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Here we show theoretically that the 2-D autocorrelation (or 2-D spectrum)
of surface scattered reflections
is the same as that of the primary reflections.
Thus by autocorrelation we will concentrate information
that could be widely distributed in time and space.
Later, we'll convert the autocorrelation to something more familiar.
Let the layered earth response from shot *s* to geophone *g*
be *u*(*s*,*g*,*t*)=*u*(0,*g*-*s*,*t*)=*u*(*g*-*s*,*t*) or in Fourier space,
or simply *u*(*g*-*s*).
When an upcoming wave hits the
earth surface at *g*_{1} it encounters a scattering object
which reflects the primary wave with a random scaling .The signal at *g*_{1} then takes off for a second flight
like a multiple reflection, but departing in all directions.
We are going to build the theoretical 2-D spectrum
of this surface scattered wave *w* from the theoretical 2-D spectrum of *u*,
the layered media primary reflection.

First we express the cascade of the two bounces. The arrival *w*
at *g*_{2} at time *t* is the sum of the time of each bounce,
and .Since this is a convolution in the time domain,
we express it as a product in the frequency domain.
Then we form the complex conjugate of this expression in preparation for
autocorrelation on the *x*-axis.

| |
(1) |

| (2) |

| (3) |

We insert the last two expressions into the expression
for spatial autocorrelation.
| |
(4) |

We will determine *A*(*s*,*x*) experimentally as described earlier.
Here we will see
its theoretical relation to the primary reflected field *u*.

| |
(5) |

| (6) |

| (7) |

| (8) |

| (9) |

We Fourier transform over *x*.
The first factor above is not a function of space.
It is merely a function of , say a filter .Thus our main result:

| |
(10) |

We see that in principle, for each shot point *s*,
we measure the spectrum of the impulse response
of the layered medium.

** Next:** Hazardous cross terms
** Up:** Claerbout: Factorizing 2-D seismic
** Previous:** SUBTRACT ANY TWO SHOT
Stanford Exploration Project

4/27/2000