Quantifying spatial distribution of snow depth errors from LiDAR using Random Forest

Remote Sensing of Environment 141 (2014) 105–115

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Quantifying spatial distribution of snow depth errors from LiDAR using Random Forest Wade T. Tinkham a,⁎, Alistair M.S. Smith a, Hans-Peter Marshall b, Timothy E. Link a, Michael J. Falkowski c, Adam H. Winstral d a

Department of Forest, Rangeland, and Fire Sciences, College of Natural Resources, University of Idaho, 975 W. 6th St., Moscow, ID 83844-1133, USA Center for Geophysical Investigation of the Shallow Subsurface, Boise State University, Boise, ID 83725, USA Department of Forest Resources, University of Minnesota, St. Paul, MN 55108, USA d Northwest Watershed Research Center, Agricultural Research Service, Boise, ID 83712, USA b c

a r t i c l e

i n f o

Article history: Received 20 November 2012 Received in revised form 29 October 2013 Accepted 30 October 2013 Available online 23 November 2013 Keywords: LiDAR Snow Snow depth Snow volume Random Forest

a b s t r a c t There is increasing need to characterize the distribution of snow in complex terrain using remote sensing approaches, especially in isolated mountainous regions that are often water-limited, the principal source of terrestrial freshwater, and sensitive to climatic shifts and variations. We apply intensive topographic surveys, multi-temporal LiDAR, and Random Forest modeling to quantify snow volume and characterize associated errors across seven land cover types in a semi-arid mountainous catchment at a 1 and 4 m spatial resolution. The LiDARbased estimates of both snow-off surface topology and snow depths were validated against ground-based measurements across the catchment. LiDAR-derived snow depths estimates were most accurate in areas of low lying vegetation such as meadow and shrub vegetation (RMSE = 0.14 m) as compared to areas consisting of tree cover (RMSE = 0.20–0.35 m). The highest errors were found along the edge of conifer forests (RMSE = 0.35 m), however a second conifer transect outside the catchment had much lower errors (RMSE = 0.21 m). This difference is attributed to the wind exposure of the first site that led to highly variable snow depths at short spatial distances. The Random Forest modeled errors deviated from the field measured errors with a RMSE of 0.09–0.34 m across the different cover types. The modeling was used to calculate a theoretical lower and upper bound of catchment snow volume error of 21–30%. Results show that snow drifts, which are important for maintaining spring and summer stream flows and establishing and sustaining waterlimited plant species, contained 30 ± 5–6% of the snow volume while only occupying 10% of the catchment area similar to findings by prior physically-based modeling approaches. This study demonstrates the potential utility of combining multi-temporal LiDAR with Random Forest modeling to quantify the distribution of snow depth with a reasonable degree of accuracy. © 2013 Elsevier Inc. All rights reserved.

1. Introduction A direct result of climate warming in mountainous regions of North America has been an accelerated arrival of spring temperatures and an approximate one-third reduction in mountain snowpack water storage that directly reduces summer downstream water availability (Cayan, Kammerdiener, Dettinger, Caprio, & Peterson, 2001; Mote, Hamlet, Clark, & Lettenmaier, 2005). As a result, predicted future shifts in the timing of spring thaw or precipitation phase (i.e. snow vs. rain) have important implications for water and land management (Elsner et al., 2010). Spatial and temporal heterogeneity of snow and specifically the variability in snow depth and density have been identified as key variables to determine hydrologic budgets and regimes (NRC, 2010); especially in complex terrain that is the source of many of the world's rivers. However, snow depth and snow water equivalent (SWE) have ⁎ Corresponding author. Tel.: +1 208 885 6327. E-mail address: [email protected] (W.T. Tinkham). 0034-4257/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.rse.2013.10.021

proven to be two of the most difficult variables to accurately quantify at scales larger than in situ measurement points (b 1 m2) (Anderton, White, & Alvera, 2004; Molotch & Bales, 2005), but are frequently needed at larger scales (e.g. Meromy, Molotch, Link, Fassnacht, & Rice, 2013). Active remote sensing has demonstrated the potential to characterize these properties over extended spatial (N km2) and temporal scales (e.g., Dozier, Green, Nolin, & Painter, 2009; Hopkinson, Collins, Anderson, Pomeroy, & Spooner, 2012; Nolin, Dozier, & Mertes, 1993; and others), however limited research has investigated the accuracies and sources of uncertainty with using these methods. An equal challenge is the creation of reference datasets for testing the accuracy of derived products and datasets (Tinkham et al., 2013). The use of both active and passive remote sensing to characterize and quantify the spatial heterogeneity of snow properties has been widespread since the 1980s (Carsey, 1992; Dozier et al., 2009; Hopkinson et al., 2012; Nolin et al., 1993; Painter, Dozier, Roberts, Davis, & Green, 2003; Pietroniro & Leconte, 2005; Trujillo, Ramirez, & Elder, 2007). Passive hyperspectral platforms and satellite sensors

