This document proposes a proof-of-concept method to estimate snow water equivalent (SWE) using GPS multipath signals. SWE represents the amount of water available for runoff and is important for hydrological studies. The author developed a theoretical model relating GPS signal properties to snow layer thickness and density. Field measurements in Montana were collected and a nonlinear algorithm was able to estimate SWE based on the model. Future work is suggested to improve the model and test the technique further.

1. Estimating Snow Water Equivalent for a
Snow-Covered Ground Reflector using GPS
Multipath Signals
Dr. Mark Jacobson
Mathematics Department
1500 University Drive, Billings, MT 59101
mjacobson@msubillings.edu

2. 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

3. II. Theory

4. Relative received power at the antenna is
rh  rv  exp(i ) 2
4 (h  t ) sin 
P  1 where, 
2 0
h = height of antenna above conducting surface, m
 elevation angle, degrees
i  1
c  2.997925  108 m/s, the speed of light in a vacuum
f  1.57542  109 Hz
0  c / f  0.1902937 m, free-space wavelength
t = snow layer thickness, m
   snow   s'  i s"  s'  1  2  d
 d  relative density of dry snow, g cm−3

5. For a single-layer of snow above a conducting
surface with horizontal polarization, x = h and
vertical 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

6. 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

7. Equipment Setup for GPS Measurements

8. In order to utilize a Quasi-Newton Algorithm (QNA)
efficiently in finding estimates of snow depth and density, we
approximate the relative complex permittivity value of dry
snow as
 snow    i    1  2 d
'
s
"
s
'
s
  
"
s
'
s
������=������
1 2
SE ������, ������snow = ������������������ ������ ������������ − ������ ������������ , ������, ������snow n = 8,168
������ − 2
������=1
From 45 input pairs, the smallest SE produced the following:
t = 6.8 cm  d  0.30  s'  1.60

13. 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

14. 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

15. 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

16. 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