In many parts of the world, snow plays a leading-order role in the hydrological cycle (USACE, 1956; Mote et al., 2018). Accurate information about the spatial and temporal distribution of snow water equivalent (SWE) is useful to many stakeholders (water resource planners, avalanche forecasters, aquatic ecologists, etc.), but can be time consuming and expensive to obtain.

Snow pillows (Beaumont, 1965) are a well-established tool for measuring SWE at fixed locations. Figure 1 provides a conceptual sketch of the variation in SWE with time over a typical water year. A comparatively long accumulation phase is followed by a short ablation phase. While simple in operation, snow pillows are relatively large in size and they need to be installed prior to the onset of the season's snowfall. This limits their ability to be rapidly or opportunistically deployed. Additionally, snow pillow installations tend to require vehicular access, limiting their locations to relatively simple topography. Finally, snow pillow sites are not representative of the lowest or highest elevation bands within mountainous regions (Molotch and Bales, 2006). In the western United States (USA), the Natural Resources Conservation Service (NRCS) operates a large network of snow telemetry (SNOTEL) sites, featuring snow pillows. The NRCS also operates the smaller Soil Climate Analysis Network (SCAN), which provides the only, and very limited, snow pillow SWE measurements in the eastern United States.

SWE can also be measured manually, using a snow coring device that measures the weight of a known volume of snow to determine snow density (Church, 1933). These measurements are often one-off measurements, or in the case of “snow courses” they are repeated weekly or monthly as a transect of measurements at a given location. The simplicity and portability of coring devices expand the range over which measurements can be collected, but it can be challenging to apply these methods to deep snowpacks due to the limited length of standard coring devices. Note that there are numerous different styles of coring devices, including the Adirondack sampler and the Mt. Rose or Federal sampler (Church and Marr, 1937). The NRCS operates a large network of snow course sites (USDA, 2011) in the western United States.

There are a number of issues that affect the accuracy of both snow pillow and snow coring measurements. With coring measurements, if the coring device is not carefully extracted, a portion of the core may fall out of the device. Or, snow may become compressed in the coring device during insertion. These effects have led to varying conclusions, with some studies (e.g., Sturm et al., 2010) showing a low SWE bias and other studies (e.g., Goodison, 1978) showing a high SWE bias. As noted by Johnson et al. (2015) a good rule of thumb is that coring devices are accurate to around ±10 %. Also, studies comparing different styles of snow samplers report statistically different results, suggesting that SWE measurements are sensitive to the design of the specific coring device, such as the presence of holes or slots, the device material, etc. (Beaumont and Work, 1963; Dixon and Boon, 2012). With snow pillows, some studies (e.g., Goodison et al., 1981) note that ice bridging can lead to low biases in measured SWE, with the snow surrounding the pillow partly supporting the snow over the pillow. Other studies (Johnson and Marks, 2004; Dressler et al., 2006; Johnson et al., 2015) note a more complex situation with SWE underreported at times, but overreported at other times. Note that when snow pillow data are evaluated, they are most commonly compared to coring measurements at the same location.

All methods of measuring SWE are challenged by the fact that SWE is a depth-integrated property of a snowpack. This is why the snowpack must be weighed, in the case of a snow pillow, or a core must be extracted from the surface to the ground. This measurement complexity makes it difficult to obtain SWE information with the spatial and temporal resolution desired for watershed-scale studies. Other snowpack properties, such as the depth h, are much easier to measure. For example, using a graduated device such as a meterstick or an avalanche probe to measure the depth takes only seconds. Automating depth measurements at a fixed location can easily be done using low-cost ultrasonic devices (Goodison et al., 1984; Ryan et al., 2008). High-spatial-resolution measurements of snowpack depth are commonly made with lidar. One example of this is the Airborne Snow Observatory program (ASO; Painter et al., 2016). The comparatively high expense of airborne lidar surveys typically limits measurements geographically (to a few basins) and temporally (weekly to monthly interval).

Given the relative ease in obtaining depth measurements, it is common to use h as a proxy for SWE. Figure 1 shows a conceptual sketch of the variation in SWE with h over a typical water year. Noting the arrows on the curve, we see that SWE is multivalued for each h. This is due to the fact that the snowpack increases in density throughout the water year, producing a hysteresis loop in the curve. A large body of literature exists on the topic of how to convert h to SWE. It is beyond the scope of this paper to provide a full review of these “bulk density equations”, where the density is given by ρb=SWE/h. Instead, we refer readers to the useful comparative review by Avanzi et al. (2015). Here, we prefer to discuss a limited number of previous studies that illustrate the spectrum of methodologies and complexities that can be used to determine ρb or SWE.