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such as Landsat and MODIS have been used to map snow surface properties such as snow cover and grain size (Dozier, 1989; Painter et al., 2003). However, studies characterizing three-dimensional properties of the snow pack have been limited, with several studies highlighting the need for higher spatial resolution data to accurately represent the complex spatial patterns of snow (Blöschl & Sivapalan, 1995; Dozier et al., 2009; Nolin et al., 1993; Painter et al., 2003). Active remote sensing platforms such as Light Detection and Ranging (LiDAR) and Synthetic Aperture Radar (SAR) are being explored and tested for their ability to estimate snow depth and SWE (Deems & Painter, 2006; Hopkinson, Pomeroy, Debeer, Ellis, & Anderson, 2010; Rango, 1993). A challenge that has been widely noted in both the scaling up from point observations and the remote sensing of snow properties is the ability to relate measurements of snow depth to SWE, which is informative but is less frequently measured due to the effort required (Liston & Elder, 2006; Sturm et al., 2010). This can be particularly difficult in heterogeneous landscapes where terrain, vegetation, and hydrometeorological variability can cause large variations in the relationship between snow depth and density across short temporal and spatial scales (Winstral and Marks, unpublished data). The preferential deposition and redistribution of snow through interactions with wind and topography (Liston et al., 2008; Pomeroy et al., 1998; Winstral, Elder, & Davis, 2002), and forest canopies (Pomeroy et al., 1998; Storck et al., 2002) can lead to heterogeneous patterns of snow depth. However, airborne LiDAR has been demonstrated as a useful tool to provide spatially-explicit snow depth and volume distribution information (Dadic, Mott, Lehning, & Burlando, 2010; Deems, Fassnacht, & Elder, 2006; Hopkinson et al., 2001) and to quantify snow depth within catchments (Cavalieri et al., 2012; Cline et al., 2009; Hopkinson, Sitar, Chasmer, & Treitz, 2004; Hopkinson et al., 2010). Recently these approaches have been reaffirmed by the use of Terrestrial Laser Scanning (TLS) systems for understanding fine spatial and temporal scale process variability such as snow ablation and depth (Egli, Jonas, Grunewald, Schirmer, & Burlando, 2012; Grünewald, Schirmer, Mott, & Lehning, 2010; Grünewald et al., 2013; Schirmer, Wirz, Clifton, & Lehning, 2011). Paired multi-temporal airborne LiDAR acquisitions collected near peak snow accumulation and during snow-free conditions have the potential to provide sub-meter resolution estimates of snow depth at catchment-wide scales and has been used to infer catchment level snow volume and snow water equivalent (Hopkinson et al., 2010; Nolin, 2010). One of the inherent benefits of LiDAR data is that it is a-spatial, meaning that the spatial resolution of LiDAR-derived products is arbitrary and dependent on the density of data collected and accuracy

of interpolation methods. This resolution is also controlled by the computational limitation that can be encountered when producing fine resolution products (1 m or less) that require multiple inputs. The application of multi-temporal LiDAR for the purpose of estimating snow depths at landscape scales is not a new concept (Dadic et al., 2010; Hopkinson et al., 2001, 2012). Studies have coupled snow-on LiDAR acquisition with other geospatial data to estimate snow depth on sea ice (Leuschen et al., 2008; Varbai & Cahalan, 2007). Dadic et al. (2010) used helicopter-based LiDAR to map snow depths of the Haut Glacier d' Arolla glacier basin to compare against snow redistribution models. The latter study used identical acquisition and processing procedures and took advantage of modern LiDARs capability to be interpolated at any spatial resolution (c.f. Evans, Hudak, Faux, & Smith, 2009; Hudak, Evans, & Smith, 2009), the data was used to produce 10 m snow depth contours for validating snow models (Dadic et al., 2010). The NASA Cold Land Processes Experiment (CLPX 2002/03) acquired LiDAR datasets for estimating snow depth at high resolution, and several studies used these data to quantify and study snow distribution and variability in six different 1 km2 regions in Colorado (Cline et al., 2009; Deems et al., 2006; Deems, Fassnacht, & Elder, 2008; Fassnacht & Deems, 2006; Trujillo et al., 2007, 2009). The recent development of the JPL Airborne Snow Observatory which has just completed the first year of a three-year demonstration mission utilized scanning LiDAR to map weekly snow depth over two mountain watersheds in California and Colorado (http://www.jpl.nasa.gov/news/news. php?release=2013-154). Several of these studies had to overcome limitations including inconsistent LiDAR acquisition parameters or processing methods between two acquisitions (Hopkinson et al., 2001, 2004), limited snow depth validation data (Fassnacht & Deems, 2006), and multitemporal datasets that do not share 100% coincident coverage (Banos, Garcia, & Alavedra, 2011; Hopkinson et al., 2012). To overcome these sub-sampling challenges, Banos et al. (2011) used LiDAR and aerial photography in a catchment in the Eastern Pyrenees in an attempt to estimate snow depth. However, the two LiDAR acquisitions only overlapped by 15% and used distributed snow modeling to simulate the depths across the rest of the catchment. Similarly, Hopkinson et al. (2012) utilized strips of snow-on LiDAR data that were partially coincident with a prior LiDAR snow-off acquisition to model land surface classes and derive an estimate of catchment SWE. One of the greatest challenges highlighted in past studies is the need to accurately georeference the snow-on and snow-off LiDAR surfaces as small errors in referencing can potentially lead to significant snow depth errors, especially in areas of high slope (Hopkinson et al., 2001, 2012).

Fig. 1. Interaction of surface features (topography/vegetation) and LiDAR surface errors (±e) with errors in snow depth (d). (A) Snow-laden canopies limit LiDAR pulse penetration leading to elevated surfaces. (B) Tree-well depressions are difficult to detect or interpolate leading to overestimates of snow volume. (C) Total surface error greater in shallow (b0.25 m) snow depths: small snow filled depressions may not be recorded if snow depth is within error of combined surface. (D) Matted vegetation layers can prevent surface returns, creating an elevated false snow-off ground surface with associated upper bound error. In some vegetation types, this matted vegetation form can in-fill with snow. (E) Rocky outcrops with large sudden drop-offs (N1 m) can (i) interact with wind to produce drifts and (ii) exhibit considerable snow-off interpolation errors.

