How scatrs is validated¶

A scattering code can go wrong in two different ways, and scatrs checks for each separately.

The numerics. The Mie series is summed with recurrences that can lose accuracy to rounding, cancellation, instability or early truncation. Such errors grow in ways that are hard to predict: one size parameter may be fine while a nearby one, at a resonance or with a refractive index near 1, is not. A few spot checks cannot find them.
The physics and the conventions. A formula with a wrong sign, a phase matrix with the wrong normalization, or a refractive index of the opposite convention gives results that are precise but wrong. These errors need a comparison with independent codes and with known physics.

scatrs addresses the first with internal validation: every output is compared with reference values computed independently at very high precision, over the whole domain where scatrs claims accuracy. It addresses the second with external validation: comparisons with the published results of other codes and with analytic limits. Both run automatically, on Linux, macOS and Windows, for every change to the code.

Internal validation: is every result as accurate as claimed?¶

Each output of scatrs.mie has a declared accuracy, a few units of double-precision rounding, times a factor that grows with the size of the sphere and with how sensitive the output is to its inputs (near a sharp resonance, even rounding \(x\) and \(m\) to double precision changes the answer). The tests check it two ways:

Against rigorous reference values. An independent program computes the Mie series in Arb ball arithmetic, which carries a guaranteed error bound with every number, at hundreds of bits of precision. It never uses scatrs’s code. Its values cover size parameters from \(10^{-6}\) to \(2\cdot10^4\) and refractive indices from near 1 to metal-like: 306 spheres for the efficiencies and amplitudes, 80 for their derivatives, and 18 for the phase matrix and Greek coefficients.
On a dense grid. The same algorithms also run in double-double arithmetic, with about 32 significant digits (itself checked against Arb). The double-precision results are compared with these on a grid of thousands of spheres, so that no region of the domain goes unchecked.

The declared accuracy of each output is at least four times the worst error found, and a test fails if any point exceeds it. For the averages over size distributions, a further test checks that the error estimate of every output covers its true error, measured against the same integral at a far tighter tolerance.

Internal validation shows the results.

External validation: does scatrs agree with established results?¶

The reference values are computed from the same equations as scatrs. They would not catch a mistake in an equation, or a convention that differs from everyone else’s. So scatrs is also compared with results from outside its own code:

Wiscombe’s MIEV0 test cases. MIEV0 is the most widely used Mie code, and its published test cases are the standard benchmark for new ones. scatrs reproduces all fifteen cases, from \(x = 0.055\) to \(10^4\) and from \(m = 0.75\) to \(10 + 10i\), to within about one unit in the last of the seven digits published.
Analytic limits. Small spheres must follow Rayleigh scattering, in their cross sections and in the sign and normalization of every element of the phase matrix and every Greek coefficient. Large spheres must remove twice their geometric cross section, and every sphere must satisfy the optical theorem.
sasktran2. For size distributions, scatrs is compared with the Mie integration that sasktran2, the radiative transfer model scatrs grew out of, still uses: a separate implementation, compared on a single distribution and on an optics-database grid.

External validation runs the first two on every build of this documentation, and Comparison with sasktran2 shows the third.

Summary¶

Check	What it would catch	Reference	Runs
Arb reference values	numerical error in any output, derivatives included	independent arbitrary-precision computation with rigorous error bounds	every change
Double-double accuracy map	numerical error anywhere in the validated domain	the same algorithms with about 32 digits	every change
Ensemble error estimates	estimates that understate the error of a size average	the same average at a far tighter tolerance	every change
MIEV0 test cases	wrong equations or conventions	Wiscombe’s published values	every change
Analytic limits	wrong normalization or sign conventions	Rayleigh scattering, the optical theorem, the extinction paradox	every change
sasktran2	disagreement with an established distribution code	sasktran2’s Mie integration	when the comparison is rerun

How strong is each accuracy figure?¶

Every accuracy figure in these pages says what supports it:

Measured and enforced: the accuracy of the Mie outputs, \(C\,(\kappa' + \mathrm{cond}) \,2^{-52}\) with measured constants \(C\), checked against the Arb references and the double-double map. These are not proofs: a point that no test samples could in principle exceed its tolerance, but the dense map and the margin of four make that unlikely.
Calibrated: the error estimates of the ensembles are estimates, not proven bounds. A test checks them against the same integrals at a tight tolerance, on a grid of nine distributions.
Tested against Arb: the accuracy of the special functions (Gauss–Legendre quadrature and the Wigner d-functions) and of the double-double arithmetic used by the Rust API.
Measured: worst-case errors and timings quoted from the tests and benchmarks, or from one-off measurements on an Apple M4 Pro, which are marked as such.

Limitations and known issues lists what is not covered: inputs outside the validated domain, and the behaviour of the outputs near sharp resonances.