= Angular resolution set by transverse spectral noise
{title2=$\Delta_c=\arcsin\sqrt{2c/[R^2(\epsilon^{-2}-\sigma^{-2})]}$}
\b[Angular resolution set by transverse spectral noise] quantifies how sharply an observed spectral vector constrains the orientation of an anisotropic two-coordinate noise model. Use the <covariance> from <transverse-noise limit of a rotating rank-one covariance model>, with known longitudinal <standard deviation> $\sigma$ and transverse <standard deviation> $0<\epsilon<\sigma$. For an observation of length $R>0$ aligned with orientation $\theta_0$, the Gaussian density gives
$$
\log\frac{p(\mathbf y\mid\theta)}{p(\mathbf y\mid\theta_0)}
=-\frac K2\sin^2(\theta-\theta_0),\qquad
K=R^2(\epsilon^{-2}-\sigma^{-2}).
$$
The determinant is independent of orientation, so the angular dependence comes entirely from the <covariance>-weighted residual.<SciPy 1.16.2 documentation>[scipy.stats.multivariate_normal, Notes, full-rank Gaussian density]{ref}
Define a local half-width by a chosen log-density drop $c>0$. For $2c\leq K$, the lobe around $\theta_0$ has half-width
$$
\Delta_c=\arcsin\sqrt{\frac{2c}{K}}.
$$
When this angle is small and $\epsilon\ll\sigma$, it is approximately $\sqrt{2c}\,\epsilon/R$. If $2c>K$, even an orthogonal orientation fails to reach that drop, so this local-width criterion does not exclude any orientation. At $R=0$ or equal noise scales, the observation has no angular preference in this model.
For an original example, choose $R=2$, $\sigma=0.8$ and $\epsilon=0.04$ in consistent signal units. Then $K=2493.75$. A drop $c=1/2$ gives half-width about $0.0200$ radians, or $1.15^{\circ}$. Reducing only the transverse <standard deviation> to $0.02$ approximately halves the width. Increasing a numerical regularization parameter is not an experimental improvement in this noise scale.
The <global orientation ambiguity in a rank-one covariance fit> remains: orientations separated by $\pi$ give the same <covariance>. A narrow local lobe does not distinguish those two orientations. Nor is a fixed log-density drop automatically a <confidence interval> with a stated coverage probability. Multiple observations, unknown noise scales and experimental nuisance parameters require their corresponding inference model.
Repeated reference spectra under stable conditions can estimate the longitudinal and transverse noise components. Their <covariance> must include shared calibration effects, as required by the measurement-propagation rule.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} The numerical example is a model calculation in spectral coordinates, rather than a specification of a particular optical instrument.
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