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vegetation (e.g. ceanothus; Fig. 2D). This matted vegetation type can form an expansive dense vegetative mat, causing LiDAR surface errors as very few pulses penetrate the matted layer to produce actual ground returns (Gould, Glenn, Sankey, McNamara, & Spaete, 2013; Hopkinson et al., 2004; Tinkham et al., 2011). A final potential source is related to surface interpolation errors around rocky outcrops, which can lead to high LiDAR surface errors masking buildups of snow on leeward sides (Fig. 1e). Each of these potential sources of error can be magnified, especially on steep slopes, if careful attention is not paid to ensuring adequate co-registration of the two surfaces. The objectives of this study are to: (i) Assess the utility of multitemporal LiDAR acquisitions to quantify the spatial distribution of snow depth within a heterogeneous landscape, and (ii) Use these data alongside the Random Forest algorithm (Breiman, 2001) to further understand the controls of vegetation and topographic features on errors in LiDAR estimated snow volume at catchment scales (10–100 ha). Coincident field snow survey data collected on the same date as the snow-on LiDAR data acquisition were used to validate the depth derived from the multi-temporal LiDAR data. Snow-on, snow-off, and error surfaces were produced at both 1 and 4 m resolution to evaluate the tradeoff in resolution with computational strain. 2. Methodology 2.1. Study site

Fig. 2. Reynolds Creek Mountain East study area map showing (A) seven cover types spatially distributed across the catchment, (B) NAIP aerial photography with location of 99 high-precision ground survey plots, and (C) LiDAR-derived digital elevation map and contour map showing the variability of terrain form.

Past investigations have both observed and speculated that several sources of error unique to snow covered landscapes can arise through the interaction of the LiDAR pulses with topography and vegetation (Fig. 1). These represent sources of error that are independent of the sensor acquisition parameters. Although not an issue in the present study, snow-laden conifer canopies can often present challenges in the classification of LiDAR point returns by preventing pulses from reaching the surface (Fig. 1A). Tree-wells in conifer systems (Fig. 1B), which are deep snow surface depressions (typically on the order of 1–2 m) caused by snow interception and longwave radiation emission from tree boles and which sometimes extend to the ground surface, pose an additional problem in the interpolation of LiDAR points. This is because pulses are less likely to penetrate to the base of some of the denser conifer crowns. Along with the influence that tree canopy interception can have, redistribution of snow released from tree canopies can lead to large variations in snow depth over short (1–2 m) distances (Hardy et al., 1997). Attenuated snow depth errors (±d) are apparent due to surface co-registration problems that can arise on scour locations and steep slopes that contain little or no snow at the time of both acquisitions (Fig. 1C). These problems occur within surface depressions and other features that contain small amounts of snow, but where the observed depth is less than the combined LiDAR surface error. Another potential source of error is in areas comprised of low-lying densely matted

This study was conducted in the Reynolds Mountain East (RME) catchment, which is a 38 ha catchment, located at latitude 43° 5′ N, longitude 116° 45′W. The catchment is part of the greater Reynolds Creek Experimental Watershed, which is maintained as a hydrometeorological research site by the USDA Agricultural Research Service, Northwest Watershed Research Center. The catchment has a general northerly aspect, with a mean slope of 8° (σ = 4.5) but reaching 30° in isolated locations and elevations ranging from 2023 to 2142 m (Fig. 2). The landscape mosaic is dominated by a shrub-steppe cover type (mountain big sagebrush, Artimesia tridentata Nutt. ssp. vaseyana and mountain snowberry, Symphoricarpos oreophilus Gray) but with large patches of meadow (Lupinus ssp., Carex ssp., and Poa ssp.) and bare-ground, and homogenous coniferous and deciduous tree stands (Douglas-fir, Pseudotsuga menziesii, quaking aspen, Populus tremuloides) occurring in topographically sheltered portions of the catchment. The creek side is lined by Salix spp. and there are small patches of ceanothus (Ceanothus prostrates Benth.; Fig. 2). The diverse mix of cover types, radiation regimes, and topographic sheltering within a relatively small area makes this site a perfect location to evaluate the performance of remote sensing methods across a variety of vegetation classes. 2.2. LiDAR acquisitions Two airborne discrete return LiDAR acquisitions were obtained from the same vendor, with the same instrumentation and acquisition parameters. The data were acquired by the same vendor, with postprocessed using the same methods and parameters on both datasets (as outlined below). This consistency of methods is not only a fundamental requirement of multi-temporal remote sensing but was essential to minimize the influence of both sensor acquisition and processing parameters on the understanding of the subsequent uncertainties within the produced multi-temporal datasets. The snow-off dataset was acquired in mid-November 2007 and the snow-on dataset was acquired on March 19, 2009 to approximately coincide with the historic peak snow pack accumulations given the obvious logistical constraints. Each acquisition used a Leica ALS50 Phase II Laser, operating at 1064 nm with up to 4 returns per pulse recorded and was flown at approximately 900 m above ground level (agl). The data was acquired with a 50% flight line overlap and had a nominal pulse density of 6 pulses m − 2 , with an off nadir scan angle of ± 15°. The vendor

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Table 1 Comparison of DTM vertical accuracy utilizing both a surface produced from a single primary filtering and a surface produced following a secondary filtering. These are further compared to the surface accuracies reported by Tinkham et al. (2011). RMSE (m)

RMSE (m) from Tinkham et al. (2011)

