Ocean state estimation, from the generative story to each box
esmax · ocean · 2 · The Gulf Stream SSH-mapping problem written once, then read box by box as operators with inputs, outputs and shapes
Abstract¶
This is the traditional formulation of estimating the ocean state from sea-level maps, which we write once and then read box by box. Steps 1 and 2 are the classical problem, whereas steps 3 to 6 are what that problem turns into when parts of it are learned.
Where it sits. The running example: the data, the QG model, and what each of the six boxes takes in and returns. The full page is a self-contained document with operator boxes (inputs, outputs, shapes and units), figures and code sketches, which opens on its own and links back here.
TL;DR: Every box is an operator¶
Forward boxes (1, 3) map a sea-level field and two physical parameters to the next field. Box 1 chains that map into a cube of states and a cube of synthetic maps.
Inverse boxes (2, 4, 5) all have one signature: a cube of daily SSH maps with their masks and errors goes in, and a cube of states plus the parameters and an uncertainty comes out. They only differ in what is inside and in what each query costs.
Box 6 takes many windows of those outputs and returns diagnostics and one decision, which is either to rerun the inner loop or to change the model.
On the page¶
The problem and the data
Notation
The mathematical objects
Simple model: the generative story
Model-based inference: the classical ways
Emulator: replace 𝓕 by a network
Emulator-based inference: same cost, new model
Amortized inference: learn the inverse
Improve: the misfit decides which loop runs
The six boxes, read from the formulation