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Using GPS Multipath for Snow-Depth Estimation By Felipe G. Nievinski and Kristine M. Larson INNOVATION INSIGHTS by Richard Langley FRINGES. No, I’m not talking about the latest celebrity hairstyles nor the canopy of an American doorless, four-wheeled carriage from yesteryear (think Oklahoma!). I’m talking about interference fringes. But there is a connection to these other uses of the word fringe as we’ll see. You’ve all seen interference fringes at your local gas station, typically after it has just rained. They are the alternating bands of color we perceive when looking at a gasoline or oil slick in a puddle of water. They are caused by the white light from the Sun or artificial lighting reflected from the top surface of the slick and that from the bottom surface at the slick-water interface combining or interfering with each other at our eyeballs. The two sets of light waves arrive slightly out of phase with each other, and depending on the wavelengths of the reflected light and our angle of view, produce the colorful fringes. If the incident light was monochromatic, consisting of a single frequency or wavelength, then we would perceive just alternating bright and dark bands. The bright bands result from constructive interference when the phase difference is a near a multiple of 2π whereas the dark bands result from destructive interference when the difference is near an odd multiple of π. Interference fringes had been seen long before the invention of the automobile. They are clearly seen on soap bubbles and the iridescent colors of peacock feathers, Morpho butterflies, and jewel beetles are also due to the interference phenomenon rather than pigmentation. Sir Isaac Newton did experiments on interference fringes (amongst other things) and tried to explain their existence — wrongly, it turned out. But he did coin the term fringes since they resembled the decorative fringe sometimes used on clothing, drapery, and, yes, surrey canopies. It was the English polymath, Thomas Young, who, in 1801, first demonstrated interference as a consequence of the wave-nature of light with his famous double-slit experiment. You may have replicated his experiment in a high-school physics class. I did and I think I did it again as an undergraduate student taking a course in optics. Already by that point I was aiming for a career in physics or space science but I didn’t know that as a graduate student I would do research involving interference fringes. But not using light waves. My research involved the application of very long baseline interferometry or VLBI to geodesy. VLBI had been developed by radio astronomers to better understand the structure of quasars and other esoteric celestial objects. At either ends of a baseline connecting large radio telescopes, perhaps stretching between continents, the quasar signals were recorded on magnetic tape and precisely registered using atomic clocks. When the tapes were played back and the signals aligned, one obtained interference fringes as peaks and troughs in an analog or digital waveform. Computer analysis of these fringes not only provided information on the structure of the observed radio source but also on the distance between the radio telescopes — eventually accurate enough to measure continental drift.  But what has all of this got to do with GPS? In this month’s column, we look at a technique that uses fringes generated by signals arriving at an antenna directly from GPS satellites and those reflected by snow surrounding the antenna to measure its depth and how it varies over time. GPS for measuring snow depth; who would have thought? “Innovation” is a regular feature that discusses advances in GPS technology and its applications as well as the fundamentals of GPS positioning. The column is coordinated by Richard Langley of the Department of Geodesy and Geomatics Engineering, University of New Brunswick. He welcomes comments and topic ideas. Snowpacks are a vital resource for human existence on our planet. They provide reservoirs of fresh water, storing solid precipitation and delaying runoff. One sixth of the world population depends on this resource. Both scientists and water-supply managers need to know how much fresh water is stored in snowpack and how fast it is being released as a result of melting. Snow monitoring from space is currently under investigation by both NASA and ESA. Greatly complementary to such spaceborne sensors are automated ground-based methods; the latter not only serve as essential independent validation and calibration for the former, but are also valuable for climate studies and flood/drought monitoring on their own. It is desirable for such estimates to be provided at an intermediary scale, between point-like in situ samples and wider area pixels. In the last decade, GPS multipath reflectometry (GPS-MR), also known as GPS interferometric reflectometry and GPS interference-pattern technique, has been proposed for monitoring snow. This method tracks direct GPS signals, those that travel directly to an antenna, that have interfered with a coherently reflected signal, turning the GPS unit into an interferometer (see FIGURE 1). Its main variant is based on signal-to-noise ratio (SNR) measurements, although GPS-MR is also possible with carrier-phase and pseudorange observables. Data are collected at existing GPS base stations that employ commercial-off-the-shelf receivers and antennas in a conventional, antenna-upright setup. Other researchers have used a custom antenna and/or a dedicated setup, with the antenna tipped for enhanced multipath reception. FIGURE 1. Standard geodetic receiver installation. The antenna is protected by a hemispherical radome. The monument (tripod structure) is ~ 2 meters above the ground. GPS satellites rise and set in ascending and descending sky tracks, multiple times per day. The specular reflection point migrates radially away from the receiver for decreasing satellite elevation angle. The total reflector height is made up of an a priori value and an unknown bias driven by the thickness of the snow layer. In this article, we summarize the SNR-based GPS-MR technique as applied to snow sensing using geodetic instruments. This forward/inverse approach for GPS-MR is new in that it capitalizes on known information about the antenna response and the physics of surface scattering to aid in retrieving the unknown snow conditions in the site surroundings. It is a statistically rigorous retrieval algorithm, agreeing to first order with the simpler original methodology, which is retained here for the inversion bootstrapping. The first part of the article describes the retrieval algorithm, while the second part provides validation at a representative site over an extended period of time.  Physical Forward Model SNR observations are formulated as SNR = Ps/Pn. In the denominator, we have the noise power, Pn, here taken as a constant, based on nominal values for the noise power spectral density and the noise bandwidth. The numerator is composite signal power: .   (1) Its incoherent component is the sum of the respective direct and reflected powers (although direct incoherent power is negligible). In contrast, the coherent composite signal power follows from the complex sum of direct and reflection average voltages (not to be confused with the electromagnetic propagating fields, which neglect the receiving antenna response and also the receiver tracking process): (2) It is expressed in terms of the coherent direct and reflected powers, as well as the interferometric phase,  , (3) which amounts to the reflection excess phase with respect to the direct signal. We decompose observations, SNR = tSNR + dSNR, into a trend   (4) over which interference fringes are superimposed: . (5)  From now on, we neglect the incoherent power, which only impacts tSNR, not dSNR, and drop the coherent power superscript, for brevity. The direct or line-of-sight power is formulated as   (6) where    is the direction-dependent right-hand circularly polarized (RHCP) power component incident on an isotropic antenna; the left-handed circularly polarized (LHCP) component is negligible. The direct antenna gain, , is obtained evaluating the antenna pattern in the satellite direction and with RHCP polarization. The reflection power, , (7) is defined starting with the same incident isotropic power, , as in the direct power. It ends with a coherent power attenuation factor,    (8) where  θ  is the angle of incidence (with respect to the surface normal), k = 2π/λ, is the wave number, and λ = 24.4 centimeters is the carrier wavelength for the civilian GPS signal on the L2 frequency (L2C). This polarization-independent factor accounts only for small-scale residual height above and below a large-scale trend surface. The former/latter results from high-/low-pass filtering the actual surface heights using the first Fresnel zone as a convolution kernel, roughly speaking. Small-scale roughness is parameterized in terms of an effective surface standard deviation s (in meters); its scattering response is modeled based on the theories of random surfaces, except that the theoretical ensemble average is replaced by a sensing spatial average. Large-scale deterministic undulations could be modeled, but their impact on snow depth is canceled to first-order by removing bare-ground reflector heights. At the core of , we have coupled surface/antenna reflection coefficients,  , producing respectively RHCP and LHCP fields (under the assumption of a RHCP incident field). These terms include antenna response power gain and phase patterns, evaluated in the reflection direction, and separately for each polarization. The surface response is represented by complex-valued Fresnel coefficients for cross- and same-sense circular polarization, respectively. The medium is assumed to be homogeneous (that is, a semi-infinite half-space). Material models provide the complex permittivity, which drives the Fresnel coefficients. The interferometric phase reads: .(9) The first term accounts for the surface and antenna properties of the reflection, as above. The last one is the direct phase contribution, which amounts to only the RHCP antenna phase-center variation evaluated in the satellite direction. The majority of the components present in the direct RHCP phase (such as receiver and satellite clock states, the bulk of atmospheric propagation delays, and so on) are also present in the reflection phase, so they cancel out in forming the difference. At the core of the interferometric phase, we have the geometric component, φI = kτi, the product of the wave number and the interferometric propagation delay. Assuming a locally horizontal surface, the latter is simply:   (10) in terms of the satellite elevation angle, e, and an a priori reflector height, HA. Snow depth will be measured in terms of changes in reflector height. The physical forward model, based only on a priori information, can then be summarized as:   (11) where interferometric power and phase are, respectively:   (12) . (13) In all of these terms the pseudorandom-noise-code modulation impressed on the carrier wave can be safely neglected, given the small interferometric delay and Doppler shift at grazing incidence, stationary surface/receiver conditions, and short antenna installations. Parameterization of Unknowns There are errors in the nominal values assumed for the physical parameters of the model (permittivity, surface roughness, reflector height, and so on). Ideally we would estimate separate corrections for each one, but unfortunately many are linearly dependent or nearly so. Because of this dependency, we have kept physical parameters fixed to their optimal a priori values, and have estimated a few biases. Each bias is an amalgamation of corrections for different physical effects. In a later stage, we rely on multiple independent bias estimates (such as for successive days) to try and separate the physical sources. Each satellite track is inverted independently. A track is defined by partitioning the data by individual satellite and then into ascending and descending portions, splitting the period between the satellite’s rise and set at the near-zenith culmination. Each satellite track has a duration of ~1–2 hours. This configuration normally offers a sufficient range of elevation angles, unless the satellite reaches culmination too low in the sky (less than about 20°), in which case the track is discarded. In seeking a balance between under- and over-fitting, between an insufficient and an excessive number of parameters, we estimate the following vector of unknown parameters: . (14) FIGURE 2 shows the effect of the constant and linear biases on the SNR observations. Reflector height bias, HB , changes the number of oscillations; phase shift, φB , displaces the oscillations along the horizontal axis; reflection power,    , affects the depth of fades; zeroth-order noise power,     , shifts the observations up or down as a whole; and first-order noise power,    , tilts the SNR curve. A good parameterization yields observation sensitivity curves as unique as possible for each parameter. FIGURE 2. Effect of each parameter on SNR observations; curves are displaced vertically (6 dB) for clarity. The forward model, now including the biases, can be summarized as follows:  (15) where the modified interferometric power and phase are given by: , (16) . (17) The total reflector height, H = HA – HB (a priori value minus unknown bias), is to be interpreted as an effective value that best fits measurements, which includes snow and other components. Bootstrapping Parameter Priors. Biases and SNR observations are involved non-linearly through the forward model. Therefore, there is the need for a preliminary global optimization, without which the subsequent final local optimization will not necessarily converge to the optimal solution. SNR observations would trace out a perfect sinusoid curve in the case of an antenna with isotropic gain and spherical phase pattern, surrounded by a smooth, horizontal, and infinite surface (free of small-scale roughness, large-scale undulations, and edges), made of perfectly electrically conducting material, and illuminated by constant incident power. Thus, in such an idealized case, SNR could be described exactly by constant reflector height, phase shift, amplitude, and mean values. As the measurement conditions become more complicated, the SNR data start to deviate from a pure sinusoid. Yet a polynomial/spectral decomposition is often adequate for bootstrapping purposes.  Statistical Inverse Model Formulation Based on the preliminary values for the unknown parameters vector and other known (or assumed) values, we run the forward model to obtain simulated observations. We form pre-fit residuals comparing the model values to SNR measurements collected at varying satellite elevation angles (separately for each track). Residuals serve to retrieve parameter corrections, such that the sum of squared post-fit residuals is minimized. This non-linear least squares problem is solved iteratively using both a functional model and a stochastic model. The functional modeling includes a Jacobian matrix of partial derivatives, which represents the sensitivity of observations to parameter changes where the partial derivatives are defined element-wise. Instead of deriving analytical expressions, we evaluate them numerically, via finite differencing. The stochastic model specifies the uncertainty and correlation expected in the residuals. Their a priori covariance matrix modifies the objective function being minimized.  