Cover type

Primary filtering

Secondary filtering

Cover type

BCAL

MCC

Aspen Ceonothus Conifer Forb Bare ground Shrub Overall

0.210 0.317 0.366 0.140 0.470 0.296 0.297

0.188 0.328 0.259 0.124 0.452 0.261 0.264

Aspen Ceanothus Conifer Forb Bare ground Shrub Overall

0.190 0.126 0.249 0.130 0.507 0.268 0.273

0.193 0.412 0.292 0.131 0.467 0.268 0.280

performed flight line calibration and tiling of the raw LiDAR data using the TerraScan and TerraMatch software, respectively (Terrasolid Ltd., Jyväskylä, Finland). Absolute vertical accuracy for the snow-off and snow-on acquisitions was reported by the vendor as 3.3 and 3.7 cm root mean square error (RMSE) respectively, these were based on 1002 and 323 real time kinematic (RTK) global positioning system (GPS) points surveyed on gravel road surfaces at the lower elevations of the watershed. This represents the RMSE of the point elevations within the point cloud. The low vertical accuracy of the acquisition is attributed to GPS base stations being installed on a ridgeline at the southern end and valley bottom at the northern end of the study site during the acquisition. Along with the xyz coordinates, GPS time, intensity, and scan angle were also recorded for each return. Data were projected into the horizontal datum and projection of NAD83 UTM Zone 11 North and the vertical datum of NAVD88. For each acquisition, the Multi-scale Curvature Classification LiDAR algorithm (MCC: Evans & Hudak, 2007) was used to classify the raw LiDAR point cloud into ground and non-ground returns using parameters from a prior study (Tinkham et al., 2011). This prior study within the catchment cross-compared two commonly applied LiDAR surface generation algorithms (MCC and the BCAL algorithm; Streutker & Glenn, 2006) and demonstrated that minimal difference in accuracy

was observed between each approach. Overall, for the cover types found within our study area the MCC approach indicated a marginal improvement within the dominant cover types and thus was selected (Tinkham et al., 2011, 2013). To further improve on the MCC accuracy we explored the use of both optimized MCC scale and curvature parameters and a secondary filtering, where the MCC algorithm is reapplied to solely the ground points identified by the initial MCC application (Tinkham et al., 2012). This refinement of the MCC parameters and the additional step of the processing led to marginal improvements in accuracy within all catchment cover types (Table 1). Across the entire catchment, the LiDAR derived DEM had a mean vertical RMSE of 29 cm, but this error varied by cover type and topographic features (Tinkham et al., 2011). This error is partially attributed to the acquisition measurement and subsequent classification and interpolation errors. The snow-on and snow-off point clouds were interpolated with a thin-plate spline to produce both 1 and 4 m surfaces that were then differenced to create spatially explicit snow depth maps (Fig. 3). The surfaces were assessed for co-location errors by verifying the vertical and horizontal position of a weather station and support cabin at opposite sides of the catchment, this process revealed horizontal and vertical differences less than 4 cm. With the level of co-registration error being less than the combined point cloud absolute error provided by the vendor, the surfaces were not adjusted. In order to correct for classification and interpolation induced errors, negative snow depths (0.00–0.10 cm), which typically occurred on a windswept ridge that was observed to be snow free on the date of the acquisition and only accounted for approximately 0.1% of the catchment, were corrected manually to a depth of 0.00 cm. Further, for snow depths b0.10 cm and for isolated outlier snow depths N4 m (of which there were b30 cells), the average of the surrounding cells was assigned. From these spatially explicit maps, snow volume was computed by multiplying the snow depth by the grid cell dimensions. 2.3. Error accounting In the summer of 2009, a total of 99 high-precision ground survey plots (0.5 m point spacing on a square grid: Tinkham et al., 2011)

Fig. 3. (A) LiDAR-derived snow depth map with locations of manual snow depth transects, illustrating the heterogeneous nature of snow distribution within the catchment, and (B) Random Forest regression predictions of spatially explicit LiDAR DEM vertical error. 1 m LiDAR-derived maps are on the left, with 4 m maps on the right.

W.T. Tinkham et al. / Remote Sensing of Environment 141 (2014) 105–115 Table 2 List of variable derived from LiDAR point cloud at 1 and 4 m resolution for use in Random Forest modeling. The five variables identified as important in the final 1 m model are identified with ‡, while the four important variable in the final 4 m model are identified with ⁎. Variable

Metric name

Metric description

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

MinHeight MaxHeight RangeHeight MeanHeight ModeHeight StDevHeight VarHeight CVHeight InterQuartHeight KurtosisHeight SkewnessHeight ADDHeight #1Returns #2Returns #3Returns #4Returns TotalReturns ‡ PCT1 PCTothers PCT1AB0.1 PCTothersAB0.1 ‡ PCTABMean PCTABMode PCT5Height ⁎ PCT10Height ⁎ PCT25Height ⁎ PCT50Height PCT75Height PCT90Height PCT95Height NormAspect ‡⁎ PlanCurve ‡ ProCurve ‡ Slope

Minimum Return Height Maximum Return Height Range of Heights Mean of Heights Mode of Heights Standard Deviation of Heights Variance of Heights Coeficient of Variation of Heights Inter-Quartile Range of Heights Kurtosis of Heights Skewness of Heights Mean Absolute Deviation Height Number of 1st Returns Number of 2nd Returns Number of 3rd Returns Number of 4th Returns Total Returns Percentage 1st Returns Percentage not 1st Returns Percentage 1st above 0.10 m Percentage not 1st above 0.10 m Percentage Returns Above Mean Percentage Returns Above Mode 5th Percentile of Heights 10th Percentile of Heights 25th Percentile of Heights 50th Percentile of Heights 75th Percentile of Heights 90th Percentile of Heights 95th Percentile of Heights Normalized aspect Curvature Perpendicular to Slope Curvature Parallel to Slope Slope in Degrees

