- Research
- Open Access
Estimating snow water equivalent for a slightly tilted snow-covered prairie grass field by GPS interferometric reflectometry
- Mark D Jacobson^{1}Email author
https://doi.org/10.1186/1687-6180-2014-61
© Jacobson; licensee Springer. 2014
- Received: 7 August 2013
- Accepted: 15 April 2014
- Published: 6 May 2014
Abstract
Snow water equivalent (SWE) measurements are necessary for the management of water supply and flood control systems in seasonal snow-covered regions. SWE measurements quantify the amount of water stored in snowpack; it can be estimated by the product of snow depth and density. In this paper, snow depth and density are estimated by a nonlinear least squares fitting algorithm. The inputs to this algorithm are global positioning system (GPS) signals and a simple GPS interferometric reflectometry model (GPS-IR) that incorporates a slightly tilted surface (GPS-IRT). The elevation angles of interest at the GPS receiving antenna are between 5° and 30°. A 1-day experiment with a snow-covered prairie grass field using GPS satellites PRN 15 and PRN 18 shows potential for inferring snow water equivalent using GPS-IRT. For this case study, the average inferred snow depth (12.4 cm) from the two satellite tracks underestimates the in situ measurements (17.6 cm ± 1.5 cm). However, the average inferred snow density (0.085 g∙cm^{-3}) from the two satellite tracks is within the in situ measurement range (0.08 g∙cm^{-3} ± 0.02 g∙cm^{-3}). Consequently, the average inferred SWE (1.05 g∙cm^{-2}) from the two satellite tracks is within the in situ calculation range (1.40 g∙cm^{-2} ± 0.36 g∙cm^{-2}). These results are also compared with the GPS-IR model.
Keywords
- Global positioning system (GPS)
- Snow depth
- Snow density
- Snow water equivalent (SWE)
- Multipath
- Reflection
1. Introduction
The amount of water stored in snowpack is one of the most important measurements for the management of water supply and flood control systems in seasonal snow-covered regions. Snow water equivalent (SWE) represents the amount of water stored in snowpack. Snow is important in agriculture for the northern Great Plains in the USA and the Canadian Prairies. For example, during the winter season, snow cover protects crops from extreme cold temperatures. In addition, snow provides moisture for these crops when the snow melts into the soil [1–3]. Improved winter-time snow measurements, such as SWE, would help resource managers to improve water use efficiencies in these areas. In particular, the spatial nature of SWE would give watershed researchers a new tool to use in scaling and in extrapolating point measurements to areas. Currently, SWE data are from point measurements. The United States Natural Resources Conservation Service (NRCS) of the Department of Agriculture (USDA) is dedicated in supporting Western US water managers in developing new techniques and products to improve water use efficiencies wherever possible.
In the mountainous Western United States, including Alaska, snow depth and SWE measurements are performed by the National Water and Climate Center (NWCC) USDA NRCS. They operate and manage the snowpack telemetry (SNOTEL) system [4–6] in high-elevation areas. This system has provided critical snow data for approximately 30 years. Currently, this network operates 730 remote sites. Although this method has a higher temporal resolution, it misses critical spatial variability because of its limited spatial footprints. In order to increase the spatial coverage of snow depth and SWE in these regions, a promising new remote-sensing technique using the reflected signals of the global navigation satellite systems (GNSS) is being investigated [7–11]. Such a remote-sensing technique is commonly termed GNSS reflectometry (GNSS-R). Specifically, it has been reported that reflected global positioning system (GPS) signals can provide useful information about snow depth [12–16] and possibly SWE [17, 18]. From recent snow depth studies [14], this promising new technique has been given the name GPS interferometric reflectometry (GPS-IR). This method is basically an L-band ground-based interferometer. In other words, a GPS receiver collects both direct and reflected GPS signals simultaneously which produces a resulting interference pattern. By processing and analyzing this pattern, the characteristics of the reflection surface can be derived, and the related geophysical parameters, such as snow depth and density, can be inferred. This paper outlines a technique for estimating dry snow depth, density, and SWE (the product of snow depth and density) for a slightly tilted snow-covered prairie grass field. For this paper, the name GPS-IR signifies using the GPS-IR technique with a horizontal ground surface (zero ground tilt) [17]. Consequently, the name GPS-IRT signifies using the GPS-IR technique with a tilted ground surface (nonzero ground tilt) along the axis of the receiving antenna's main beam center. In situ snow depth and density measurements are compared with the inferred GPS-IRT and GPS-IR results. The resulting SWE results are also compared.
