Wavelength and wavenumber

Radiative transfer models often compute spectra on a wavenumber grid while instrument line shapes are characterized in wavelength. A Gaussian in wavelength is not a Gaussian in wavenumber, so rather than approximating shapes, to_wavenumber() (and to_wavelength(), reciprocal(), affine()) use each element in the other coordinate exactly: knots are mapped and the integrator follows the shape through the change of variable.

The Jacobian

jacobian is required, because it is a physical choice:

  • jacobian=True multiplies by \(|d\lambda/d\nu|\) so that integrals over \(\nu\) equal integrals over \(\lambda\). Use it for kernels: an instrument response given in wavelength averages over wavelength, whatever grid the spectrum is computed on. Use it also for bases of spectral densities (radiance per nm).

  • jacobian=False keeps values, \(f(\nu) = f(\lambda(\nu))\): bases of functions (transmittance, reflectance, sun-normalized radiance).

Example

A model spectrum on a wavenumber grid, measured by a spectrometer whose line shape is a 0.5 nm Gaussian in wavelength:

import numpy as np

import rebasis as rb

nu = np.linspace(19_500, 20_500, 100_001)          # cm⁻¹
f = 1 + 0.3 * np.sin(nu / 5.0)                     # e.g. a transmittance
hires = rb.Space.hats(nu)

centers_nm = np.arange(490.0, 510.0, 0.2)
ils = rb.Space.gaussians(centers_nm, fwhm=0.5).to_wavenumber(jacobian=True)
measured = rb.measure(ils, hires) @ f
measured[:5]

array([0.95945101, 1.05167411, 1.03105507, 0.94230675, 0.9793073 ])

The same measurement computed the other way round, in wavelength with the basis mapped instead, agrees to the conditioning of the coordinates:

other = rb.measure(rb.Space.gaussians(centers_nm, fwhm=0.5), hires.to_wavelength(jacobian=False)) @ f
print(f"max difference: {np.abs(measured - other).max():.1e}")

max difference: 1.5e-13

Parameters stay in the kernel’s own coordinate: measure_derivative() with "center" gives derivatives with respect to the line center in nm, and "scale" with respect to the width in nm.