109

predicted the LiDAR error across the bare ground DEM. Optimal subsets of variables were selected for use in the prediction of vertical DEM error, to produce the most parsimonious and accurate model. Subsets were derived by iteratively running the RF algorithm and selecting the model that provided the highest accuracy but used the fewest predictor variables as identified by a Model Improvement Ratio (MIR) (Falkowski, Evans, Martinuzzi, Gessler, & Hudak, 2009; Murphy, Evans, & Storfer, 2009). The MIR procedure takes the mean decrease in the MSE, standardized from zero to one to indicate variable importance. From predefined threshold levels, model variables are then selected, with all variables above the threshold retained in each iterative model. From these candidate models the final model was determined based on the smallest mean square error of prediction and largest percentage variation explained. Before running the model selection procedure, a screening process based upon Gram-Schmidt QR-Decomposition was used to remove multi-collinear predictor variables (Gentle, Hardle, & Mori, 2005; Golub & Van Loan, 1996). The regression models came from 5000 bootstrap replicates of the calculated plot level MSE training dataset (n = 99) with replacement using a 36% data-withhold sample. With each of the bootstrap replicates producing an individual tree against which the data-withhold sample is then used in the calculation of the mean squared error (MSE) at each node and within the tree. Overall error and accuracy of the model was calculated by averaging error rates across all trees in the forest; this is similar to estimating error and accuracy through a cross-validation procedure (Cutler et al., 2007). From the final RF models, spatially explicit predictions of LiDARderived DEM vertical MSE were generated across the catchment using the AsciiGridPredict function of the yaImpute package within the R statistical software program (Crookston & Finley, 2008). These spatial maps of predicted LiDAR vertical errors were then converted to RMSE, as this is the more common statistic used when reporting LiDAR accuracy (Fig. 3).

2.4. Snow depth and volume validation

were stratified by cover and terrain type to capture the variability that topography and vegetation have on the accuracy of LiDAR derived surface (Fig. 2). The 0.5 m survey point spacing was selected to produce a validation dataset of higher point spacing and precision then the LiDAR dataset. Each survey plot was conducted over a 16 m2 area and was then sampled to correspond to a 1 × 1 m and 4 × 4 m grid spacing, with 9 points/m2, allowing the LiDAR to be assessed at two resolutions. Each plot was surveyed using a Topcon GTS-236w laser total station that had been georefenced using both a Topcon Hyper-pro real-time kinematic (RTK) global positioning system (Topcon Corp., Livermore, CA, USA) and USGS monuments. Vertical and horizontal precision of the RTK was 15 and 10 mm, respectively, with the survey points estimated to be within 2–3 cm horizontally and vertically. For each plot, all survey points were differenced from the snow-off LiDAR-derived DEM to determine error, the errors were squared and all points were averaged to determine the mean squared error (MSE) of the plot. Although less commonly applied, MSE is the appropriate statistic to use in this case as it can be estimated without bias using the point data. In order to account for the error associated with the LiDAR-derived estimates of snow depth, a spatially explicit map of LiDAR DEM vertical error was developed using the survey plot MSEs (n = 99) at both a 1 and 4 m grid scale, along with thirty-four variables derived from the snow-off LiDAR point cloud (Table 2) within a regression using the Random Forest algorithm (Breiman, 2001). Random Forest is an ensemble regression that consists of multiple decision trees, outputting the unweighted average over the forest. This was performed by running a model selection procedure in the R statistical software package (Liaw & Wiener, 2002) to develop a Random Forest (RF) regression tree that

The spatially explicit predicted errors of the snow-off DEM were propagated through to the snow depth maps to determine the range of possible snow depths within each grid cell following Eq. (1), where the snow depths come from the differenced LiDAR DEMs and the predicted errors come from the RF regression predictions of vertical error. The range of possible snow volume within a grid cell was

Table 3 Comparison of individual manual snow survey measures with LiDAR-derived snow depths. Performed with both 1 m and 4 m processing. The comparison highlights the conservative nature of the predicted error derived from the decision tree modeling, showing that accuracy assessment can be conducted in a spatially explicit manner. LiDAR-survey (m)

Mean RF model RMSE (m)

Cover type

n

Mean

StDev

RMSE

1 m processing Shrub Willow Deciduous Conifer edge Conifer outside Overall

105 183 102 154 94 544

−0.08 0.09 0.06 0.08 −0.04 0.05

0.13 0.2 0.13 0.36 0.21 0.24

0.15 0.21 0.14 0.37 0.21 0.25

0.37 0.28 0.26 0.18 – 0.27

4 m processing Shrub Willow Deciduous Conifer edge Conifer outside Overall

105 183 102 154 94 544

−0.06 0.11 0.09 0.1 0.15 0.07

0.15 0.21 0.14 0.37 0.23 0.26

0.16 0.24 0.17 0.38 0.28 0.27

0.28 0.34 0.3 0.27 – 0.3

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Fig. 4. Comparison of coincident manual and LiDAR estimated snow depths by cover types. In all cases the given regression functions performed slightly better than a linear function. The RMSE are arrived at by differencing the individual point level measures with the LiDAR estimated depths. Across all of the cover types the RMSE is 0.25 and 0.28 m for the 1 and 4 m surfaces respectively. The linear lines represent a 1:1 line.

performed in a similar manner, but the depth values were first multiplied by the area of a single grid cell. Range of Snow Depth ¼ Snow Depth r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  Predicted Erroroff þ ðPredicted Erroron Þ2

ð1Þ Due to the simplified terrain and surface roughness in snow covered landscapes, a conservative estimate of the snow-on LiDAR DEM error is assumed to equal the snow-off LiDAR DEM error. In this case, Eq. (1) simplifies to √(Predicted Error2 × 2). We acknowledge that the snowon surface would be expected to exhibit lower LiDAR DEM errors due to the high reflectivity and simplicity of the snow-on surface (Deems & Painter, 2006) and that this conservative assumption may produce

inflated estimates of the total error. The most optimistic scenario, although unrealistic, of snow volume error can be considered by assuming no snow-on LiDAR DEM error, which simplifies Eq. (1) to √(Predicted Error2). Calculation of the error by both routes effectively produces an upper and lower bound estimate of the error. The catchment was stratified by both redistribution terrain classes and cover types for assessment of distributed snow depth and volume. For this analysis, redistribution classes were defined as high, moderate, and low snow depth zones. High and low zones are classified as snow depths ± 1 standard deviation of the mean catchment snow depth (114 ± 93 cm for both 1 and 4 m grid). The seven catchment cover types ranging from bare-ground to mature trees is shown in Fig. 2A. A coincident snow survey was designed to be temporally coincident with the snow-on LiDAR acquisition to serve as a manual validation of LiDAR-derived snow depth map. The coincident snow survey was georeferenced using differentially corrected high precision GPS

