Rain Rate Estimation Bounds and Weather-Adaptive Pilot Allocation for LEO Satellite ISAC
Rain attenuates Ku-band satellite signals by up to 20~dB, encoding precipitation information along the Earth-space slant path. This paper derives the Bayesian Cramér-Rao bound (BCRB) for rain rate estimation from LEO broadband OFDM downlinks. Using corrected ITU-R P.838-3 coefficients, the standard CRB yields a minimum detectable rain rate $R_{\min} \approx 4.3\mmh$ for a single link at the $38^\circ$ reference elevation. We derive the prior Fisher information in closed form for log-normal rain ($c_v = 1.05$, from 186{,}292 samples) and show that a single-snapshot BCRB reduces $R_{\min}$ to $1.1\mmh$; exploiting temporal correlation ($ρ= 0.95$) over a 30-min window further tightens it to $0.95\mmh$, while multi-link fusion across $N = 215$ links lowers the operating-point RMSE \emph{lower bound} at $R = 20\mmh$ to approximately $0.07\mmh$. Building on these bounds, we formulate a weather-adaptive pilot allocation that minimizes the BCRB subject to a hard spectral-efficiency constraint, characterize its three-regime structure (full-sensing, throughput-tracking, outage), and pair it with a CUSUM rain onset detector achieving sub-10-min delay for $R \geq 20\mmh$. A closed-form analysis of dynamic LEO slant geometry identifies a sensing-optimal elevation at the P.618-validity floor of $15^\circ$ that yields a $1.58\times$ geometric improvement over the $38^\circ$ baseline, exposing a structural anti-correlation between sensing- and communication-optimal elevations along an orbital pass. Validation against 9.4~million radar samples from 215 Ku-band GEO satellite links ($r = 0.72$, RMSE~$= 1.24\dB$) and 113 rain gauges confirms the underlying attenuation model; the bounds transfer to LEO constellations under matched OFDM signal parameters, with dedicated LEO validation left for future work.