2. Model
The total received signal power levels in both simulation and measurement results have been normalized to maximum signal levels for each satellite track. This normalization is referred to as the relative received power. Reflected signals from the air-snow interface arrive at the GPS receiver both coherently and incoherently. The proportions of these signals depend upon on the roughness of the air-snow surface. Smooth surfaces produce specular reflection signals which occur primarily within the first Fresnel zone about the specular point. This simple model neglects the scattering from rough surfaces and the incoherent component of the simulated received reflected power. Conversely, the measured received signal power at the GPS receiver includes these factors. A 1-day experiment attempts to partially answer the following question: Can this simple model be used to estimate snow depth and density for a slightly tilted snow-covered prairie grass field?
where 1 is the normalized direct coefficient for the RHCP field component, r_{h} is the reflection coefficient for the horizontally polarized field component; r_{v} is the reflection coefficient for the vertically polarized field component; $\mathit{\varphi}=\frac{4\mathit{\pi}\left[\mathit{h}+{\mathit{t}}_{3}-\left({\mathit{t}}_{1}+{\mathit{t}}_{2}\right)\right]\mathit{sin}{\mathit{\theta}}^{\prime}}{{\mathit{\lambda}}_{0}}$ is the phase shift difference in physical path length between the direct path and the air-snow interface reflected path [20]; $\mathit{i}=\sqrt{-1}$; h is the height of antenna above the frozen-soil surface (m); t_{1} is the snow layer thickness (m); t_{2} is the effective prairie grass layer thickness (m); t_{3} is the frozen-soil penetration depth (m), i.e., the effective reflector depth; θ′ is the local elevation angle (degrees) as given in (1); c = 2.997925 × 10^{8} m/s is the speed of light in a vacuum; f = 1.57542 GHz is the GPS L1 frequency; and λ_{0} = c/f = 0.1902937 m is the GPS L1 free-space wavelength.
Fixed input parameters for the model
ϵ _{2} | ϵ _{3} | t_{2}(cm) | t_{3}(cm) | h(cm) |
---|---|---|---|---|
1.5 - i 0 | 4.4 - i 0 | 5.3 | 5.0 | 71.5 |
The input parameters are relative complex permittivity value of crested wheatgrass (ϵ_{2}), the relative complex permittivity value of frozen soil (ϵ_{3}), the prairie grass layer thickness (t_{2}), the frozen-soil penetration depth (t_{3}), and the height of the antenna above the frozen-soil surface (h).
It must be noted that the 5-cm frozen-soil penetration depth is obtained from dry-ground assumptions [14, 21, 22]. This penetration depth is more likely deeper for frozen ground. Furthermore, soil penetration depth is still unknown for active systems like GPS-IR; however, it known for passive systems [23, 24]. Therefore, 5 cm is only an estimate for the penetration depth of frozen ground when using GPS signals.
These inferred tilt angles provide the optimal fit between theory and measurement (see the next section). Furthermore, a small negative tilt (terrain is sloping downhill towards the axis of the receiving antenna's main beam center) was observed at the measurement site. The small tilt angle difference between the two GPS satellites occurs because the satellites' azimuth angle tracks are not along the axis of the receiving antenna's main beam (see the next section). In other words, the reflected GPS signals from each satellite are from different sections of the snow-covered prairie grass field which, in turn, have slightly different slopes.
3. SWE estimation
Specifications for GPS receiving antenna
Specification | Value |
---|---|
Frequency range | 1,575.42 ± 1.023 MHz |
Gain | +3.0 dBi, minimum at 90° |
-4.0 dBi, minimum at 20° | |
Polarization | RHCP |
Axial ratio | +4.0 dB, maximum at 90° |
+6.0 dB, maximum at 10° | |
Half-power beamwidth | 140° |
Front-to-back ratio | +15 dB |
The measured data from the snow-covered prairie grass field are fitted with (2) in a quasi-Newton algorithm (QNA) [25, 26]. The measurements were performed on January 20, 2012 with partly cloudy skies between 20:33 and 21:38 GPS time for satellite PRN 15 and between 22:20 and 23:30 GPS time for satellite PRN 18. The azimuth angles were at approximately 51° for PRN 15 for the 1.1-h measurement. On the other hand, the azimuth angles increased from 80° to 100° for PRN 18 for the 1.2-h measurement. Fresh snow was deposited on the prairie grass field from a 2-day storm that occurred from January 18 to 19, 2012. The snow storm was preceded by cold air temperatures of approximately -20°C. This produced very low snow density values [14, 24].