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snow-on surfaces at small spatial extents. This 1 m data will be more beneficial in understanding fine-scale topographically-driven hydrological processes that occur over short spatial scales, such as flow pathways in ephemeral water courses, and for the validation of distributed snow modeling in heterogeneous landscapes. However, for regional-scale assessments of snow cover the gain in precision in using the 1 m grid scale should be considered against the computational benefits of using the 4 m grid scale. The analysis also highlights the cover type dependent accuracy that has been noted in past studies (Hopkinson et al., 2004). 3.2. Random Forest predicted error

Fig. 5. Comparison of coincident manual and LiDAR estimated snow depths for the Conifer Outside transect at the 1 and 4 m scales. The linear lines represent a 1:1 line.

locations (±0.5 m), containing 545 snow probe depth measurements, stratified by 4 of the investigated cover types (Fig. 3). The snow survey was used to both assess the mean depth measurements with the coincident 1 and 4 m LiDAR grid cells and to evaluate the predicted accuracy of the snow depth within each cover type. 3. Results and discussion 3.1. Snow depth validation Table 3 shows that when comparing LiDAR-derived snow depths to ground-based snow depth surveys, the LiDAR based estimates at the 1 m and 4 m cell sizes were most accurate in the deciduous and shrub transects (RMSE = 0.14–0.17 m), compared to the conifer edge transect (RMSE = 0.37–0.38 m). However, a second transect within another conifer stand (conifer outside transect) located just outside the catchment produced RMSE of 0.21 and 0.28 m for the 1 and 4 m surfaces respectively (Fig. 5). The discrepancy between observed errors within different cover types is attributed to the simplified nature of the snow surface within the shrub and meadow cover types, supporting the simplified snow surface error proposed by Deems and Painter (2006); as compared to the LiDAR filtering challenge introduced by the complex vertical structure of the snow covered tree cover types. The results show contradictory results to Hopkinson et al. (2004), who showed conifer cover types with little understory vegetation to produce smaller errors than other treed cover types. When comparing these results with a second conifer transect with similar understory conditions located just outside the catchment, errors are found that are similar to the prior study (Fig. 5, Table 2). The high errors exhibited for the conifer edge transect within the catchment are attributed to the location which is near the edge of a snow drift, where several stinger drifts (~3 m long and 0.5 m wide) extending into the conifer stand were observed by the field sampling crew. Across all transects, it is believed that coregistration errors between the manual snow survey and LiDAR surfaces may be leading to inflated snow depth error observations. Overall the 1 m grid size provides similar or better snow depth accuracies in all cover types when compared to the 4 m grid size (Table 2 and Fig. 4). We attribute this improvement to the ability of the 1 m grid size to capture variability in both the ground and

The final models produced a RMSE of prediction for the random forest trees of 0.24 and 0.21 m at the 1 and 4 m grid sizes, respectively, with correlation coefficients of 0.59 and 0.78 (p b 0.0001) between the observed and predicted DEM errors at the 1 and 4 m resolutions, respectively. Although Table 2 shows that at coincident snow probe locations the 1 m surface provided more accurate representations of terrain, modeling these accuracies at the fine scale includes more variability leading to a need for more training data as scales decrease to capture the increase heterogeneity. The comparison of the RF predicted RMSE and the observed survey grid plot level RMSE led to an overall RMSD of 0.14 m (range of 0.07–0.28 m) across all 99 plots (Tinkham et al., 2013), providing a conservative estimate for the errors commonly associated with different vegetation and terrain features (Tinkham et al., 2013). This conservative modeling of errors is evident in comparing the LiDAR snow depth with the manual snow measurements (Table 3). This shows that the in situ and LiDAR estimated snow depths are similar between cover types, but also highlights that the predicted errors at both the 1 and 4 m grid size are approximately 10% greater than the observed error from the ground surveys (Table 3, Tinkham et al., 2013). When comparing the snow survey calculated depth errors against the Random Forest modeled error there is a RMSE ranging from 0.09 to 0.34 m, with the 1 m surface producing predicted errors more closely representing the field calculated errors (Fig. 6). A possible source of error that could arise in the Random Forest modeling may come from the training data that was used to train the predictive model of LiDAR surface error. The model is limited by the training data's ability to fully represent the range of conditions; this may have led to errors in the current study as survey plots were not sampled on roads, stream beds, or within the willow cover type. The linear high error features that are evident within the 4 m predicted error map are coincident with roads and stream beds within the catchment (Fig. 3). Without sampling these catchment features, the regression tree was not able to distinguish them as unique cover types and was forced to produce uncertain predictions of error. Differences seen between the two resolutions (Fig. 3) are attributed to the training data's ability to represent the catchment level textural heterogeneity, when scaling up by a factor of sixteen from 1 to 4 m grid size, remotes sensing theory says variability will be reduced (Moody & Woodcock, 1995). This means that both resolutions will capture different features within the landscape, leading to them representing the error structure differently when modeled. Within the 1 m error prediction map the same features seem to be more reasonably represented, while areas within the bare ground cover type were predicted to have excessively high errors (Fig. 3). These excessive errors are attributed to training plots that were located upon rocky outcroppings, which were shown to cause DEM vertical RMSE of up to 1.75 m (Tinkham et al., 2011) and although this type of feature occupies a small area, they are an important component of the variability within the catchment. Within the bare ground cover type these outcroppings raised the average plot level RMSE to 0.44 m while the median was only 0.21 m. However, given the RF model uses the individual plot data and not the cover type mean or median values the snow volume errors should not be overly influenced by individual

