Basic operators and adjoints |

(17) |

(18) |

Sampling the time axis gives a matrix equation that we should call ``causal summation,'' but we often call it ``causal integration.'' Equation (19) represents causal integration for and leaky integration for .

(which may also be regarded as a numerical representation of the

When it comes time to think about the adjoint, however,
it is easier to think of equation (19) than of equation (20).
Let the matrix of equation (19) be called .
Transposing to get and applying it to
gives us something back in the space of ,
namely
.
From it we see that the adjoint calculation,
if done recursively,
needs to be done backward, as in:

A module to do these jobs is `leakint`.
The code for anticausal integration is not obvious
from the code for integration and the adjoint coding tricks we
learned earlier.
To understand the adjoint, you need to inspect
the detailed form of the expression
and take care to get the ends correct.
Figure 5 illustrates the program for .

t = 0.; |

causint
Figure 5. in1 is an input pulse.
C in1 is its causal integral.
C' in1 is the anticausal integral of the pulse.
A separated doublet is in2.
Its causal integration is a box,
and its anticausal integration is a negative box.
CC in2 is the double causal integral of in2.
How can an equilateral triangle be built?
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Later, we consider equations to march wavefields up toward the Earth surface, a layer at a time, an operator for each layer. Then, the adjoint starts from the Earth surface and marchs down, a layer at a time, into the Earth.

Basic operators and adjoints |

2014-09-27