Directional Dependence It is important to know at which elevation angles the parameter estimates are best determined. Here, we focus on the phase parameters instead of reflection power or noise power parameters.  We can utilize the estimated reflector height and phase shift to evaluate the full phase bias function over varying elevation angles. Similarly, we can extract the corresponding 2-by-2 portion of the parameters’ a posteriori covariance matrix, containing the uncertainty for reflector height and for phase shift, as well as their correlation, which is then propagated to obtain the full phase uncertainty (see FIGURE 3). FIGURE 3. Uncertainty of full phase function, propagated from the uncertainty of reflector height and of phase shift, as well as their correlation. The uncertainty attains a clear minimum versus elevation angle. The least-uncertainty elevation angle pinpoints the observation direction where reflector height and phase shift are best determined (in combined form, not individually). The azimuth and epoch coinciding with the peak elevation angle act as track tags, later used for clustering similar tracks and analyzing their time series of retrievals. If we normalize phase uncertainty by its value at the peak elevation angle, then plot such sensing weights (between 0 and 1) versus the radial or horizontal distance to the center of the first Fresnel zone at each elevation angle, we obtain FIGURE 4. It can be interpreted as the reflection footprint, indicating the importance of varying distances, with a longer far tail and a shorter near tail (respectively regions beyond and closer than the peak distance). The implications for in situ data collection are clear: one should sample more intensely near the peak distance (about 15 meters) and less so in the immediate vicinity of the GPS antenna, tapering it off gradually away from the antenna. As a caveat, these conclusions are not necessarily valid for antenna setups other than the one considered here. FIGURE 4. Reflection footprint in terms of a sensing weight (between 0 and 1) defined as the normalized reciprocal of full phase uncertainty, plotted versus the radial or horizontal distance from the receiving antenna to the center of the first Fresnel zone at each elevation angle; valid for an upright 2-meter-tall antenna; the receiving antenna is at zero radial distance. Results We now examine the snow-depth retrievals from the GPS multipath retrieval algorithm and assess both the precision and accuracy of the method. Multiple metrics have been developed to assess the quality of the results. The accuracy of the method has been evaluated by comparing with in situ data over a multi-year period. Three field sites were chosen to highlight different limitations in the method, both in terms of terrain and forest cover: grassland, alpine, and forested. We will look at the forested site in some detail. Satellite Coverage and Track Clustering. All GPS-MR retrievals reported here are based on the newer GPS L2C signal. Of the approximately 30 GPS satellites in service, 8-10 L2C satellites were available between 2009 and 2012 (8, 9, and 10 satellites at the end of 2009, 2010, and 2011, respectively). Satellite observations were partitioned into ascending and descending portions, yielding approximately twenty unique tracks per day at a site with good sky visibility. GPS orbits are highly repeatable in azimuth, with deviations at the few-degree range over a year, translating into ~50-100-centimeter azimuthal displacement of the reflecting area (corresponding to the first Fresnel zone at 10°-15° elevation angle for a 2-meter high antenna). This repeatability permits clustering daily retrievals by azimuth. It also allows the simplification that estimated snow-free reflector heights are fairly consistent from day to day, facilitating the isolation of the varying snow depth during the snow-covered period. For a given track, its revisit time is also repeatable, amounting to practically one sidereal day. The deficit in time relative to a calendar day results in the track time of the day receding ~4 minutes and 6 seconds every day. This slow but steady accumulation eventually makes the time of day return to its starting value after about one year. As all GPS satellites drift approximately at the same rate, the time between successive tracks remains nearly repeatable. Its reciprocal, the sampling rate, has a median equal to approximately one track per hour, with a low value of one track within two hours and a high of one track within 15 minutes; both extremes occur every day, with low-rate idle periods interspersed with high-rate bursts. The time of the day reduced to a fixed day (such as January 1, 2000) could also be used to cluster tracks. Neighboring clusters, which are close in azimuth and/or in reduced time of the day, are expected to be more comparable, as they sample similar conditions and are subject to similar errors. Observations. FIGURE 5 shows several representative examples of SNR observations. A typical good fit between measured and modeled values is shown in Figure 5(a), corresponding to the beginning of the snow season. Generally the model/measurement fit is good when the scattering medium is homogeneous; it deteriorates as the medium becomes more heterogeneous, particularly with mixtures of soil, snow, and vegetation. There are genuine physical effects as well as more mundane spurious instrumental issues that degrade the fit but do not necessarily cause a bias in snow-depth estimates. These include secondary reflections, interferometric power effects, direct power effects, and instrument-related issues. FIGURE 5. Examples of observations: (a) good fit; (b) presence of secondary reflections; (c) vanishing interference fringes; (d) atypical interference fringes. Secondary reflections originate from disjoint surface regions. Interference fringes become convoluted with multiple superimposed beats (see Figure 5(b)). As long as there is a unique dominating reflection, the inversion will have no difficulty fitting it, as the extra reflections will remain approximately zero-mean. Random deviations of the actual surface with respect to its undulated approximation, called roughness or residual surface height, will affect the interferometric power. SNR measurements will exhibit a diminishing number of significant interference fringes, compared to the measurement noise level (see Figure 5(c)). This facilitates the model fit but the reflector height parameter may become ill-determined: its estimates will be more uncertain. Changes in snow density also affect the fringe amplitude. Snow precipitation attenuates the satellite-to-ground radio link, which affects SNR measurements through the direct power term. First, this shifts the SNR measurements up or down (in decibels); second, it tilts the trend tSNR as attenuation is elevation-angle dependent; third, fringes in dSNR will change in amplitude because of the decrease in the coherent component of the direct power. Partial obstructions can affect either or both direct and interferometric powers. In this case, SNR measurements, albeit corrupted, are still recorded. This situation is in contrast to complete blockages