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Fig. 6. Comparison of Random Forest modeled errors against LiDAR-derived snow depth errors from snow depth transects within the catchment at the 1 and 4 m scales. Across all transects within the catchment the difference between individual observed and predicted errors for the 1 and 4 m surfaces was a RMSE of 0.22 and 0.24 m, respectively. The linear lines represent a 1:1 line. Overall, observed and predicted upper bound snow depth errors demonstrated correlations of 0.24 and 0.22 (p b 0.001) at the 1 and 4 m scales, respectively, with similar values for the lower bound errors.

outliers (Tinkham et al., 2013). Studies attempting a similar analysis should be careful to capture a representative sample of the variability seen within cover types; improved field surveying (e.g. installing a greater number of plots with smaller spatial extents) may have provided an even stronger representation of the surface errors. The limited agreement between the predicted and observed snow depth errors is in part attributed to co-registration errors of the manual snow measurements and fine scale fluctuations of snow depth within the catchment (Fig. 6). 3.3. Catchment level snow volume At the catchment level the total snow volume error at both grid scales was estimated to have an upper and lower bound of approximately 30

and 22%, respectively, with the bare ground and ceanothus cover types exhibiting the greatest errors (Tables 4, 5). With a mean snow depth of 0.16 m within the bare ground cover type, the modeled surface errors quickly mask the snow signal and quickly lead to these high errors. However, together these two cover types account for less than 5% of the catchment snow volume. Both the low and high redistribution zones only occupied approximately 10% of the catchment area but exhibited contrasting snow volume error levels. Although the low snow depth zone error magnitude appears excessive, these exposed rocky areas only exhibited a mean snow depth of 5 cm and thus represent a very small fraction (~2%) of the catchment-wide snow volume. In contrast, high (drift) zones account for approximately 30% (±5 and 6% at the 1 and 4 m scales, respectively)

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Table 4 Distribution and quantification of snow depth and volume across the catchment, stratified by redistribution zones. Upper and lower bound error assessment performed with 1 m sampling and 4 m sampling. Snow volume (m3)

Snow depth (m)

Catchment area

Distribution regime

Mean

Min

Max

StDev

Minimum

Maximum

% Error

Hectares

%

Upper 1 m distribution High Moderate Low Total

3.29 1 0.05 1.14

1.05 0 0 0

6.64 3.09 1.33 6.64

0.97 0.46 0.07 0.94

114,528 193,799 1 308,328

142,061 407,911 21,317 571,289

10.7% 35.6% 100% 29.9%

3.9 30 3.7 37.6

10.4% 79.7% 9.9% 100%

Upper 4 m distribution High Moderate Low Total

3.29 1 0.05 1.14

2.08 0.21 0 0

6.15 2.07 0.2 6.15

0.97 0.46 0.07 0.94

111,999 192,870 9 304,877

142,863 409,875 13,242 565,979

12.1% 36.0% 99.9% 30.0%

3.9 29.9 3.7 37.5

10.3% 79.7% 9.9% 100%

Lower 1 m distribution High Moderate Low Total

3.29 1 0.05 1.14

1.35 0 0 0

6.45 2.79 1 6.45

0.97 0.46 0.07 0.94

118,560 222,888 17 341,465

138,029 375,967 15,649 529,646

7.6% 25.6% 99.8% 21.6%

3.9 30 3.7 37.6

10.4% 79.7% 9.9% 100%

Lower 4 m distribution High Moderate Low Total

3.29 1 0.05 1.14

1.08 0 0 0

6.15 2.07 0.2 6.15

0.97 0.46 0.07 0.94

116,519 219,796 32 336,348

138,343 377,296 9935 525,575

8.6% 26.4% 99.3% 22.0%

3.9 29.9 3.7 37.5

10.3% 79.7% 9.9% 100%

Table 5 Distribution and quantification of snow depth and volume across the catchment, stratified by cover type. Upper and lower bound error assessment performed with 1 m sampling and 4 m sampling. Snow volume (m3)

