The Modeling Ladder

Part 0: The Forward Problem

Learning objectives

Order the forward operators from cheap and approximate to costly and complete
State the fit-for-purpose rule: use the cheapest physics that still answers the question
Name what each rung adds that the rung below it cannot produce
Match a modeling task to the rung that fits it

Two Ways to Get It Wrong

Faced with a modeling task, there are two easy mistakes. The first is to always reach for the fullest physics available, a full elastic anisotropic simulation for a job a one-line convolution would have finished before lunch. The second is to always reach for the cheapest, and then wonder why the diffraction you were hired to study never showed up. Good modeling avoids both by matching the physics to the question.

The Ladder

The forward operators form a ladder. Each rung adds a capability the rung below it cannot produce, and each rung costs more than the one below.

1D convolution. A reflectivity series convolved with a wavelet, at one location. Gives you a trace: well ties, tuning, quick looks.
2D convolution. The same idea across a model, trace by trace. Gives you a stacked section with structure: faults, folds, stratigraphy, and training images by the thousand.
The acoustic wave equation. Solve $\partial^2 u/\partial t^2 = c^2 \nabla^2 u$ on a grid. Now the wave actually propagates, so you get diffractions, multiples, spreading, and pre-stack shot records: everything convolution leaves out.
Elastic and AVO. Carry shear as well as compression. Now amplitude varies with angle, which is where fluids and lithology hide.
Anisotropic. Let velocity depend on direction. Now shale mis-ties and, crucially, fracture signatures that vary with azimuth become visible.

The Rule

Climb only as high as the task requires. The right rung is the cheapest one that still answers the question: any lower and you miss something the task depends on; any higher and you have paid for physics that does not change your answer. Pick a task below and the ladder shows you the fit, and points to the part of this course where you build that rung.

Notice that fit for purpose is not a compromise. For fault-detection training images, 2D convolution is not the budget option that a wave-equation solver would improve on; it is the correct choice, because the label the network learns is the geometry you drew, and the wave equation would spend a thousand times the compute without changing that geometry. Matching the rung to the task is the whole craft, and the rest of this course is you learning to build every rung so the choice is yours.