Derivatives with respect to line-shape parameters

measure_derivative() returns the measurement matrix \(W\) and its derivative \(\partial W / \partial \theta\), row \(i\) with respect to the parameter of kernel \(i\). For a spectrum \(f\), dW @ f is the Jacobian of the measurements with respect to per-channel parameters (it is diagonal), or with respect to one parameter shared by all channels, such as a common spectral shift.

parameter	kernels
center	all (spectral shift)
scale	all but deltas: width·∂/∂width, every width scaled together (squeeze or stretch)
sigma	Gaussian, Voigt, smoothed box
gamma	Lorentzian, Voigt
eta	pseudo-Voigt
power, asymmetry	super-Gaussian
half_width	smoothed box

The derivatives are exact derivatives of what measure() computes. The kernel derivative is treated as a distribution: a smooth part \(\partial K / \partial\theta\), integrated with the same quadrature as \(W\), plus point terms where the kernel jumps (box edges, and the truncation edges of profiles, which move with the parameter). The normalization is differentiated too: \(\partial W = (\partial A - W\, \partial N) / N\).

import numpy as np

import rebasis as rb

x = np.linspace(758, 772, 28001)
hires = rb.Space.hats(x)
f = 1 - 0.8 * np.exp(-(((x - 764.3) / 0.05) ** 2))  # one absorption line

centers = np.arange(760.0, 769.0, 0.2)
build = lambda shift: rb.Space.super_gaussians(centers + shift, fwhm=0.4, power=3.0)

W, dW = rb.measure_derivative(build(0.0), hires, "center")
jacobian = dW @ f  # ∂g_i / ∂(center shift)

h = 1e-6
finite_difference = (rb.measure(build(h), hires) @ f - rb.measure(build(-h), hires) @ f) / (2 * h)
print(f"largest relative difference: {np.abs(jacobian - finite_difference).max() / np.abs(jacobian).max():.1e}")

largest relative difference: 5.2e-09

Computing \(W\) and \(\partial W\) together costs about twice measure() alone.