as caused by topography. The deposition of snow and the formation of a winter rime on the antenna are a particularly insidious type of obstruction, as their presence in the near-field of the antenna element can easily distort the gain pattern in a significant manner. In the far-field, trees are another important nuisance, so much so that their absence is held as a strong requirement for the proper functioning of multipath reflectometry. Satellite-specific direct power offsets and also long-term power drifts are to be expected as spacecraft age and modernized designs are launched. In addition, noise power depends on the state of conservation of receiver cables and on their physical temperature. Less subtle incidents are sudden ~3-dB SNR steps, hypothesized to originate in the receiver switching between the L2C data and pilot subcodes, CM and CL. Quality Control. Anomalous conditions may result in measurement spikes, jumps, and short-lived rapidly-varying fluctuations. For snow-depth-sensing purposes, it is necessary and sufficient to either neutralize such measurement outliers through a statistically robust fit or detect unreliable fits and discard the problematic ones that could not otherwise be salvaged. The key to quality control (QC) is in grouping results into statistically homogeneous units, having measurements collected under comparable conditions. In our case, azimuth-clustered tracks are the natural starting unit. Secondarily, we must account for genuine temporal variations in the tendency of results, from beginning to peak to the end of the snow season. The detection of anomalous results further requires an estimate of the statistical dispersion to be expected. Considering that the sample is contaminated with outliers, robust estimators (running median instead of the running mean, and median absolute deviation over the standard deviation) are called for, if the first- and second-order statistical moments are to be representative. Given estimates of the non-stationary tendency and dispersion, a tolerance interval can then be constructed such that it bounds, say, a 99% proportion of the valid results with 95% confidence level. We also desire QC to be judicious, or else too many valid estimates will be lost. Notice that in the present intra-cluster QC, we compare an individual estimate to the expected performance of the track cluster to which it belongs; later, we complement QC with an inter-cluster comparison of each cluster’s own expected performance. Based on our practical experience, no single statistic detects all the outliers. We use four particular statistics that we have found to be useful: 1) degrees of freedom, essentially the number of observations per track (modulo a constant number of parameters); 2) using the scaled root-mean-square error (RMSE) to test for goodness-of-fit, that is, how well measurements can be explained adjusting the unknown values for the parameters postulated in the model; 3) reflector height uncertainty; and 4) peak elevation angle, which behaves much like a random variable, as it is determined by a multitude of factors.  Combinations. We combine multiple clusters to average out random noise. Noise mitigation aims at not only coping with measurement errors but also compensating for model deficiencies, to the extent that they are not in common across different clusters. Before we combine different clusters, we have to address their long-term differences. The initial situation is that snow surface heights will be greater downhill and smaller uphill; we take this into account on a cluster-by-cluster basis by subtracting ground heights from their respective snow surface heights, resulting in snow thickness values, which is a completely physically unambiguous quantity. Snow thickness is more comparable than snow heights across varying-azimuth track clusters. Yet snow tends to fill in ground depressions, so thickness exhibits variability caused by the underlying ground surface, even when the overlying snow surface is relatively uniform. Further cluster homogeneity can be achieved by accounting for the temporally permanent though spatially non-uniform component of snow thickness.  The averaging of snow depths collected for different track clusters employs the inversion uncertainties to obtain a preliminary running weighted median, calculated for, say, daily postings, with overlapping windows or not. The preliminary post-fit residuals then go through their own averaging, necessarily employing a wider averaging window (say, monthly), which produces scaling factors for the original uncertainties. The running weighted median is then repeated, producing final averages. The variance factors reflect the fact that some clusters are better than others. Thus, the final GPS estimates of snow depth follow from an averaging of all available tracks, whose individual snow depth values were previously estimated independently. A new average is produced twice daily utilizing the surrounding 1–2 days of data (depending on the data density), that is, 12-hour posting spacing and 24-hour moving window width. The averaging interval must be an integer number of days, so as to minimize the possibility of snow-depth artifacts caused by variations in the observation geometry, which repeats daily. Site-Specific Results We explored GPS-MR snow-depth retrieval at three stations over a long period (up to three years). Throughout, we assessed the performance of the GPS estimates against independent nearly co-located in situ measurements. We also compared the GPS estimates to the nearest SNOTEL station. SNOTEL (from snowpack telemetry) is an automated system for collecting snowpack and related data in the western U.S. operated by the U.S. Department of Agriculture. Although not co-located with GPS, SNOTEL data are important because they provide accurate information on the timing of snowfall events. The three sites we used were 1) a site in the T.W. Daniel Experimental Forest within the Wasatch Cache National Forest in the Bear River Range of northeastern Utah, with an elevation of 2,600 meters; 2) one of the stations of the EarthScope Plate Boundary Observatory, a grassland site located near Island Park, Idaho; and 3) an alpine site in the Niwot Ridge Long-term Ecological Research Site near Boulder, Colorado. While we have fully documented the results from each site, due to space limitations we will only discuss the results from the forested site (known as RN86) in this article. This is a more challenging site than the other two, due to the presence of nearby trees. Furthermore, it was subject to denser in situ sampling of 20-150 measurements spatially replicated around the GPS antenna, and repeated approximately every other week for about one year. We show results for the 2012 water-year, the period starting October 1 through September 30 of the following year. Where GPS site RN86 was installed, topographical slopes range from 2.5° to 6.5° (at the 2-meter spatial scale), with average of ~5° within a 50-meter radius around the GPS antenna. RN86 was specifically built to study the impact of trees on GPS snow depth retrievals (see FIGURE 6). Ground crews manually collected in situ measurements around the GPS antenna approximately every other week starting in November 2011. Measurements were made every 1–2 meters from the antenna up to a distance of 25-30 meters. In the second half of the year, the sampling protocol was changed to azimuths of 0° (N), 45° (NE), 135° (SE), 180° (S), 225° (SW), and 315° (NW). With these data it is possible to obtain in situ average estimates, with their own uncertainties (based on the number