Why Model? The Forward Problem

Two Directions Through the Same Physics

Seismic work runs in two directions. Nature runs it forward: a real earth, probed by a real source, produces the data we record. Interpretation and inversion run it backward: given data, infer the earth. The backward direction is the hard one, and it is where careers are spent, because many different earths can explain the same data.

Forward modeling is the deliberate act of running the physics in the easy direction: you specify an earth model, you specify the physics that connects rock to waveform, and you compute the seismic response. Written compactly, if $m$ is the earth model and $F$ is the physics, the synthetic data are $d = F(m)$.

Why Generate Seismic at All?

Because for a synthetic, and only for a synthetic, you know the answer. The earth that produced the data is not a hypothesis; it is the input you typed. That single property makes synthetic seismic the working currency of the field: it calibrates well ties, it stress-tests processing and imaging algorithms on targets whose true position is known, it supplies labeled training data for machine learning at any volume you like, and it lets you rehearse a survey or a monitoring program before spending money on it.

The figure above is the forward problem at its smallest useful size. The earth is three layers; the physics is the convolutional model you will build in Part 2; the data are one trace. Make the sand slow like a gas sand and the top reflection flips soft. Thin the sand below a dominant period and the top and base responses begin to interfere. Nothing about the data is mysterious, because you made the earth yourself.

The Question This Course Keeps Asking

The physics $F$ comes in grades. A convolution is nearly free. A finite-difference solution of the wave equation costs real compute but produces diffractions, multiples, and everything else waves actually do. An anisotropic, fracture-aware model costs thought as well as compute. The craft of synthetic modeling is not always choosing the most complete physics; it is choosing physics that is fit for purpose: complete enough that your synthetic answers the question you are asking, and no more expensive than that. The next section turns that idea into a tool you will use for the rest of the course.