Many studies express ρb as an increasing function (often linear) of h. In some cases (e.g., Lundberg et al., 2006) a second equation is added where ρb attains a constant value when a threshold h is exceeded. A single linear equation captures the process of densification of the snowpack during the accumulation phase, but performs poorly during the ablation phase, where depths are decreasing but densities continue to increase or approach a constant value.

Other approaches choose to parameterize ρb in terms of time, rather than h. Pistocchi (2016) provides a single equation while Mizukami and Perica (2008) provide two sets of equations, one set each for early and late seasons. Each set contains four equations, each of which is applicable to a particular “cluster” of stations. This clustering was driven by observed densification characteristics and the resulting clusters are relatively spatially discontinuous. Jonas et al. (2009) take the idea of region- (or cluster-) specific equations and extend it further to provide coefficients that depend on time and elevation as well. They use a simple linear equation for ρb in terms of h and the slope and intercept of the equation are given as monthly values, with three elevation bins for each month (36 pairs of coefficients). There is an additional contribution to the intercept (or “offset”) which is region-specific (one of seven regions).

These classifications, whether based on region, elevation, or season, are valuable since they acknowledge that all snow is not equal. McKay and Findlay (1971) discuss the controls that climate and vegetation exert on snow density, and Sturm et al. (2010) address this directly by developing a snow density equation where the coefficients depend upon the “snow class” (five classes). Sturm et al. (1995) explain the decision tree, based on temperature, precipitation, and wind speed, that leads to the classification. The temperature metric is the “cooling degree month” calculated during winter months only. Similarly, only precipitation falling during winter months was used in the classification. Finally, given the challenges in obtaining high-quality, high-spatial-resolution wind information, vegetation classification was used as a proxy. Using climatological values (rather than values for a given year), Sturm et al. (1995) were able to develop a global map of snow classification.

There are many other formulations for snow density that increase in complexity and data requirements. Meloysund et al. (2007) express ρb in terms of sub-daily measurements of relative humidity, wind characteristics, air pressure, and rainfall, as well as h and estimates of solar exposure (“sun hours”). McCreight and Small (2014) use daily snow depth measurements to develop their regression equation. They demonstrate improved performance over both Sturm et al. (2010) and Jonas et al. (2009). However, a key difference between the McCreight and Small (2014) model and the others listed above is that the former cannot be applied to a single snow depth measurement. Instead, it requires a continuous time series of depth measurements at a fixed location. Further increases in complexity are found in energy-balance snowpack models (SnowModel, Liston and Elder, 2006; VIC, Liang et al., 1994; DHSVM, Wigmosta et al., 1994, others), many of which use multilayer models to capture the vertical structure of the snowpack. While the particular details vary, these models generally require high-temporal-resolution time series of many meteorological variables as input.

Despite the development of multilayer energy-balance snow models, there is still a demonstrated need for bulk density formulations and for vertically integrated data products like SWE. Pagano et al. (2009) review the advantages and disadvantages of energy-balance models and statistical models and describe how the NRCS uses SWE (from SNOTEL stations) and accumulated precipitation in their statistical models to make daily water supply forecasts. If SWE information is desired at a location that does not have a SNOTEL station, and is not part of a modeling effort, then bulk density equations and depth measurements are an excellent choice.

The present paper seeks to generalize the ideas of Mizukami and Perica (2008), Jonas et al. (2009), and Sturm et al. (2010). Specifically, our goal is to regress physical and environmental variables directly into the equations. In this way, environmental variability is handled in a continuous fashion rather than in a discrete way (model coefficients based on classes). The main motivation for this comes from evidence (e.g., Fig. 3 of Alford, 1967) that density can vary significantly over short distances on a given day. Bulk density equations that rely solely on time completely miss this variability and equations that have coarse (model coefficients varying over either vertical bins or horizontal grids) spatial resolution may not fully capture it either.

Our approach is most similar to Mizukami and Perica (2008), Jonas et al. (2009), and Sturm et al. (2010) in that a minimum of information is needed for the calculations; we intentionally avoid approaches like Meloysund et al. (2007) and McCreight and Small (2014). This is because our interests are in converting h measurements to SWE estimates in areas lacking weather instrumentation. The following sections introduce the numerous data sets that were used in this study, outline the regression model adopted, and assess the performance of the model.

A comparison of the three regression models (one-equation model, Eq. 2; two-equation model, Eqs. 3–5; multivariable two-equation model, Eqs. 5, 7–8) is provided in Fig. 6. The left column shows scatter plots of modeled SWE to observed SWE for the validation data set with the 1:1 line shown in black. Figure 6a, c, and e show distributions of the model residuals. The vertical lines in Fig. 6b, d, and f show the mean error, or model bias. Visually, it is clear that the one-equation model performs relatively poorly with a large negative bias. This large negative bias is partially overcome by the two-equation model (Fig. 6c, d). The cloud of points is closer to the 1:1 line and the vertical black line indicating the mean error is closer to zero. In Fig. 6e, f, we see that the multivariable two-equation model yields the best result by far. The residuals are now evenly distributed with a small bias. Several metrics of performance for the three models, including R2 (Pearson coefficient), bias, and root-mean-square error (RMSE), are provided in Table 2. Figure 7 shows the distribution of model residuals for the multivariable two-equation model as a function of DOY.

