are needed for accurate Kirchho sums;
for multishot simulation, a number of shot gathers can be computed simultaneously,
and the expense of the traveltime and amplitude computations amortized.
for iterative inversion, say via conjugate gradient or related methods, the traveltime
and amplitude tables can be stored on disk, and re-used.
The grid decimation has already been mentioned. We have implemented only decimation by factors of 2 and 4, and used only decimation by 2. In principle the choice of
decimated grid should be governed by accuracy considerations, and the slowness eld interpolated appropriately; we intend that a future version of our code will have this feature.
In any case the gain in eciency is important: decimation by 2 decreases CPU time in
the eikonal and transport solves by a factor of 4, and also decreases the storage and i/o
costs.
Because of the use of decimated grids, a considerable number of traveltime and amplitude elds can be held in core at one time. Therefore, for uniform and commensurable shot
and receiver spacing, these elds can be used repeatedly in the summations. For example,
if shot and receiver spacing are the same, each newly computed receiver travel time table
can be used to add a trace to every shot gather currently in core to which the receiver
location belongs. Thus the cost becomes asymptotically independent of the number of
shots. For example a simulation of a single shot for the 15 Hz source mentioned above,
with 8m spatial grid and 2ms time step (as in Figure 13) requires approximately 100s
for an IBM RS6000/380 to execute our mildly optimized nitte dierence code, running
at about 45 Mops. The Kirchho simulation of one shot using 32m grids to compute
20
Kirchho Simulation
Trip 94
traveltimes and amplitudes, and 16m grids for Kirchho summation, required about 49s.
Kirchho simulation of 25 shots, retaining traveltime and amplitude tables for all 25 shots
in memory, required 228s, or about 9s per shot. Simulation of 50 shots with 60 traces per
shot came to about the same cost. So to provide accurate simulations, the combination
of coarser sampling, grid decimation, and amortization of traveltime and amplitude computations resulted in an order of magnitude cost advantage for the Kirchho simulation
over nite dierence. We are certain that further signicant optimization opportunities
are available in the Kirchho code, whereas the nite dierence code is already performing
at a near-optimal level.
Finally, we have provided the capability for the simulator to use precomputed traveltime
and amplitude tables stored on disk. For example the second and subsequent iterations of
a conjugate gradient loop for solution of the least squares linearized inverse problem using
the Kirchho simulation and migration tools will be able to take advantage of this option.
The cost for the 50 shot, 60 trace example dropped to less than 4s per shot. Because
the current implementation involves some identiable ineciencies when operating in this
mode, we project that the actual cost of this mode, i.e. Kirchho summation plus disk
i/o, for the parameters and scales of our examples is roughly 2s per shot.
21
Symes, Versteeg, Sei, Tran
Trip 94
3100
3000
2900
2800
2700
2600
2500
2400
2300
2200
2100
0
500
1000
1500
2000
2500
3000
3500
4000
Figure 1: Exact and computed traveltimes for constant velocity medium. Solid line
= exact; dashed line = nite dierence traveltime, x = 16m; dot-dash line = nite
dierence traveltime, x = 32m; dotted line = nite dierence traveltime, x = 64m.
22
4500
Kirchho Simulation
Trip 94
-4
9
x 10
8
7
6
5
4
3
2
1
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Figure 2: Relative errors in computed traveltimes for constant velocity medium.
Dashed line = relative error in nite dierence traveltime, x = 16m; dot-dash line
= relative error in nite dierence traveltime, x = 32m; dotted line = relative error in
nite dierence traveltime, x = 64m.
23
Symes, Versteeg, Sei, Tran
Trip 94
-3
8.8
x 10
8.6
8.4
8.2
8
7.8
7.6
7.4
7.2
7
6.8
0
500
1000
1500
2000
2500
Figure 3:
3000
3500
4000
4500
Exact and computed amplitudes for constant velocity medium. Solid line
= exact; dashed line = nite dierence amplitude, x = 16m; dot-dash line = nite
dierence amplitude, x = 32m; dotted line = nite dierence amplitude, x = 64m.
24
Kirchho Simulation
Trip 94
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
500
1000
1500
2000
2500
3000
3500
4000
Figure 4: Relative errors in computed amplitudes for constant velocity medium.
Dashed line = relative error in nite dierence amplitude, x = 16m; dot-dash line
= relative error in nite dierence amplitude, x = 32m; dotted line = relative error in
nite dierence amplitude, x = 64m.
25
4500
Symes, Versteeg, Sei, Tran
Trip 94
Plane 1
Trace 150
0
200
250
300
350
0
2
1.9375
1.875
1.8125
1.75
1000
1000
1.6875
1.625
1.5625
1.5
2032
Trace 150
2032
200
250
300
350
Plane 1
Figure 5: Gascloud velocity model. Horizontal trace spacing is 16m. Depth axis is in
m. Velocity unit is m/ms. First shotpoint is at trace 200 (oset 3200m). Cable extends
to right of shotpoint roughly to trace 311 (oset 4980). Shot and receiver depths are 32m.
