Skew Voigt¶

Convolution with a Gaussian LSF¶

The skew Voigt profile extends the pseudo-Voigt with a skewness parameter \(\alpha\) that shifts flux toward the red or blue wing while preserving normalisation.

Definition¶

Given a pseudo-Voigt profile \(V(x)\) (centred at zero, normalised to 1, even in \(x\)) with Gaussian component FWHM \(\Gamma_g\) and Lorentzian component FWHM \(\Gamma_l\), define the Voigt FWHM via the Thompson et al. (1987) approximation

\[ \Gamma_V = C_1\,\Gamma_l + \sqrt{C_2\,\Gamma_l^2 + \Gamma_g^2}, \qquad C_1 = \tfrac{1+\delta}{2},\quad C_2 = \bigl(\tfrac{1-\delta}{2}\bigr)^{\!2},\quad \delta = 0.099\ln 2, \]

and the erf scale \(w_0 = \Gamma_V/(2\sqrt{\ln 2}) = \sigma_V\sqrt{2}\) where \(\sigma_V = \Gamma_V/(2\sqrt{2\ln 2})\) is the equivalent Gaussian sigma. The parametrisation satisfies the exact limits \(\Gamma_V\to\Gamma_g\) as \(\Gamma_l\to 0\) and \(\Gamma_V\to\Gamma_l\) as \(\Gamma_g\to 0\) (since \(C_1+\sqrt{C_2}=1\) exactly).

The skew Voigt profile is then

\[ V_\text{skew}(x) = V(x)\,\bigl[1 + \text{erf}\!\left(\tfrac{\alpha x}{w_0}\right)\bigr] = V(x)\,\bigl[1 + \text{erf}\!\left(\tfrac{\alpha x}{\sigma_V\sqrt{2}}\right)\bigr]. \]

Normalisation. Since \(V(x)\) is even and \(\text{erf}(\alpha x/w_0)\) is odd, their product is odd and integrates to zero, so \(\int V_\text{skew}\,dx = \int V\,dx = 1\) for any \(\alpha\). For the pure-Gaussian case this reduces exactly to the skew-normal distribution with shape parameter \(\alpha\) and dispersion \(\sigma_g\).

Approximation after LSF convolution¶

Let the LSF be \(G_\text{lsf}(x) = \mathcal{N}(0,\sigma_\text{lsf}^2)\). The convolution splits into a symmetric part and a skew correction:

\[ (V_\text{skew} * G_\text{lsf})(x) = \underbrace{(V * G_\text{lsf})(x)}_{V'(x)} + \int_{-\infty}^{\infty} V(t)\,\text{erf}\!\left(\tfrac{\alpha t}{w_0}\right) G_\text{lsf}(x-t)\,dt. \]

The symmetric part \(V' = V * G_\text{lsf}\) is the standard LSF-convolved pseudo-Voigt (Thompson et al. 1987 approximation with Gaussian width \(\Gamma_{cg} = \sqrt{\Gamma_g^2+\Gamma_\text{lsf}^2}\)). The skew correction does not have a closed form for the mixed Voigt case, so the convolved profile is approximated as

\[ (V_\text{skew} * G_\text{lsf})(x) \;\approx\; V'(x)\,\Bigl[1 + \text{erf}\!\Bigl(\frac{\alpha_\text{eff}\,x}{w_0'}\Bigr)\Bigr], \]

where \(w_0' = \Gamma_V'/(2\sqrt{\ln 2})\) uses the post-convolution Voigt FWHM

\[ \Gamma_V' = C_1\,\Gamma_l + \sqrt{C_2\,\Gamma_l^2 + \Gamma_{cg}^2}, \qquad \Gamma_{cg} = \sqrt{\Gamma_g^2 + \Gamma_\text{lsf}^2}, \]

and \(\alpha_\text{eff}\) is the effective skewness derived below.

Gaussian-body exact formula¶

For the pure-Gaussian component (\(\Gamma_l = 0\)) the skew correction integral factors exactly via the Gaussian product identity. The result is:

\[ \alpha_\text{gauss} = \frac{\alpha\,w_0}{\sqrt{w_0'^{\,2} + 2\,\alpha^2\,\sigma_\text{lsf}^2}}. \]

With the \(w_0\) definitions above this simplifies to

\[ \alpha_\text{gauss} = \frac{\alpha\,\Gamma_g}{\sqrt{\Gamma_g^2 + (1+\alpha^2)\,\Gamma_\text{lsf}^2}}. \]

Properties:

\(\alpha_\text{gauss} \to \alpha\) as \(\sigma_\text{lsf} \to 0\) (no LSF)
\(\alpha_\text{gauss} \to 0\) as \(\sigma_\text{lsf} \to \infty\) (LSF washes out skew)
\(|\alpha_\text{gauss}| < |\alpha|\) always (convolution reduces skewness)

FXIG2 boost correction for the Lorentzian component¶

When \(\Gamma_l > 0\), the Lorentzian contribution causes \(\alpha_\text{eff}\) to exceed \(\alpha_\text{gauss}\). A multiplicative boost \(B \geq 1\) was fit numerically over a grid of \((\textrm{lor}, \alpha, \eta) \in [0, 8] \times [0.3, 10] \times [0.1, 3]\), where:

\[ \textrm{lor} = \frac{\Gamma_l/2}{\sigma_g}, \qquad \eta = \frac{\sigma_\text{lsf}}{\sigma_g}, \qquad \xi = \frac{\textrm{lor}}{\eta} = \frac{\Gamma_l/2}{\sigma_\text{lsf}}. \]

\(\xi\) is the ratio of Lorentzian half-width to LSF sigma; when \(\xi\) is large the Lorentzian wings are resolved by the LSF and carry more skew. The boost is modelled as:

\[ \ln B = \frac{k\,\xi^a\,\eta^b}{(1 + q\,\xi^c)\,|\alpha|^d}, \]

with fitted parameters \((k, a, b, c, q, d) = (0.27045,\;0.53872,\;1.0461,\;1.7778,\;1.1286,\;0.34693)\).

Boundary conditions satisfied:

\(\ln B = 0\) at \(\textrm{lor} = 0\) (Lorentzian absent, Gaussian formula exact)
\(\ln B = 0\) at \(\eta = 0\) (no LSF, \(\alpha_\text{eff} = \alpha\) trivially)

Final formula¶

\[ \boxed{ \alpha_\text{eff} = \alpha_\text{gauss} \cdot \exp(\ln B) = \frac{\alpha\,w_0}{\sqrt{w_0'^{\,2} + 2\,\alpha^2\,\sigma_\text{lsf}^2}} \cdot \exp\!\left(\frac{k\,\xi^a\,\eta^b}{(1+q\,\xi^c)\,|\alpha|^d}\right) } \]

Accuracy (numerical validation, \(\Gamma_l \in [0,8\sigma_g]\), \(\eta \in [0.1, 3]\), \(\alpha \in [0.3, 10]\)): median profile error 0.51%, 95th-percentile 1.58%, maximum 2.23%, versus 1.27% / 5.91% for \(\alpha_\text{gauss}\) alone.

Pixel integration¶

The skew Voigt is not analytically integrated over pixels. Both the Gaussian skew correction (expressible via Owen’s T function) and the Lorentzian skew correction require numerical work that would add two further approximation layers on top of the Thompson pseudo-Voigt.

Instead, unite evaluates the approximation pointwise at the pixel midpoint and multiplies by the pixel width (midpoint-rule approximation), which is adequate when pixels are small relative to the profile width.