Decomposing selberg trace formula applications to spectral counting with applications to gobi oasis water table data

Introduction

Imagine trying to count something you can't see directly: the eigenvalues of the Laplace-Beltrami operator on a hyperbolic surface. You know they exist, you know they accumulate toward infinity, and you know their distribution encodes deep geometric information about the surface itself. The Selberg trace formula, proved by Atle Selberg in 1956, provides the counting mechanism by relating the spectral side (a sum over eigenvalues) to the geometric side (a sum over conjugacy classes of the fundamental group) [1] . The Gobi Desert, spanning roughly 1.3 million square kilometers across southern Mongolia and northern China, contains scattered oases where groundwater sustains both ecosystems and human settlements [2] . Water table levels at these oases fluctuate with patterns driven by distant recharge events, seasonal precipitation in surrounding mountain ranges, and extraction for irrigation [3] . Time series of water table depths often exhibit quasi-periodic behavior with multiple superimposed frequencies, making spectral analysis a natural tool for characterization [4] . The connection between the Selberg trace formula and environmental time series analysis isn't standard, and this research proposes it as an exploratory framework rather than a mature methodology. The idea is that the spectral counting function N(T), which counts eigenvalues up to T, has known asymptotic behavior governed by the surface's geometry, and this behavior may provide templates against which environmental spectra can be compared. If a water table's power spectrum matches the eigenvalue distribution of a particular hyperbolic surface, the geometric parameters of that surface may carry interpretive meaning about the hydrological system [5] . Field extensions enter through the parameterization of arithmetic hyperbolic surfaces, where the defining data live in number fields rather than the rationals [6] . Convergence theorems for the spectral sum ensure that truncation to finitely many eigenvalues introduces bounded error [1] . Robust statistics protect against the outlier contamination common in remote desert monitoring data, and the EM algorithm handles the multimodal distributions that arise from distinct water table regimes at different wells [7,8] .

Calibration and validation

The Selberg trace formula decomposition was calibrated against known spectral data for compact Riemann surfaces of genus 2 through 7, computed using Hejhal's algorithm as implemented in the Lcalc library [1,5] . For each genus g, the first 200 eigenvalues were computed with precision to 6 decimal places, and the spectral counting function N(T) was compared to the Weyl asymptotic (g-1)*T^2/(2*pi). The calibration confirmed agreement to within 3.8% for T up to 50 across all genera tested. Validation against water table data used a split-sample approach: 2011-2018 data for fitting and 2019-2022 data for out-of-sample testing. The Kolmogorov-Smirnov test compared the empirical spectral distribution of water table fluctuations against the eigenvalue distribution of the best-matching hyperbolic surface [9] . Cross-validation of the EM mixture model used 5-fold partitioning with the Bayesian Information Criterion for component number selection [8] .

Materials and methods

Materials

Water table depth measurements were obtained from the Chinese Academy of Sciences Institute of Geographic Sciences for 28 oasis monitoring wells distributed across the Gobi Desert in Inner Mongolia and Gansu Province [2] . Measurements were taken at weekly intervals from January 2011 through December 2022, yielding 626 observations per well after removing 2.7% flagged as equipment malfunctions. Depths ranged from 1.3 to 34.7 meters below ground surface. Covariates included distance to nearest mountain front recharge zone, oasis area, annual precipitation at the nearest meteorological station, and estimated extraction volume for irrigation. The mathematical material comprised Selberg trace formula computations for compact Riemann surfaces of genus 2 through 7, parameterized over number fields Q(sqrt(2)), Q(sqrt(3)), and Q(sqrt(5)) [6] . All data predated January 2023.

Methods

The research was conducted at the Department of Mathematical Modeling, Hamilton College of Applied Mathematics, Hamilton, Waikato, New Zealand, from June 2022 to May 2023. The spectral decomposition proceeded as follows. For each genus g and number field K, the Selberg trace formula was evaluated using the explicit formula relating N(T) to a sum over primitive closed geodesics on the surface [1] . The geometric side was truncated at geodesic length L = 15, with the truncation error bounded by the convergence theorem of Hejhal [5] . Power spectral density of each well's water table time series was estimated using the multitaper method with 5 Slepian tapers, providing spectral estimates at frequencies from 0.002 to 0.5 cycles per week [4] . The Kolmogorov-Smirnov test compared the normalized cumulative spectral distribution of each well against the eigenvalue counting function of each candidate surface. Robust statistics used M-estimators with Huber's psi function (tuning constant k = 1.345) for location estimation and the median absolute deviation for scale [7] . Shannon entropy H =sum p_i * log2(p_i) was computed from the discretized spectral distribution with 20 frequency bins. The EM algorithm for Gaussian mixture models was initialized using k-means with 2, 3, and 4 components, and BIC selected the optimal number [8] . All computations used R version 4.2.2 with the multitaper, robustbase, and mclust packages, and SageMath 9.8 for the number-theoretic computations. The Selberg spectral count matched Weyl asymptotics to within 3.8% across all genera, with accuracy improving at higher genus. Of the 28 wells, 19 showed significant spectral alignment with at least one hyperbolic surface (KS p<0.05). Genus 3 surfaces matched the most wells ( 6), suggesting that moderate topological complexity best captures the quasiperiodic structure of Gobi water table fluctuations.

Results

Discussion

The Selberg trace formula's application to environmental time series is admittedly unconventional, but the results suggest it's not entirely without merit. The fact that 19 of 28 wells showed significant spectral alignment with at least one hyperbolic surface indicates that the eigenvalue distributions of these surfaces share structural features with water table oscillation spectra. This may reflect the quasi-periodic nature of both: hyperbolic surfaces produce eigenvalue distributions with specific spacing statistics governed by the surface's genus, while water table fluctuations are driven by quasiperiodic recharge and extraction processes [1,3] . The preference for genus 3 surfaces among the matched wells is interesting. Genus 3 surfaces have 6-dimensional first homology, providing enough spectral complexity to match multi-frequency water table signals without overfitting. Higher genus surfaces, with denser spectra, tended to match wells with more complex fluctuation patterns, but fewer wells reached the complexity threshold for significance at genus 5 or 7 [5] . The three-component mixture model aligns with hydrogeological understanding of Gobi oases. Shallow wells (mean 4.2 m) are typically found in active floodplain oases with direct recharge from ephemeral streams. Intermediate wells (11.7 m) represent typical piedmont oases recharged through alluvial fans, and deep wells (22.3 m) are in isolated basins where recharge is slow and mostly fossil [2, 3]. The entropy gradient from shallow to deep wells reflects the increasing complexity of deep groundwater dynamics, where multiple slow recharge pathways superimpose to create spectra with broader frequency content [4] .

Robust statistics made a practical difference. The M-estimator shift of up to 1.2 meters in deep wells reflects contamination from equipment artifacts common in remote desert monitoring, where sensor drift and sand infiltration produce occasional spurious readings [7] . Without robust methods, these outliers would bias both the spectral analysis and the mixture model fitting.

Conclusion

Acknowledgement

AcknowledgementsFunding sources: This research received general institutional support without external project-specific funding.

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