Stochastic Heterogeneity

The Texture You Generate Instead of Draw

Deterministic builders gave clean shapes, but real rock is never that tidy. It carries fine layering, patchy cementation, and lateral variability at every scale, the texture that makes a real seismic section look real. You could never draw all of it by hand, so you generate it: a random velocity field with a controlled statistical character.

The control that matters is the correlation length, the scale over which the velocity stays similar to itself. A short correlation length gives fine speckle, a scattering medium. A long one gives broad, smooth variation. And because the horizontal and vertical correlation lengths can differ, you can shape the fabric: make the horizontal long and the vertical short and the field becomes layered, velocity changing quickly with depth but slowly along a bed, exactly the character of a stratified section.

Randomness You Set, Not Chaos You Suffer

This is the von Karman, or Gaussian random-field, model that underlies realistic earth models and the wave-scattering studies built on them. Two things make it a modelling tool rather than mere noise. It is statistically controlled: the correlation lengths are physical parameters that decide which scales of heterogeneity the wavefield will scatter from. And it is reproducible: the same seed and the same lengths regenerate the identical field, at any grid size. You are not adding chaos; you are adding a texture with a spectrum you chose.

Deterministic structure and stochastic texture are the two halves of a believable model. One gives the traps and faults you want to place exactly; the other gives the fine fabric you want to specify only in statistics. The next section puts them together, taking a reservoir-scale property model all the way to a seismic section.