Estimating Snow Water Equivalent for aSnow-Covered Ground Reflector using GPS Multipath Signals Dr. Mark Jacobson Mathematics Department 1500 University Drive, Billings, MT 59101 firstname.lastname@example.org
I. Introduction Proof-of-concept method for estimating snow water equivalent (SWE) at the GPS frequency of 1.5 GHz SWE is the single most important parameter for hydrological studies because it represents the amount of water potentially available for runoff SWE estimations are used for the management of water supply and flood control systems The U.S. D.O.A. operates and manages the Snowpack Telemetry (SNOTEL) system, 730 sites GPS signals could provide a new and economical technique for estimating SWE
Relative received power at the antenna is rh rv exp(i ) 2 4 (h t ) sin P 1 where, 2 0h = height of antenna above conducting surface, m elevation angle, degreesi 1c 2.997925 108 m/s, the speed of light in a vacuumf 1.57542 109 Hz0 c / f 0.1902937 m, free-space wavelengtht = snow layer thickness, m snow s i s" s 1 2 d d relative density of dry snow, g cm−3
For a single-layer of snow above a conductingsurface with horizontal polarization, x = h andvertical polarization, x = v iZ x tan 1 sin rx Zh iZ x tan 1 cos 2 2 Zv cos 2 t cos 2 sin 0
III. Measurements and Computation March 31, 2007 T = -1.7 C, t = 7.6 cm, h = 45.1 cm, no snow density data III. Measurements and Computation March 31, 2007 T = -1.7 C, t = 7.6 cm, h = 45.1 cm
In order to utilize a Quasi-Newton Algorithm (QNA)efficiently in finding estimates of snow depth and density, weapproximate the relative complex permittivity value of drysnow as snow i 1 2 d s " s s " s s = 1 2SE , snow = − , , snow n = 8,168 − 2 =1From 45 input pairs, the smallest SE produced the following: t = 6.8 cm d 0.30 s 1.60
IV. Conclusions Theoretical results and GPS measurements are in good agreement using a nonlinear QNA Estimating SWE may be possible using a nonlinear least squares technique “Inferring Snow Water Equivalent for a Snow-Covered Ground Reflector Using GPS Multipath Signals”, Remote Sensing, Vol. 2, 2426-2441, October 2010
V. Future Work Try a QNA for a snow layer above frozen soil Collect more in situ measurements of snow depth, snow density, and frozen soil permittivity Try other nonlinear least-squares algorithms: Levenbeg-Marquardt and Conjugate Gradient Incorporate 2 or more snow layers in the theoretical model
V. Future Work (continued) Incorporate the antenna pattern in the theoretical model Incorporate surface roughness of snow and frozen soil in the theoretical model Use a horizontally-mounted (zenith-pointing) GPS antenna Investigate this technique for GPS antennas housed on an aircraft or satellite
Acknowledgment Montana State University Billings – Dr. Tasneem Khaleel, Dean CAS – Dr. Maggie McBride, Math Dept. – RACE Grant Ron and Jeanne Jacobson, my parents Wade Dotson, Trimble Navigation C. McFarland and T. McFarland, land owners Anonymous reviewers of paper
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