The techniques for measuring snow depth (t_{1}) and snow density (ρ_{d}) are given in [17]. Briefly, snow depths (t_{1}) were measured with a metal ruler. ρ_{d} measurements required the following tools: (1) a 60-cm-long polyvinyl chloride (PVC) tube with a 7.6-cm inside diameter (one end of this tube was beveled to enhance the cutting affect through the snow), (2) a 5-cm-thick circular Styrofoam (The Dow Chemical Company, Midland, MI, USA) with a 25-cm diameter, (3) a portable digital scale with a resolution of 0.1 g and an accuracy of 0.5 g, (4) a 4-L plastic ziplock bag, and (5) a rigid sheet of cardboard. The average snow temperatures for the PRN 15 and PRN 18 tracks were -9.5°C and -5.8°C, respectively. The snow temperatures were measured with a digital thermometer with an attached probe with a resolution of 0.1°C and an accuracy of 1°C. This probe was placed at approximately 1 cm below the air-snow interface at the site location. This is a one-point temperature measurement over time. Eleven pairs of snow depth and density were taken during this experiment.
The measured snow depth and density ranges are approximately 17.6 cm ± 1.5 cm and 0.08 g∙cm^{-3} ± 0.02 g∙cm^{-3}, respectively; the calculated SWE range is approximately 1.40 g∙cm^{-2} ± 0.36 g∙cm^{-2}[17]. For this paper, the elevation angles were restricted to be between 5° and 30° in order to maximize the multipath effects from the snow layer. This occurs because the electrical path length of the GPS signal through the snow increases as the elevation angle decreases. This elevation range is approximately less than the Brewster angle (≥ 29° [27]) for the snow-covered prairie grass field. Therefore, the reflected wave is not purely RHCP but is right-hand elliptically polarized (RHEP) [19]. For this elevation angle range (similar to that chosen by [12, 14–18]), there are 7,292 data points for PRN 15 and 8,097 data points for PRN 18. Satellite PRN 15 had fewer data points than PRN 18 because the GPS receiver stopped receiving data from PRN 15 between elevation angles of 19.2° and 18.1°. This occurred because the portable generator ran out of gas. Fortunately, these missing data points did not seriously affect this study.
We use the geometric optics approximation (far field, Fraunhofer diffraction) to estimate the first Fresnel zone dimensions. The first Fresnel zone dimensions are used to determine if the receiving antenna's power pattern, relative to the specularly reflected path, is within the area of the snow-covered prairie grass field. With an antenna height of 48.6 cm above the air-snow interface and an elevation angle of 5°, the first Fresnel zone is calculated to have a major axis length of approximately 26 m and a minor axis length of approximately 2.3 m [10]. The size of this ellipse is largest here at 5° and becomes smaller and closer to the antenna as the satellite rises. For example, the entire first Fresnel zone is only a few meters from the antenna when the elevation angle is at 30°. The snow-covered prairie grass field is approximately topographically flat for approximately 60 m in the direction of the axis of the receiving antenna's main beam (major axis) and also for approximately 100 m in the direction perpendicular (minor axis) to the axis of the receiving antenna's main beam. However, there is a small negative ground surface slope (tilt < 5°) with respect to the horizontal when approaching the receiving antenna along its major axis; this small negative slope was neglected in [17]. Therefore, the first Fresnel zone lies entirely on a slightly tilted snow-covered prairie grass field.
where P is given in (2), and Norm is the normalization constant that minimizes the difference, in a least squares sense between theory and measurement. In this case study, the initial guess value of Norm is set to 2.5 in the QNA for each paired combination of snow depth and density. The resulting Norm value produced by the QNA for each paired combination is between 2.0 and 3.0. Each final Norm value minimizes the errors in the constraints [26]. Essentially, the Norm value vertically shifts the theoretical curve to match the measurement data in a least squares sense. The normalization constant, Norm, is used as the input to the fitting function PdB in a QNA. A snow depth range of approximately 10 to 15 cm and a snow density range from 0.04 to 0.12 g∙cm^{-3} are chosen to bracket the measured values of this experiment. For each GPS satellite, 112 different paired combinations of snow depth and density provided the necessary resolution and range to estimate the absolute minimum between theory and measurement using a QNA.
where y is the relative measured power value in dB; PdB is the normalized fitting function in dB; θ_{ i } are the elevation angles in degrees; θ_{t}, t_{1}, t_{2}, t_{3}, ρ_{d}, ϵ_{2}, ϵ_{3}, T, and f are the parameters given in Section 2; n is the number of data points for each satellite track, and min is the abbreviation for minimize.
