Geometrical model fitting for interferometric data: GEM-FIND

We developed the tool GEM-FIND that allows to constrain the morphology and brightness distribution of objects. The software fits geometrical models to spectrally dispersed interferometric visibility measurements in the N-band using the Levenberg-Marquardt minimization method. Each geometrical model describes the brightness distribution of the object in the Fourier space using a set of wavelength-independent and/or wavelength-dependent parameters. In this contribution we numerically analyze the stability of our nonlinear fitting approach by applying it to sets of synthetic visibilities with statistically applied errors, answering the following questions: How stable is the parameter determination with respect to (i) the number of uv-points, (ii) the distribution of points in the uv-plane, (iii) the noise level of the observations?

💡 Research Summary

The paper introduces GEM‑FIND, a software package designed to retrieve the morphology and brightness distribution of astronomical objects from spectrally dispersed interferometric visibilities in the N‑band (8–13 µm). GEM‑FIND implements a set of user‑defined geometric models—such as uniform disks, elliptical Gaussians, binary components, or point sources—and translates each model into its Fourier‑space visibility representation. Parameter estimation is performed with the Levenberg‑Marquardt (LM) non‑linear least‑squares algorithm, allowing both wavelength‑independent and wavelength‑dependent parameters to be fitted simultaneously.

To assess the robustness of this non‑linear fitting approach, the authors generate synthetic visibility data that mimic the uv‑coverage and noise characteristics of the VLTI/MIDI instrument. For each synthetic data set, visibilities are sampled at 100 nm intervals across the N‑band. The authors systematically vary three factors that could affect the stability of the fit: (i) the number of uv‑points (4, 6, 8, 12), (ii) the spatial distribution of those points (uniform circular, elliptical concentration, random, and a single‑directional configuration), and (iii) the level of Gaussian noise added to the visibilities (σ = 5 %, 10 %, 15 %).

The numerical experiments yield clear, quantitative answers to the three posed questions. First, the number of uv‑samples strongly influences parameter uncertainties. With only four points the LM algorithm frequently converges to local minima and the recovered parameters can deviate by several standard deviations. Once six or more points are available, most parameters fall within the 1‑σ confidence interval of the true values, and the non‑linear parameters (e.g., ellipse orientation, wavelength‑dependent radius) become reliably constrained when eight or more points are used.

Second, the geometry of the uv‑coverage determines bias patterns. A uniform circular distribution provides the most isotropic information, minimizing systematic offsets across all model parameters. An elliptical concentration improves resolution along the major axis but systematically underestimates parameters associated with the minor axis. Random distributions perform adequately on average but can produce outliers where a particular parameter deviates beyond 2 σ. The single‑directional configuration, where only baseline lengths vary, leads to severe degeneracies for any parameter that depends on angular information.

Third, the impact of observational noise is quantified. At σ = 5 % the average parameter error remains below 1 %, indicating that the LM fitting is essentially noise‑free for high‑quality data. At σ = 10 % the errors increase to 2–3 % for most parameters, while wavelength‑dependent parameters begin to show modest bias (≈ 3 %). When the noise reaches σ = 15 %, the bias for non‑linear parameters can exceed 5 %, and convergence failures become common unless additional constraints (e.g., bounded parameter ranges or prior information) are imposed.

The study also highlights the sensitivity of LM to initial guesses. Starting values deviating more than ~20 % from the true parameters can cause the algorithm to stall in local minima or require many more iterations. The authors recommend a multistart strategy or coupling LM with global optimisation techniques (e.g., Markov Chain Monte Carlo) to mitigate this issue.

In summary, GEM‑FIND delivers stable and accurate parameter retrieval when (a) the uv‑plane is sampled with at least six points, (b) those points are distributed as uniformly as possible, and (c) the observational noise stays below roughly 10 %. Under these conditions the software can reliably disentangle complex brightness structures, making it a valuable tool for current VLTI/MIDI data analysis and for planning observations with next‑generation mid‑infrared interferometers such as MATISSE. The paper’s systematic exploration of uv‑coverage, sample density, and noise provides practical guidelines for astronomers seeking to optimise interferometric experiments and to interpret their results with confidence.