The Mathematics Behind Seismic Processing
From the algebra you half remember to the transforms processing leans on. Every step of the sequence is a piece of mathematics wearing field clothes; this path teaches the mathematics first, then shows it at work.
You can follow every equation a processing sequence rests on, from convolution and the Fourier family to least squares and the wave equation, and say in plain words what each step of the flow assumes.
The algebra you half remember
A seismic trace is a function before it is anything else, and reading graphs fluently is most of the daily job.
Decibels, Q, and every gain curve you will ever apply live on logarithmic scales; be at home there.
Every wavelet you will ever see is built from sines and cosines; radians and the sum identities are the grammar of that sentence.
One complex number carries an amplitude and a phase at once, which is exactly what a seismic frequency component is.
A sampled trace is a finite sum, and the geometric series is the Z-transform waiting for you three stages from now.
Matrices turn a system of equations into one object you can reason about, and processing is multichannel from the first trace.
The analysis toolkit
Rates, areas, and convergence are the machinery under every transform and every misfit function on this path.
Every filter and every stack is a linear map, and eigenvalues are how you see what an operator does before you run it.
Misfit surfaces live in thousands of dimensions; the Jacobian is the local map you navigate them with.
Decomposing a signal into sines is the single most used idea in seismic processing, and here it is built honestly from the integral up.
Noise is a random variable, and the central limit theorem is the exact reason stacking works at all.
The wave equation is the physical law the whole processing sequence serves; meet it as mathematics before you meet it as data.
The processing bootcamp
The convolutional model is the founding sentence of processing: the earth speaks through the wavelet, and phasors make that sentence algebra.
Aliasing is the one mistake acquisition cannot undo; know the sampling theorem cold and you will never be fooled by a folded frequency.
Digital filters live in the z-plane, and the geometric series you learned two stages ago is the entire trick.
One trio under deconvolution, statics, and inversion alike: model the noise, write the misfit, descend the gradient.
Thirty minutes of PDE that migration will spend the rest of this path cashing in.
Where the math earns its keep
Wiener's least-squares filter turns the convolutional model inside out; this is where the algebra starts paying rent.
A noise attenuation suite is Fourier reading comprehension under time pressure: know which domain each noise type separates in.
NMO is the Pythagorean theorem at reservoir scale, and semblance picking is statistics wearing a velocity label.
SRME and Radon are the transform toolkit hunting multiples in the domain where they separate from primaries.
Migration runs the wave equation backwards to put energy where it belongs; RTM is that idea with the physics left fully on.
Misfit, gradient, regularization: the mathematical skeleton that every inversion in geophysics shares, stated once and cleanly.
The whole sequence compressed onto one card, with the mathematics you now own standing behind every box.