Inferred values of snow depth, snow density, SWE, and SE using the GPS-IRT (nonzero ground tilt) method
Method | Snow depth (cm) | Snow density (g∙cm^{-3}) | SWE (g∙cm^{-2}) | SE (dB) |
---|---|---|---|---|
GPS-IRT PRN 15 | 13.0 (18.3) | 0.08 (0.12) | 1.04 (2.20) | 0.377 (0.642) |
GPS-IRT PRN 18 | 11.7 (17.5) | 0.09 (0.14) | 1.05 (2.45) | 0.760 (0.798) |
GPS-IRT AVG | 12.4 (17.9) | 0.085 (0.13) | 1.05 (2.33) | 0.573 (0.721) |
In situ | 17.6 ± 1.5 | 0.08 ± 0.02 | 1.40 ± 0.36 |
With this in mind, we list several model deficiencies that limit the accuracy of retrieving the snow parameters. First, although the approximation of snow by a planar homogenous layer (with respect to a slightly tilted ground) is reasonable, the approximation of prairie grass by a similar homogeneous planar layer (with respect to a slightly tilted ground) is questionable. Second, approximating the frozen soil as a uniform medium with a constant dielectric permittivity is also questionable. A more realistic frozen-soil model requires a soil dielectric permittivity dependence with depth, and a relationship between the soil dielectric permittivity and soil moisture, as well as soil material composition [22, 28, 29]. Third, the approximation of a slightly tilted, flat ground surface is questionable since the actual ground surface topography is more complicated than this. The true ground surface topography affects the reflected signals, which, in turn, affects the retrievals of snow depth and density [12]. Therefore, the ground surface topography should be measured and modeled before the satellites will be covered with snow. By using this modeled ground surface topography in the theoretical model, more accurate retrievals of snow depth and density may be possible.
4. Conclusions
We investigated a nonlinear least squares fitting technique for inferring snow depth and density for a snow-covered prairie grass field using GPS-IRT during a 1-day experiment. The product of these two parameters provides an estimate of the SWE, which helps agricultural resource managers in estimating the amount of moisture for crops when the snow melts into the soil. In addition, SWE estimates are extremely important for hydrological studies in seasonal snow-covered regions because it represents the amount of water potentially available for runoff.
A QNA produced an average inferred snow depth (12.4 cm), from two satellite tracks, that underestimates the in situ measurements (17.6 cm ± 1.5 cm). However, the average inferred snow density (0.085 g∙cm^{-3}), from two satellite tracks, is within the in situ range (0.08 g∙cm^{-3} ± 0.02 g∙cm^{-3}). Consequently, the average inferred SWE (1.05 g∙cm^{-2}), from two satellite tracks, is within the in situ calculation range (1.38 g∙cm^{-2} ± 0.36 g∙cm^{-2}). These results show that there may be potential for estimating snow depth, snow density, and SWE using GPS-IRT. However, caution must be exercised when attempting retrievals of snow depth, snow density, and SWE from this simple model.
For comparison purposes, the GPS-IRT (nonzero ground tilt) model estimates snow density and SWE better than the GPS-IR (zero ground tilt) model. However, the GPS-IR model did better in estimating snow depth than the GPS-IRT model. Snow depth and density retrieval results from these models indicate that terrain slopes may be an important factor in the physical model for retrieving snow depth, snow density, and SWE. Furthermore, these retrieval results also indicate that these simple models may be limited in producing accurate retrievals of snow depth, density, and SWE. Further investigation is needed to quantify the usefulness of these models in estimating snow depth, snow density, and SWE.
Continuing research will explore the feasibility of using this technique to infer SWE for snow layers above different types of vegetation, including no vegetation (bare soil). This will require further measurements for different snow depths and densities in open and mountainous terrains. Also, the received signals from different GPS satellites at a specific site and over an appropriate time period need to be compared and analyzed. In addition, potential biases in the retrieved parameters might be reduced if the receiving antenna's height is increased from 0.5 m to approximately 2 m (typical antenna height for snow-sensing GPS stations [12–15]). An increased antenna height will produce more power oscillations at the receiving antenna. Continuing theoretical developments will incorporate the following: more snow layers, surface roughness of snow and frozen soil, realistic ground surface topography, realistic prairie grass and frozen soil compositions, the configuration of a horizontally mounted (zenith-pointing) GPS antenna, and the antenna beam pattern including its antenna phase center.
In conclusion, if the GPS-IRT technique can be used in estimating SWE, then it may be more cost-effective than the current techniques. In particular, it may expand the spatial coverage of SWE measurements that are not currently provided by SNOTEL sites. For example, there are hundreds of geodetic GPS receivers operating in snowy regions in the USA [15]. Therefore, some of these GPS receivers could possibly be used to estimate SWE. Furthermore, low-cost GPS receivers could potentially be placed in agricultural snow-covered areas to estimate SWE for crops.
Declarations
Acknowledgements
The author would like to thank Montana State University Billings' (MSUB) Research and Creative Endeavor Grant Committee and Tasneem Khaleel of MSUB for their funding. In addition, the author would like to thank M. McBride of MSUB, J. Jacobson (my mother), W. Dotson of Keri Systems, and C. McFarland and T. McFarland (land owners) for their critical involvement in this research. The author would also like to thank the anonymous reviewers for their superb comments and suggestions.
Authors’ Affiliations
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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.