Snow depth (m) Cover type

Min

Max

StDev

Minimum

Maximum

% Error

Upper bound 1 Bare-ground Ceanothus Conifer Deciduous Meadow Shrub Willow Total

m 0.16 0.97 1.43 1.25 1.91 1.1 1.32 1.14

Mean

0 0 0 0 0 0 0 0

3.14 3.52 4.75 6.15 6.64 5.71 4.26 6.64

0.26 0.43 0.41 1.25 1.31 0.66 0.34 0.94

1653 2608 16,732 59,024 67,361 135,856 25,096 308,328

29,111 5188 27,231 99,118 98,239 267,869 44,532 571,289

89.3% 33.1% 23.9% 25.4% 18.6% 32.7% 27.9% 29.9%

Upper bound 4 Bare-Ground Ceanothus Conifer Deciduous Meadow Shrub Willow Total

m 0.16 0.96 1.43 1.91 1.25 1.1 1.32 1.14

0 0 0 0.16 0 0 0 0

1.92 2.73 3.18 6.15 5.51 5.34 3.82 6.15

0.26 0.43 0.4 1.32 1.25 0.66 0.34 0.94

1900 1377 15,739 65,335 59,859 139,907 20,760 304,877

20,993 6695 28,211 99,834 97,732 263,214 49,301 565,979

83.4% 65.9% 28.4% 20.9% 24.0% 30.6% 40.7% 30.0%

Lower bound 1 Bare-ground Ceanothus Conifer Deciduous Meadow Shrub Willow Total

m 0.16 0.97 1.43 1.25 1.91 1.1 1.32 1.14

0 0 0 0 0 0 0 0

2.85 3.44 4.64 5.98 6.45 5.6 4.19 6.45

0.26 0.43 0.41 1.25 1.31 0.66 0.34 0.94

2362 2947 18,259 64,190 71,846 153,929 27,933 341,465

22,558 4796 25,691 93,033 93,709 248,176 41,673 529,646

81.0% 23.9% 16.9% 18.3% 13.2% 23.4% 19.7% 21.6%

Lower bound 4 Bare-Ground Ceanothus Conifer Deciduous Meadow Shrub Willow Total

m 0.16 0.96 1.43 1.91 1.25 1.1 1.32 1.14

0 0 0 0.16 0 0 0 0

2.43 3.04 3.52 7.15 6.05 6.15 4.39 7.15

0.26 0.43 0.4 1.32 1.25 0.66 0.34 0.94

2511 1868 17,518 70,130 64,529 155,240 24,550 336,348

16,823 5852 26,377 94,748 92,014 244,673 45,088 525,575

74.0% 51.6% 20.2% 14.9% 17.6% 22.4% 29.5% 22.0%

of the total snow volume (Table 4). This result is similar to that found previously by Marks, Winstral, and Seyfried (2002), who showed that drifts contained 25% of the SWE in only 10% of the catchment area based on physically-based snowpack modeling results. The consistency between the two approaches confirms the utility of multi-temporal LiDAR for the assessment and validation of catchment scale snow volume distributions. This is of particular significance given that drifts act as water sources late into the growing season and allow for pockets of water limited species (e.g. aspen and fir trees) to be established or sustained. This results in a heterogeneous vegetation structure across the landscape that in turn influences the radiative and hydrometerological regime and creates biological “hotspots” (Baumeister & Callaway, 2006; Campbell & Bartos, 2001; Reba, Marks, Winstral, Link, & Kumar, 2011). Relatively high snow densities in the drift zones could potentially amplify the importance of this observed high snow volume by providing more water per unit snow depth than observed in shallower snow that has not been condensed through redistribution processes (Sturm et al., 2010). Although the effect of rocky outcroppings is potentially significant, as large quantities of snow could potentially accumulate over time, these features are limited in extent within Reynolds Mountain East (b1% of area), rarely accumulate snow depths N10 cm and can be seen as isolated patches of high error values along the northeast side of the catchment. 4. Conclusions This study presented a robust and comprehensive evaluation of the efficacy of multi-temporal LiDAR to quantify snow depths and the use of Random Forest to estimate the associated accuracies across seven different vegetative cover classes. The utility of Random Forest for the assessment of accuracies was well demonstrated both in the present study and in related works (Tinkham et al., 2013). Given the relatively fast analysis of multi-temporal LiDAR produced results of snow volume distribution that agreed with a prior intensive 8 year field measurement and modeling study (Marks et al., 2002) the potential of multi-temporal LiDAR for this application has been re-affirmed. While it is apparent that challenges remain in quantifying snow volume from multi-temporal LiDAR (Nolin, 2010), advances in the acquisition and processing of

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LiDAR are making these surmountable. However, the contrasting errors found between the two manually measured snow depth transects within the conifer cover type highlight how difficult it can be to validate snow depth in areas were depth varies at very small spatial scales. The current study determined a theoretical upper bound of 30% error in catchment wide snow volume through Random Forest modeling, highlighting the range of accuracies within different cover types and snow redistribution zones. Although the lower bound error of 21.6% represents a considerable improvement, clear strategies are apparent that could further reduce this lower bound. For example, the adoption of cover specific classification algorithms may allow the errors to be lowered in heterogeneous landscapes, or enhanced understanding of how pulse intensity can be used to penetrate dense vegetation. Ultimately, even with improved LiDAR processing for DEM creation the snow volume error will always be dominated by the LiDAR DEM error in the underlying snow-off surface, which in the current study represents the 21.6%. These estimates could be enhanced by building off studies applying high precision Terrestrial LiDAR Systems (TLS) or ground based topological surveys from total stations to validate aerial LiDAR estimates of snow depth (Grünewald et al., 2010; Prokop, 2008; Schirmer et al., 2011). It is likely that this would result in very small errors of snow volume. Under such a schema, robust validation of the error in the snow-on surface would be required, perhaps through a snow-on high precision topographical survey referenced to monument controls. Clearly, the complexity of the surfaces at this site and the shallow snow depths represent an effective worst-case scenario to test the utility of multi-temporal LiDAR and Random Forest for snow volume assessments. Through performing similar analysis of landscapes with either more continuous or simpler cover types, it is likely that this lower bound error of 21.6% could be greatly diminished. The magnitude of the upper and lower bound errors in this study are driven by the LiDAR DEM error; a high percentage when compared to the shallow snow depths at our site. Clearly, repeating this application on sites with snow packs an order of magnitude deeper than what we observed (e.g., 5–6 m as observed in the Sierra Nevada and Cascade Ranges) would make the LiDAR DEM errors a small percentage of the total depth, likely reducing the upper and lower error bounds by a similar degree. The ability to perform this type of analysis at extended spatial scales will help to advance understanding of the importance of snow distribution on water resources, surface energy balance, and vegetation dynamics. The application of these techniques is of potentially high importance in semiarid regions that rely on shallow snow packs for annual water supply. The next step in investigating the spatial distribution of snow at landscape levels will require the incorporation of snow density and/or SWE to improve the estimation of water storage in complex terrain. Acknowledgments We would like to thank Dr. Andrew Robinson for his guidance and review of the statistical methodologies utilized in this work. This project was funded by the UMAC and the Idaho Space Grant Consortium, which are both in turn funded by NASA. Additional funding and support was provided through the Agricultural Research Service Northwest Watershed Research Center. Funding support was provided by Idaho NSF EPSCoR and under awards NSF EPS-0814387, NSF EPS-0701898, NSF CBET-0854553, and a NASA New Investigator Award NNX10AO02G. References Anderton, S. P., White, S. M., & Alvera, B. (2004). Evaluation of spatial variability of snow water equivalent for a high mountain catchment. Hydrological Processes, 18, 435–453. Banos, I. M., Garcia, A.R., & Alavedra, J. M. (2011). Assessment of airborne LiDAR for snowpack depth modeling. Boletín de la Sociedad Geológica Mexicana, 63, 95–107.

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