This paper presents a measurement-based framework for characterizing altitude-dependent spectral behavior of signals received by a tethered Helikite unmanned aerial vehicle (UAV). Using a multi-year spectrum measurement campaign in an outdoor urban environment, power spectral density snapshots are collected over the 89 MHz--6 GHz range. Three altitude-dependent spectral metrics are extracted: band-average power, spectral entropy, and spectral sparsity. We introduce the Altitude-Dependent Spectral Structure Model (ADSSM) to characterize the spectral power and entropy using first-order altitude-domain differential equations, and spectral sparsity using a logistic function, yielding closed-form expressions with physically consistent asymptotic behavior. The model is fitted to altitude-binned measurements from three annual campaigns at the AERPAW testbed across six licensed and unlicensed sub-6 GHz bands. Across all bands and years, the ADSSM achieves low root-mean-square error and high coefficients of determination. Results indicate that power transitions occur over narrow low-altitude regions, while entropy and sparsity evolve over broader, band-dependent altitude ranges, demonstrating that altitude-dependent spectrum behavior is inherently multidimensional. By explicitly modeling altitude-dependent transitions in spectral structure beyond received power, the proposed framework enables spectrum-aware UAV sensing and band selection decisions that are not achievable with conventional power- or threshold-based occupancy models.