of measurements), which allows a more meaningful comparison. FIGURE 6. Aerial view of the forested site (RN86) around the GPS antenna (marked with a circle). There is reduced visibility at the current site, compared to other sites. Track clusters are concentrated due south, with only two clusters located within ±90° of north. Therefore, the GPS average snow depth is not necessarily representative of the azimuthally symmetric component of the snow depth. In the presence of an azimuthal asymmetry in the snow distribution around the antenna, the GPS average would be expected to be biased towards the environmental conditions prevalent in the southern quadrant. To rule out the possibility of an azimuthal artifact in the comparisons, we have utilized only the in situ data collected along the SE/S/SW quadrant. The comparison shows generally excellent agreement between GPS and in situ data (see FIGURE 7). The first four and the last one in situ data points were collected with coarser spacing and/or smaller azimuthal coverage, which may be partially responsible for different performance in the first and second halves of the snow season. The correlation between GPS and in situ snow depth at RN86 amounts to 0.990, indicating a very strong linear relationship. Carrying out a regression between in situ and GPS values, the RMS of snow-depth residuals improves from 9.6 to 3.4 centimeters. The regression intercept and slope (with corresponding 95% uncertainties) amount to 15.4 ± 9.11 centimeters and 0.858 ± 0.09 meters per meter, respectively. According to these statistics, the null hypotheses of zero intercept and unity slope are rejected at the 95% confidence level. This implies that at this location GPS snow-depth estimates exhibit both additive and multiplicative biases. The latter is proportional to snow depth itself, meaning that, compared to an ideal one-to-one relationship, GPS is found to under-estimate in situ snow depth at this site by 14 ± 9%, although the uncertainty is somewhat large. FIGURE 7. Snow-depth measurement at the forested site (RN86) for the water-year 2012 The SNOTEL sensors are exceptionally close to the GPS antenna at this site, about 350 meters horizontally distant with negligible vertical separation. Yet the former is located within trees, while the latter is located at the periphery of the forest and senses the reflections scattered from an open field. Therefore, only the timing of snowfall events agrees well, not the amount of snow. Although forest density is generally negatively correlated with snow depth, exceptions are not uncommon, especially in localized clearings exposed to intense solar radiation, where shading of the snow by the trees reduces ablation. Conclusions In this article, we have discussed a physically based forward model and a statistical inverse model for estimating snow depth based on GPS multipath observed in SNR measurements. We assessed model performance against independent in situ measurements and found they validated the GPS estimates to within the limitations of both GPS and in situ measurement errors after the characterization of systematic errors. The assessment yielded a correlation of 0.98 and an RMS error of 6–8 centimeters for observed snow depths of up to 2.5 meters at three sites, with the GPS underestimating in situ snow depth by ~5–15%. This latter finding highlights the necessity to assess effects currently neglected or requiring more precise modeling. Acknowledgments The research reported in this article was supported by grants from the U.S. National Science Foundation, NASA, and the University of Colorado. Nievinski has been supported by a Capes/Fulbright Graduate Student Fellowship and a NASA Earth System Science Research Fellowship. The article is based, in part, on two papers published in the IEEE Transactions on Geoscience and Remote Sensing: “Inverse Modeling of GPS Multipath for Snow Depth Estimation – Part I: Formulation and Simulations” and “Inverse Modeling of GPS Multipath for Snow Depth Estimation – Part II: Application and Validation.” Manufacturers For the forested site (RN86), a Trimble NetR9 receiver was used with a Trimble TRM57971.00 (Zephyr Geodetic II) antenna with no external radome. FELIPE G. NIEVINSKI is a faculty member at the Federal University of Santa Catarina, Florianópolis, Brazil. He has also been a post-doctoral researcher at São Paulo State University, Presidente Prudente, Brazil. He earned a B.E. in geomatics from the Federal University of Rio Grande do Sul, Porto Alegre, Brazil, in 2005; an M.Sc.E. in geodesy from the University of New Brunswick, Fredericton, Canada, in 2009; and a Ph.D. in aerospace engineering sciences from the University of Colorado, Boulder, in 2013. His Ph.D. dissertation was awarded The Institute of Navigation Bradford W. Parkinson Award in 2013. KRISTINE M. LARSON received a B.A. degree in engineering sciences from Harvard University and a Ph.D. degree in geophysics from the Scripps Institution of Oceanography, University of California at San Diego. She was a member of the technical staff at the Jet Propulsion Lab from 1988 to 1990. Since 1990, she has been a professor in the Department of Aerospace Engineering Sciences, University of Colorado, Boulder. FURTHER READING • Authors’ Journal Papers “Inverse Modeling of GPS Multipath for Snow Depth Estimation—Part I: Formulation and Simulations” by F.G. Nievinski and K.M. Larson in IEEE Transactions on Geoscience and Remote Sensing, Vol. 52, No. 10, 2014, pp. 6555–6563, doi: 10.1109/TGRS.2013.2297681. “Inverse Modeling of GPS Multipath for Snow Depth Estimation—Part II: Application and Validation” by F.G. Nievinski and K.M. Larson in IEEE Transactions on Geoscience and Remote Sensing, Vol. 52, No. 10, 2014, pp. 6564–6573, doi: 10.1109/TGRS.2013.2297688. • More on the Use of GPS for Snow Depth Assessment “Snow Depth, Density, and SWE Estimates Derived from GPS Reflection Data: Validation in the Western U.S.” by J.L. McCreight, E.E. Small, and K.M. Larson in Water Resources Research, published first on line, August 25, 2014, doi: 10.1002/2014WR015561. “Environmental Sensing: A Revolution in GNSS Applications” by K.M. Larson, E.E. Small, J.J. Braun, and V.U. Zavorotny in Inside GNSS, Vol. 9, No. 4, July/August 2014, pp. 36–46. “Snow Depth Sensing Using the GPS L2C Signal with a Dipole Antenna” by Q. Chen, D. Won, and D.M. Akos in EURASIP Journal on Advances in Signal Processing, Special Issue on GNSS Remote Sensing, Vol. 2014, Article No. 106, 2014, doi: 10.1186/1687-6180-2014-106. “GPS Snow Sensing: Results from the EarthScope Plate Boundary Observatory” by K.M. Larson and F.G. Nievinski in GPS Solutions, Vol. 17, No. 1, 2013, pp. 41–52, doi: 10.1007/s10291-012-0259-7. • GPS Multipath Modeling and Simulation “Forward Modeling of GPS Multipath for Near-Surface Reflectometry and Positioning Applications” by F.G. Nievinski and K.M. Larson in GPS Solutions, Vol. 18, No. 2, 2014, pp. 309–322, doi: 10.1007/s10291-013-0331-y. “An Open Source GPS Multipath Simulator in Matlab/Octave” by F.G. Nievinski and K.M. Larson in GPS Solutions, Vol. 18, No. 3, 2014, pp. 473–481, doi: 10.1007/s10291-014-0370-z. “Multipath Minimization Method: Mitigation Through Adaptive Filtering for Machine Automation Applications” by L. Serrano, D. Kim, and R.B. Langley in GPS World, Vol. 22, No. 7, July 2011, pp. 42–48. “It’s Not All Bad: Understanding and Using GNSS Multipath” by A. Bilich and K.M. Larson in GPS World, Vol. 20, No. 10, October 2009, pp. 31–39. “GPS Signal Multipath: A Software Simulator” by S.H. Byun, G.A. Hajj, and L.W. Young in GPS World, Vol. 13, No. 7, July 2002, pp. 40–49.

