Wave-equation migration

Wave-equation migration of reflection seismic data to produce images of the subsurface entails four basic operations:

•Summation of all shots•Wave-field extrapolation (phase shift operator)•Cross-correlation of source (U) with data (D)•Zero lag extraction by R(x,z,)

Thankfully, all these operators commute which allows the correlation in migration to satisfy the correlation required to produce the reflection response of the subsurface from the transmission records. This is the case if the transmission records are used as both the source and receiver wave-fields.

Standard Migration

This shows the commutability of the correlation and extrapolation operators (and coincidentally the equivalence of shot-profile and source-receiver migration) due to the seperability of the exponential operator.Extracting the zero time of the wavefield R at any depth level gives the image at that depth.

Passive MigrationThe reciprocity theory tells us that another factorization of R, besides UD is the cross-correlation of T, or:

Direct migration of passive data uses the transmission wave-field, T, for both upgoing, U, and downgoing, D, wave-fields in the same structure.

R = U DR = R eR = U D eR = U D e = U e (D e )

0

1

0

0

0*

*

*

*

1

1

0

0 0

0 0

0

+i Kz z

+i Kz z

+i Kz(U) z + i Kz(D) z

+i Kz(U) z -i Kz(D) z

Shot-gather from cross-correlation

passive transmission data

equivalent reflection data

Correlating every trace with every other squares the number of traces from the experiment. However, only the correlation lags corresponding to the depth of the deepest reflector of interest need be kept. This decimates the time axis by several orders of magnitude.

The case presented next door explains the use of a modified shot-profile migration algorithm to image the subsurface with telesiesmic coda energy. However, the theory of passive seismic imaging extends directly to allow us to migrate the raw data without imposing (incorrect) assumptions during pre-processing steps such as deconvolution or rotation.

Using a wave-equation based migration algorithm, and performing the correlations after the extrapolation step, the physics of wave propagation is honored for all, however complicated, energy available within the data set. This extends the imaging process to higher frequency local noise, as well as removing ambiguities associated with human interpretation of data before migration.

The use of depth migration requires a velocity model. To image converted modes, both shear and compresional models are needed. Images produced with this technique however show remarkable tolerance to produce reasonable images despite gross errors in velocity, as well as provide a tool to update the velocity model to accommodate errors in the output model space.

Application to the coda

Theory dictates that a truly identical data set, including amplitude accuracy, is generated by correlating the transmission records. This holds true only if the distribution of source energy is spatially even. Irregularity of the strength and distribution of subsurface energy leads to variations of the illumination of the model space.

Direct migration of raw transmission data

Migration with a true velocity model (rather than 1D) images yeilds crisp images even of the steeply dipping flanks of the syncline. However, if the location is subject to difficulties such as inter-bed multiples, inappropriate energy can mask the true reflectors just like conventional reflection.

hidden primary

multiple

Application to the shallow subsurface

Cross-correlation of 72 channel acquisition on the beach of Monterey, California lead to too few channels in any direction to find hyperbolas. The wave-front healing capacity of wave-field propagation allows infill with zero-traces that will interpolate the data during migration. This leads to garbage at shallow depth, but produces an interpretable result at greater depth. Deconvolution prior to migration as well as simple band-pass versions of data were used from several different times of the day.

hollow pipe

hollow pipehollow pipe

water table?

ambient energy and recording geometry

r1 r2 r1*r1 r1*r2

raw data

equivalent shot-gatherafter correlations

t

lag

Noise to data via cross-correlation

Because every trace records both the incident wave-field, which is the source, and the energy returning from subsurface reflectors, all traces have ‘source’ energy as well as ‘data’ information. This is similar to the case of surface related multiples. The correlation of every trace with every other builds hyperbolas from subsurface reflectors as well as removes the unknown time offset and phase characteristics of the probing energy.

r1 r2

x0 3500 7000 0 3500 7000

100 200 300 400

100 200 300 400

day 1 day 2

100 200 300 400

x

x0 3500 7000 0 3500 7000

x

100 200 300 400

100 200 300 400 500 600

I thank Deyan Dragonov of Delft University for modeling transmission panels, and Jeff Shragge and Biondo Biondi for many discussions.

R = UD = T T0 00* *

0 0

The importance of velocity

Application fo CASC-like synthetic

Passive Seismic ImagingS11E-0334 Brad Artman,Stanford Universitybrad@sep.stanford.edu