Transit-Based Surface Mapping

• Transit-based surface mapping is a method that extracts detailed spatial information from transit data, applying photometric models for exoplanetary spot mapping and sensor-driven techniques in urban settings.
• The approach integrates forward modeling, Bayesian inference, and agent-based segmentation to achieve high-resolution mapping, supporting applications from stellar surface analysis to autonomous urban navigation.
• It enables temporal trend analysis by assimilating continuous transit and sensor data, facilitating real-time monitoring of stellar evolution and urban infrastructure changes.

Transit-based surface mapping refers to the extraction and interpretation of spatially and temporally resolved surface or infrastructure properties through the analysis of transit observations. The term encompasses several domains, notably exoplanetary science—where it arises from detailed modeling of variations in stellar brightness during planetary transits—and urban or autonomous systems mapping—where vehicles and sensors traversing a surface generate local observations that are assimilated into digital surface representations. Rigorous implementations span methods for stellar spot characterization, agent-based semantic segmentation, network-based accessibility mapping, and terrain modeling via stochastic meshes.

1. Principles of Transit Photometry in Surface Mapping

In exoplanetary systems, transit-based surface mapping exploits high-cadence photometric data to resolve features occulted by a transiting planet. Fine-scale modulations (localized "bumps") in the transit light curve arise when the planet crosses photospheric spots on the host star. Detailed forward modeling incorporates limb darkening (often quadratic), spot geometry (radius $S$, contrast $f_i$, longitude), and planetary trajectory (fixed latitude chord), converting photometric deficits

$D = (1 - f_i) S^2$

into spatial spot maps. The mapping is parameterized through spatial resolution along the transit chord; as demonstrated for CoRoT-2, this can attain $2^\circ$ longitudinal precision over dozens of transits (Silva-Valio et al., 2011).

Spot longitudes are converted between inertial (topocentric) and rotating reference frames via

$$\beta_{\rm rot} = \beta_{\rm topo} - 360^\circ \cdot (n P_{\rm orb} / P_{\rm star})$$

where $P_{\rm orb}$ is the planetary orbital period and $P_{\rm star}$ the stellar rotation period. This enables tracking spot drift and the assessment of latitudinal differential rotation.

2. Algorithms for Transit-Based Mapping in Urban and Autonomous Systems

Transit-based mapping generalizes to terrestrial domains through agent- and sensor-driven frameworks:

Stochastic Triangular Mesh (STM):

Mobile robots create a $2.5$-D terrain mesh composed of surfels, each defined by three vertex heights $(h_0, h_\alpha, h_\beta)$ and roughness parameter $\nu$:

$$\gamma = (1 - \alpha - \beta) h_0 + \alpha h_\alpha + \beta h_\beta + \epsilon, \qquad \epsilon \sim \mathcal{N}(0, \nu)$$

Incremental Bayesian updates leverage variational message passing (VMP) and loopy belief propagation (LBP), integrating noisy sensor data (LiDAR, stereo vision) with robot pose uncertainties—often in landmark-relative (IRF) frames. Surfels are interconnected to enforce cross-element continuity (Lombard et al., 2019).

Ego-Trajectory and TrackletMapper:

Robots automatically annotate ground surface types (sidewalks, roads, crossings) by projecting both their own SLAM-derived trajectories and detected tracklets of other agents into camera images:

$u = K \cdot T_C^B \cdot T_B^W \cdot \hat{p}$

$u$ is the pixel coordinate, $K$ the intrinsic matrix, and $T$ the relevant transformations. Segmentation models (e.g., DeepLabv3+) are trained on sparse labels and densified via self-distillation, with log-odds aggregation producing robust semantic maps (Zürn et al., 2022).

3. Model-Free and Probabilistic Inference of Surface Properties

Model-Free Stellar Mapping:

Transit imaging approaches dispense with stellar atmosphere or limb-darkening assumptions. Instead, phase-folded and median-combined light curves define reference surface intensities, while deviations in individual transits trigger iterative brightness map updates:

$$S_o^{n+1}(x, y, \phi) = S_o^n(x, y)\left[1 - (1 - P(x, y, \phi))(F_t(\phi) - F_s(\phi))c \right]$$

with regularization (first-order Tikhonov) smoothing the resulting maps:

$$\Phi = \sum_{x,y} W(x, y)[\Delta S(x, y)]^2 + \Lambda \sum_{x,y}\left[ \left( \frac{d \Delta S(x, y)}{dx} \right)^2 + \left( \frac{d \Delta S(x, y)}{dy} \right)^2 \right]$$

This methodology is validated against both synthetic and archival observational data (e.g., FORS2/VLT) (Aronson, 2019).