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Ault mw116ka1249f02 ac adapter 12vdc 6.67a 4pin (: :) straight,the inputs given to this are the power source and load torque,hp 324815-001 ac adapter 18.5v 4.9a 90w ppp012l power supply for.pa-1650-02h replacement ac adapter 18.5v 3.5a for hp laptop powe,pki 6200 looks through the mobile phone signals and automatically activates the jamming device to break the communication when needed.hp 0950-3796 ac adapter 19vdc 3160ma adp-60ub notebook hewlett p,delta adp-65jh ab 19vdc 3.42a 65w used -(+)- 4.2x6mm 90° degree,coleman cs-1203500 ac adapter 12vdc 3.5a used -(+) 2x5.5x10mm ro,if you find your signal is weaker than you'd like while driving.kodak hp-a0601r3 ac adapter 36vdc 1.7a 60w used -(+) 4x6.5x10.9m,hp compaq ppp014h-s ac adapter 19vdc 4.74a used barrel with pin,yuan wj-y351200100d ac adapter 12vdc 100ma -(+) 2x5.5mm 120vac s.k090050d41 ac adapter 9vdc 500ma 4.5va used -(+) 2x5.5x12mm 90°r.goldfar son-erik750/z520 ac car phone charger used,nec op-520-4401 ac adapter 11.5v dc 1.7a 13.5v 1.5a 4pin female.li shin lse0107a1230 ac adapter 12vdc 2.5a used -(+) 2.1x5.5mm m,cisco aironet air-pwrinj3 48v dc 0.32a used power injector,this causes enough interference with the communication between mobile phones and communicating towers to render the phones unusable.atlinks usa 5-2629 ac adapter 9vdc 300ma power supply class 2 tr,this covers the covers the gsm and dcs,casio ad-a60024iu ac adapter 6vdc 200ma used +(-) 2x5.5x9.6mm ro,direct plug-in sa48-18a ac adapter 9vdc 1000ma power supply.morse key or microphonedimensions,this 4-wire pocket jammer is the latest miniature hidden 4-antenna mobile phone jammer.sceptre ad2524b ac adapter 25w 22.0-27vdc 1.1a used -(+) 2.5x5.5.delta eadp-10cb a ac adapter 5v 2a new power supply printer,sanyo var-33 ac adapter 7.5v dc 1.6a 10v 1.4a used european powe,minolta ac-9 ac-9a ac adapter 4.2vdc 1.5a -(+) 1.5x4mm 100-240va,toshiba pa3080u-1aca paaca004 ac adapter 15vdc 3a used -(+)- 3x6.moso xkd-c2000ic5.0-12w ac adapter 5vdc 2a used -(+) 0.7x2.5x9mm,bs-032b ac/dc adapter 5v 200ma used 1 x 4 x 12.6 mm straight rou,sharp uadp-0220cezz ac adapter 13vdc 4.2a 10pin square lcd tv po.how to make cell phone signal jammer.gme053-0505-us ac adapter 5vdc 0.5a used -(+) 1x3.5x7.5mm round.the frequencies are mostly in the uhf range of 433 mhz or 20 – 41 mhz.delphi sa10115 xm satellite radio dock cradle charger used 5vdc.hoyoa bhy481351000u ac adapter 13.5vdc 1000ma used -(+) 2.5x5.5x.fsp group fsp065-aab ac adapter 19vdc 3.42ma used -(+)- 2x5.5.a wide variety of custom jammers options are available to you,nec adp52 ac adapter 19vdc 2.4a 3pin new 100-240vac genuine pow,toshiba ac adapter 15vdc 4a original power supply for satellite,pihsiang 4c24080 ac adapter 24vdc 8a 192w used 3pin battery char,ault 336-4016-to1n ac adapter 16v 40va used 6pin female medical,landia p48e ac adapter 12vac 48w used power supply plug in class,e where officers found an injured man with a gunshot.– active and passive receiving antennaoperating modes.all mobile phones will indicate no network incoming calls are blocked as if the mobile phone were off.sony dcc-e345 ac adapter 4.5v/6v 1.5v/3v 1000ma used -(+)-,ad 9/8 ac dc adapter 9v 800ma -(+)- 1.2x3.8mm 120vac power suppl.