26
Kirchho Simulation
Trip 94
Plane 1
Trace 150
0
200
250
300
350
0
0.335706
0.25178
0.167853
0.0839265
0
1000
1000
-0.0839265
-0.167853
-0.25178
-0.335706
2032
Trace 150
2032
200
250
Plane 1
Figure 6:
Gascloud reectivity prole. Depth axis is in m.
27
300
350
Symes, Versteeg, Sei, Tran
Trip 94
Plane 1
Trace 1
0
400
350
300
250
200
51
101
151
0
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000
1100
1100
1200
1200
1300
1300
1400
1400
1500
1500
1600
1600
1700
1700
1800
1800
1900
1900
2000
2048
2000
2048
150
100
50
0
Trace 1
51
101
Plane 1
Figure 7:
Finite dierence traveltimes, 32m grid.
28
151
Kirchho Simulation
Trip 94
Plane 1
Trace
0
200
250
300
0
1.5e-02
1.3e-02
1.1e-02
9.4e-03
7.5e-03
1000
1000
2000
2048
2000
2048
5.6e-03
3.8e-03
1.9e-03
0.0e+00
Trace
200
250
Plane 1
Figure 8:
Finite dierence amplitudes, 32m grid.
29
300
Symes, Versteeg, Sei, Tran
Trip 94
Plane
1
Trace
0
1
11
21
31
0
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000
1100
1100
1200
1200
1300
1300
1400
1400
1500
1500
1600
1600
1700
1700
1800
1800
1900
1900
1996
1996
Trace
1
Plane
1
11
21
31
Figure 9: Kirchho simulation of shot gather. 32m grid for traveltime and amplitude
calculations, 16m grid for summation.
30
Kirchho Simulation
Plane
1
Trace
0
1
11
21
Trip 94
31
0
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000
1100
1100
1200
1200
1300
1300
1400
1400
1500
1500
1600
1600
1700
1700
1800
1800
1900
1900
1996
1996
Trace
1
Plane
1
Figure 10:
11
21
31
(2,4) nite dierence simulation of shot gather, using 8m grid.
31
Symes, Versteeg, Sei, Tran
Trip 94
-3
3
x 10
2
1
0
-1
-2
-3
-4
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Figure 12: Kirchho simulation, 15 Hz source, 32m grid for traveltime and amplitude
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Figure 13:
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Figure 15: Kirchho simulation, 20 Hz source, 32m grid for traveltime and amplitude
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Kirchho Simulation
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Figure 16:
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Figure 17:
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Trace 30, 20 Hz source: solid = nite dierence, dashed = Kirchho.
38
Kirchho Simulation
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Figure 18: Kirchho simulation, 12.5 Hz source, 32m grid for traveltime and amplitude calculations, 16m grid for summation.
39
Symes, Versteeg, Sei, Tran
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Figure 19:
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Finite dierence simulation, 12.5 Hz source, x = z = 8m, t = 2ms.
References
[1] A.J. BERKHOUT. Seismic Migration. Elsevier, Amsterdam, 1984.
[2] G. BEYLKIN. Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform. J. Math. Phys., 26:99{108, 1985.
[3] G. BEYLKIN and R. BURRIDGE. Linearized inverse scattering problem of acoustics
and elasticity. Wave Motion, 12:15{22, 1990.
[4] N. BLEISTEIN. On the imaging of reectors in the earth. Geophysics, 52:931{942,
1987.
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[5] S. CAO and B. KENNETT. Reection seismograms in a 3-d elastic model: an
isochronal approach. Journal of Geoph. Res., 96:63{80, 1990.
[6] V. CERVENY and A. COPPOLI D.M. Ray born synthetic seismograms for complex
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1993.
[8] F. COLLINO. Analyse Numerique de Modeles de Propagation d'Ondes. Application
a la Migration et a l'Inversion des Donnees Sismiques. PhD thesis, Universite Paris
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[9] M.A. DABLAIN. The application of high-order dierencing to the scalar wave equation. Geophysics, 51:54{66, 1986.
[10] ANDREAS EHINGER. FARS{ a prestack shot record migration program. First
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[11] I.M. GEL'FAND and G.E. SHILOV. Generalized Functions, volume I. Academic
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[12] S. GEOLTRAIN and J. BRAC. Can we image complex structures with rst-arrival
traveltime? Geophysics, 58:564{575, 1993.
[13] S.H. GRAY and W.P. May. Kirchho migration using eikonal equation traveltimes.
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[14] P. LAILLY. Migration methods: partial but ecient solutions to the seismic inverse
problem. In Santosa et al., editors, Inverse Problems of Acoustic and Elastic Waves.
SIAM, Philadelphia, 1984.
[15] A.R. LEVANDER. Fourth order nite dierence P-SV seismograms. Geophysics,
53:1425{1434, 1988.
[16] T.J. MOSER. Migration using the shortest path method. Geophysics, 59:1110{1120,
1994.
[17] S. OSHER and J. SETHIAN. Fronts propagating with curvature dependent speed:
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[19] R. RICHTMYER and K. W. MORTON. Dierence Methods for Initial Value Problems. Wiley/Interscience, New York, 1967.
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78:2062{2076, 1988.
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[21] B.S. WHITE. The stochastic caustic. SIAM J. Appl. Math., 44:127{149, 1982.
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