Hierarchical Bayesian Stellar Mapping:

The StarryStarryProcess framework models stellar surfaces via spherical harmonics, spot parameters drawn from hyperpriors, and planetary transits to resolve local features. The spot latitude distribution employs the Beta function in $\cos \varphi$,

$$p(\cos \varphi | \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} (\cos \varphi)^{\alpha-1} (1 - \cos \varphi)^{\beta - 1}$$

Joint inference over the surface map and transit geometry is performed using marginal likelihoods combining Gaussian process priors and transit design matrices (Sagynbayeva et al., 30 Apr 2025).

4. Network and Geographic Transit Surface Mapping

Transit Network Isochrone Mapping:

GTFS2STN transforms static GTFS schedules into spatiotemporal hypergraphs, duplicating stops across time slices, connecting nodes with "transit," "stop" (waiting), and "walking" links. The system supports comprehensive accessibility analysis, with shortest path (Dijkstra) searches performed over the network to produce isochrone contours:

• Transit link: $(i, t) \to (j, t + \Delta t_{transit})$
• Walking link: $(i, t) \to (k, t + \Delta t_{walk})$ if $d(i, k) \leq \text{buffer}$
• Waiting link: $(i, t) \to (i, t + \Delta t)$

Comparative studies with platforms such as Mapnificent highlight the impact of integrated waiting time and user-customizable feed inputs (Liu et al., 4 May 2024).

Geographically Accurate Transit Map Generation:

LOOM utilizes GTFS data to construct line graphs representing shared segments, followed by integer linear programming (ILP) to minimize map artifacts (crossings, separations), leveraging graph reductions for large-scale tractability. This enables the production of maps that reflect the true spatial topology of transit networks, suited for overlays and tiling in mapping services (Bast et al., 2017).

5. Temporal Evolution and Trend Analysis

Stellar Surface Evolution:

Longitude-binned spot coverage time series are extracted; periodicities are identified with Lomb–Scargle periodograms, revealing spot evolution timescales (9–53 days, mean $\sim$31 days) and global oscillations at $\sim$17.7 days—potentially indicative of magnetic cycles or star–planet interactions. Temporal domain comparisons are linked to differential rotation estimates via

$$P_{CoRoT-2} = \frac{360^\circ}{80.5 - 2.9 \cos^2(90^\circ - \alpha)}, \quad \Delta\Omega \simeq 0.042\,\text{rad/d}$$

as evidenced by differences in rotation periods sampled at distinct latitudes (Silva-Valio et al., 2011).

Urban Infrastructure Trend Mapping:

Time series application of transformer-based segmentation models (leveraging pre-trained encoders such as SAM) to historical aerial imagery enables quantitative detection of urban infrastructure changes. Ratios of predicted to ground-truth areas for streets, parking, and pedestrian regions are used to estimate multi-decade trends in cities such as Madrid and Vienna—e.g., up to 20% decrease in parking and 30-40% increase in road and sidewalk surface in central districts (Pliego et al., 19 Mar 2025).

6. Innovations and Application Contexts

Transit-based surface mapping provides unique advantages:

• Direct probing of spatial features otherwise inaccessible in integrated signal analyses (e.g., nullspace restrictions in rotational light curves (Sagynbayeva et al., 30 Apr 2025)).
• Scalability to large datasets without manual annotation, facilitated via agent-based labeling, model-free mapping, and self-distillation techniques.
• Integration with real-time monitoring and visualization platforms, including VR infrastructure for rail networks (Wang et al., 2021).
• Cross-domain extensibility: from astrophysical parameter inference to autonomous navigation, urban planning, and historical trend analysis.

7. Summary Table of Methodological Approaches

Domain	Approach	Key Features
Stellar Mapping	Spot-fitting transit photometry	Spatial spot parameter inference, differential rotation
Urban Mapping	TrackletMapper	Multimodal agent-trajectory labeling, self-distillation
Terrain Mapping	STM meshes	Bayesian surfel updates, uncertainty propagation
Accessibility	GTFS2STN, LOOM	Isochrone generation, line-graph ILP optimization
Historical Trends	Transformer segmentation	Automated geospatial dataset synthesis, temporal analysis

Transit-based surface mapping encompasses a rigorously quantitative suite of methods in which the spatial and temporal characteristics of a surface—whether stellar, terrestrial, or urban—are reconstructed from transit-occluded observations or vehicle trajectories, leveraging mathematical modeling, probabilistic inference, and network-based algorithms. These methods yield detailed physical and semantic maps, support evolutionary and trend analyses, and underpin a growing spectrum of scientific and operational applications.