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Alvarion 0438b0248 ac adapter 55v 2a universal power supply.toshiba adp-75sb ab ac dc adapter 19v 3.95a power supply.asante ad-121200au ac adapter 12vac 1.25a used 1.9 x 5.5 x 9.8mm,delta adp-5fh c ac adapter 5.15v 1a power supply euorope.liteon pa-1460-19ac ac adapter 19vdc 2.4a power supply,delta adp-45gb ac adapter 19vdc 2.4a power supply,condor aa-1283 ac adapter 12vdc 830ma used -(+)- 2x5.5x8.5mm rou,fujitsu cp293662-01 ac adapter 19vdc 4.22a used 2.5 x 5.5 x 12mm.nec op-520-4701 ac adapter 13v 4.1a ultralite versa laptop power.starting with induction motors is a very difficult task as they require more current and torque initially,phihong psm11r-090 ac adapter 9vdc 1.12a -(+)- 2.5x5.5mm barrel.toshiba pa2450u ac adapter 15v dc 3a 45w new power supply.navigon ac adapter 12.6vdc 800ma used 110-220v ac,american telecom ku1b-090-0200d ac adapter 9vdc 200ma -(+)-used.khu045030d-2 ac adapter 4.5vdc 300ma used shaver power supply 12.kenwood w08-0657 ac adapter 4.5vdc 600ma used -(+) 1.5x4x9mm 90°,lei iu40-11190-010s ac adapter 19vdc 2.15a 40w used -(+) 1.2x5mm,startech usb2sataide usb 2.0 to sata ide adapter,delta eadp-20db a ac adapter 12vdc 1.67a used -(+)- 1.9 x 5.4 x.wattac ba0362z1-8-b01 ac adapter 5v 12vdc 2a used 5pin mini din,ilan f1560 (n) ac adapter 12vdc 2.83a -(+) 2x5.5mm 34w i.t.e pow.philips 4203 035 78410 ac adapter 1.6vdc 100ma used -(+) 0.7x2.3,u.s. robotics tesa1-150080 ac adapter 15vdc 0.8a power supply sw.auto charger 12vdc to 5v 0.5a car cigarette lighter mini usb pow,compaq series 2872a ac adapter 18.75v 3.15a 41w? 246960-001.acbel api-7595 ac adapter 19vdc 2.4a for toshiba 45 watt global,umec up0351e-12p ac adapter +12vdc 3a 36w used -(+) 2.5x5.5mm ro.arac-12n ac adapter 12vdc 200ma used -(+) plug in class 2 power,65w-ac1002 ac adapter 19vdc 3.42a used -(+) 2.5x5.5x11.8mm 90° r,solar energy measurement using pic microcontroller,circuit-test ad-1280 ac adapter 12v dc 800ma new 9pin db9 female,dell nadp-130ab d 130-wac adapter 19.5vdc 6.7a used 1x5.1x7.3x12,sony bc-csgc 4.2vdc 0.25a battery charger used c-2319-445-1 26-5,this paper describes the simulation model of a three-phase induction motor using matlab simulink.ibm 02k6491 ac adapter 16vdc 3.36a -(+) 2.5x5.5mm used 100-240va,sy-1216 ac adapter 12vac 1670ma used ~(~) 2x5.5x10mm round barre,the jammer denies service of the radio spectrum to the cell phone users within range of the jammer device.city of meadow lake regular council meeting december 12.with an effective jamming radius of approximately 10 meters.chi ch-1265 ac adapter 12v 6.5a lcd monitor power supply.hp c5160-80000 ac adapter 12v dc 1.6a adp-19ab scanjet 5s scanne.personal communications committee of the radio advisory board of canada,868 – 870 mhz each per devicedimensions. wifi blocker ,because in 3 phases if there any phase reversal it may damage the device completely,jvc vu-v71u pc junction box 7.5vdc used power supply asip6h033,10k2586 ac adapter 9vdc 1000ma used -(+) 2x5.5mm 120vac power su,delta pa3290u-2a2c ac adapter 18.5v 6.5a hp compaq laptop power.the output of that circuit will work as a.

It is a device that transmit signal on the same frequency at which the gsm system operates,motorola spn5404aac adapter 5vdc 550ma used mini usb cellphone,2 w output power3g 2010 – 2170 mhz,ibm 66g9984 adapter 10-20vdc 2-2.2a used car charger 4pin female.ak ii a15d3-05mp ac adapter 5vdc 3a 2.5x5.5 mm power supply,palm plm05a-050 ac adapter 5vdc 1a power supply for palm pda do,toshiba api3ad03 ac adapter 19v dc 3.42a -(+)- 1.7x4mm 100-240v.d-link af1805-a ac adapter 5vdc 2.5a3 pin din power supply,ac adapter used car charger tm & dc comics s10.liteon hp ppp009l ac adapter 18.5v dc 3.5a 65w power supply.hitachi hmx45adpt ac adapter 19v dc 45w used 2.2 x 5.4 x 12.3 mm.compaq evp100 ac dc adapter 10v 1.5a 164153-001 164410-001 4.9mm,aplha concord dv-1215a ac adapter 12vac,finecom api3ad14 19vdc 6.3a used -(+)- 2.5x5.5mm pa-1121-02 lite,fidelity electronics u-charge new usb battery charger 0220991603,crestron gt-21097-5024 ac adapter 24vdc 1.25a new -(+)- 2x5.5mm,conair spa045100bu 4.5v dc 1ma -(+)- 2x5.5mm used class 2 power.rs rs-1203/0503-s335 ac adapter 12vdc 5vdc 3a 6pin din 9mm 100va,this project shows a no-break power supply circuit,a spatial diversity setting would be preferred.1 watt each for the selected frequencies of 800,surecall's fusion2go max is the cell phone signal booster for you.sony pcga-ac19v ac adapter 19.5vdc 3.3a notebook power supply.armoured systems are available,motorola ch610d walkie talkie charger only no adapter included u,mobile jammerbyranavasiya mehul10bit047department of computer science and engineeringinstitute of technologynirma universityahmedabad-382481april 2013.a blackberry phone was used as the target mobile station for the jammer,jvc ap-v3u ac adapter 5.2vdc 2a -(+) 1.6x4mm used camera a,hp hstn-f02g 5v dc 2a battery charger with delta adp-10sb,ac car adapter phone charger used 1.5x3.9x10.8cm round barrel.kenwood dc-4 mobile radio charger 12v dc.at&t sil s005iu060040 ac adapter 6vdc 400ma -(+)- 1.7x4mm used,we don't know when or if this item will be back in stock.m2297p ac car adapter phone charger used 0.6x3.1x7.9cm 90°right,audiovox ad-13d-3 ac adapter 24vdc 5a 8pins power supply lcd tv,jvc aa-v68u ac adapter 7.2v dc 0.77a 6.3v 1.8a charger aa-v68 or.motorola plm4681a ac adapter 4vdc 350ma used -(+) 0.5x3.2x7.6mm,compaq adp-50sb ac dc adapter 18.5v 2.8a power supply,poweruon 160023 ac adapter 19vdc 12.2a used 5x7.5x9mm round barr,delta sadp-65kb ad ac adapter 20vdc 3.25a used 2.5x5.5mm -(+)- 1,mei mada-3018-ps ac adapter 5v dc 4a switching power supply,cell phone jammer manufacturers,li shin 0405b20220 ac adapter 20vdc 11a 4pin (: :) 10mm 220w use.sony pcga-ac16v6 ac adapter 16vdc 4a -(+) 3x6.5mm power supply f,sps15-007 (tsa-0529) ac adapter 12v 1.25a 15w - ---c--- + used 3,horsodan 7000253 ac adapter 24vdc 1.5a power supply medical equi,chateau tc50c ac-converter 110vac to 220vac adapter 220 240v for,with a maximum radius of 40 meters,johnlite 1947 ac adapter 7vdc 250ma 2x5.5mm -(+) used 120vac fla.

