Mathematical Methods for Geophysics
The graduate mathematics your methods quietly assume, built properly once: several variables, the integral theorems, holomorphic functions, Fourier, and the classical PDEs. The last stage walks the whole kit into the acoustic wave equation, and out comes a synthetic seismogram.
You can differentiate maps between spaces and wield the Jacobian, read the divergence and Stokes theorems as physical statements about flux and circulation, handle holomorphic functions and their series, move a signal between time and frequency with honest convergence claims, tell Laplace, heat, and wave problems apart on sight, and discretize the acoustic wave equation into a stable synthetic shot record.
Several variables
A velocity model is a function of three variables and a wavefield is a function of four; the Jacobian is the derivative of anything that maps space to space, and it is where calculus becomes usable.
The inverse function theorem says when a change of coordinates is legal, and the implicit function theorem is why a horizon defined by an equation is honestly a surface. Working tools, both.
The integral theorems
Gradient, divergence, and curl are the three verbs of field physics; every conservation law and every wave equation on this path is a sentence written with them.
Both theorems say the same deep thing: what happens inside a region is fully recorded on its boundary. That one idea underwrites every flux argument in geophysics.
Conservation of mass in a small box is the divergence theorem read aloud, and it is exactly half of the derivation of the acoustic wave equation waiting at the end of this path.
Complex analysis
Complex differentiability is a brutally strong condition, and the Cauchy-Riemann equations are the reason a single complex function can carry amplitude and phase as one object, the way a spectrum does.
Cauchy's formula rebuilds an analytic function from its boundary values, power series make it computable, and conformal maps bend one solved problem into another; this is the machinery under the transforms you are about to invert.
Fourier and differential equations
The formulas matter, and the convergence chapter matters more: Gibbs ringing at a discontinuity is a theorem, not a software bug, and you will see it on sections for the rest of your career.
The ODE catalog is small: decay, oscillation, resonance. It repays learning cold, because every geophone, every filter, and every damped system you will ever model obeys one of its entries.
Physics keeps three canonical PDEs and geophysics runs on all of them: potential fields obey Laplace, diffusion obeys heat, and seismic obeys wave. Meet them side by side once and you will never mistake their characters.
Applied to waves
The prerequisite kit of a working processing shop: phasors that carry amplitude and phase at once, the transform taken slowly, and the sampling theorem that decides what your data can even contain.
Newton and Hooke give you the acoustic wave equation in a page, and a finite-difference stencil turns it into arithmetic a machine can run; this is the moment the whole path becomes executable.
A simulator will lie to you politely: violate the CFL condition and it explodes, coarsen the grid and it disperses. Both failure modes are theorems, and you should trigger each one on purpose exactly once.
The last two ingredients of a usable synthetic: edges that do not reflect energy back into the model, and the shot-record geometry the field actually delivers. After this, seismograms are something you manufacture.