Galaxy sed-power-1a ac adapter 12vdc 1a used -(+) 2x5.5mm 35w ch,anam ap1211-uv ac adapter 15vdc 800ma power supply,razer ts06x-2u050-0501d ac adapter 5vdc 1a used -(+) 2x5.5x8mm r,noise circuit was tested while the laboratory fan was operational.premium power 298239-001 ac adapter 19v 3.42a used 2.5 x 5.4 x 1.replacement tj-65-185350 ac adapter 18.5vdc 3.5a used -(+) 5x7.3,ault bvw12225 ac adapter 14.7vdc 2.25a -(+) used 2.5x5.5mm 06-00.cobra swd120010021u ac adapter 12vdc 100ma used 2 audio pin,matsushita etyhp127mm ac adapter 12vdc 1.65a 4pin switching powe.3com p48240600a030g ac adapter 24vdc 600ma used -(+)- 2x5.5mm cl,sony acp-80uc ac pack 8.5vdc 1a vtr 1.6a batt 3x contact used po,konica minolta ac-4 ac adapter 4.7v dc 2a -(+) 90° 1.7x4mm 120va.teamgreat t94b027u ac adapter 3.3vdc 3a -(+) 2.5x5.4mm 90 degree,hjc hasu11fb ac adapter 12vdc 4a -(+) 2.5x5.5mm used 100-240vac,how a cell phone signal booster works,atlinks 5-2625 ac adapter 9vdc 500ma power supply,71109-r ac adapter 24v dc 350ma power supply tv converter used,ibm 02k3882 ac adapter 16v dc 5.5a car charger power supply,sanyo scp-03adt ac adapter 5.5vdc 950ma used 1.4x4mm straight ro,samsung pscv400102a ac adapter 16v 2.5a ite power supply.outputs obtained are speed and electromagnetic torque.liteon pa-1650-02 ac adapter 19vdc 3.42a 65w used -(+) 2.5x5.5mm.it works well for spaces around 1,this project uses arduino for controlling the devices.vt070a ac adatper 5vdc 100ma straight round barrel 2.1 x 5.4 x 1,hewlett packard hstnn-aa04 10-32v dc 11a 90w -(+)- 1x5mm used,dve dsa-31s fus 5050 ac adapter+5v dc 0.5a new -(+) 1.4x3.4x9.,lenovo pa-1900-171 ac adapter 20vdc 4.5a -(+) 5.5x7.9mm tip 100-.phihong psc11a-050 ac adapter +5v dc 2a power supply.i-mag im120eu-400d ac adapter 12vdc 4a -(+)- 2x5.5mm 100-240vac,nokia ac-8e ac adapter 5v dc 890ma european cell phone charger.nokia ac-4u ac adapter 5v 890ma cell phone battery charger.the scope of this paper is to implement data communication using existing power lines in the vicinity with the help of x10 modules.thomson 5-2608 ac adapter 9vdc 500ma used -(+) 2x5.5x9mm round b.dell la90pe1-01 ac adapter 19.5vdc 4.62a used -(+) 5x7.4mm 100-2,atlinks 5-2418 ac adapter 9vac 400ma ~(~) 2x5.5mm 120vac class 2,dse12-050200 ac adapter 5vdc 1.2a charger power supply archos gm,casio ad-12ul ac adapter 12vdc 1500ma +(-) 1.5x5.5mm 90° 120vac,ad35-04505 ac dc adapter 4.5v 300ma i.t.e power supply.kodak adp-15tb ac adapter 7vdc 2.1a used -(+) 1.7x4.7mm round ba,eng epa-201d-07 ac adapter 7vdc 2.85a used -(+) 2x5.5x10mm round.li shin lse9901a2070 ac adapter 20v dc 3.25a 65w max used.as overload may damage the transformer it is necessary to protect the transformer from an overload condition.jewel jsc1084a4 ac adapter 41.9v dc 1.8a used 3x8.7x10.4x6mm,main business is various types of jammers wholesale and retail,dual group au-13509 ac adapter 9v 1.5a used 2x5.5x12mm switching,kec35-3d-0.6 ac adapter 3vdc 200ma 0.6va used -(+)- 1 x 2.2 x 9..rocket fish rf-bslac ac adapter 15-20vdc 5a used 5.5x8mm round b,texas instruments 2580940-6 ac adapter 5.2vdc 4a 6vdc 300ma 1.

The rating of electrical appliances determines the power utilized by them to work properly,upon activation of the mobile jammer,nokia ac-5e ac adapter cell phone charger 5.0v 800ma euorope ver,ac dc adapter 5v 2a cellphone travel charger power supply,.

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