It is useful to also consider the model errors in a nondimensional way. Therefore, an RMSE was computed at each station location and normalized by the winter precipitation (PPTWT) at that location. Figure 8 shows the probability density function of these normalized errors. The average RMSE is approximately 15 % of PPTWT with most values falling into the range of 5 %–30 %. The spatial distribution of these normalized errors is shown in Fig. 9. For the SNOTEL stations, it appears there is a slight regional trend, in terms of stations in continental climates (Rockies) having larger relative errors than stations in maritime climates (Cascades). The British Columbia stations also show higher relative errors.

The results presented in this study show that the regression equation described by Eqs. (5), (7)–(8) is an improvement (lower bias and RMSE) over other widely used bulk density equations. The key advantage is that the current method regresses in relevant parameters directly, rather than using discrete bins (for snow class, elevation, month of year, etc.), each with its own set of model coefficients. The comparison (Figs. 10–11; Table 4) to the model of Sturm et al. (2010) reveals a peculiar behavior of that model for the taiga snow class, with a large negative bias in the Sturm estimates. Inspection of the coefficients provided for that class (Table 3) shows that the model simply predicts that ρb=ρmax⁡=0.217 for all conditions.

When our multivariable two-equation model, developed solely from western North American data, is applied to northeastern US locations, it produces SWE estimates with smaller RMSE values and larger biases than the western stations. When comparing the SWE–h scatter plots of the SNOTEL data (Fig. 4b) to those of the east coast data sets (left column; Fig. 12), it is clear that the northeast data generally have more scatter. This is confirmed by computing the correlation coefficients between SWE and h for each data set. It is unclear if this disparity in correlation is related to measurement methodology or is instead a signal-to-noise ratio issue. Comparing Figs. 4 and 12 shows the considerable difference in snowpack depth between the western and northeastern data sets. When the western data set is filtered to include only measurement pairs where h<1.5 m, the correlation coefficient is reduced to a value consistent with the northeast data sets. This suggests that the performance of the current (or other) regression model is not as good at shallow snowpack depths. This is also suggested upon examination of the time series of observed ρb=SWE/h for a given season at a snow pillow site. Very early in the season, when the depths are small, the density curve has a lot of variability. Later in the season, when depths are greater, the density curve becomes much smoother. Very late in the season, when depths are low again, the density curve becomes highly variable again.

Measurement precision and accuracy affect the construction and use of a regression model. Upon inspection of the snow pillow data, it was observed that the precision of the depth measurements was approximately 25 mm and that of the SWE measurements was approximately 2.5 mm. To test the sensitivity of the model coefficients to the measurement precision, the depth values in the training data set were randomly perturbed by ±25 mm and the SWE values were randomly perturbed by ±2.5 mm and the regression coefficients were recomputed. This process was repeated numerous times and the mean values of the perturbed coefficients were obtained. These adjusted coefficients were then used to recompute the SWE values for the validation data set and the bias and RMSE were found to be -10.5 and 72.7 mm. This represents a roughly 10 % increase in RMSE, but a considerable increase in bias magnitude (see Table 4 for the original values). This sensitivity of the regression analysis to measurement precision underscores the need to have high-precision measurements for the training data set. Regarding accuracy, random and systematic errors in the paired SWE–h data used to construct the regression model will lead to uncertainties in SWE values predicted by the model. As noted in the introduction, snow pillow errors in SWE estimates do not follow a simple pattern. Additionally, they are complicated by the fact that the errors are often computed by comparing snow pillow data to coring data, which itself is subject to error. Lacking quantitative information on the distribution of snow pillow errors, we are unable to quantify the uncertainty in the SWE estimates.

Another important consideration has to do with the uncertainty of depth measurements that the model is applied to. For context, one application of this study is to crowd-sourced, opportunistic snow depth measurements from programs like the Community Snow Observations (CSO; Hill et al., 2018) project. In the CSO program, backcountry recreational users submit depth measurements, typically taken with an avalanche probe, using a smartphone in the field. The measurements are then converted to SWE estimates, which are assimilated into snowpack models. These depth measurements are “any time, any place” in contrast to repeated measurements from the same location, like snow pillows or snow courses. Most avalanche probes have centimeter-scale graduated markings, so measurement precision is not a major issue. A larger problem is the considerable variability in snowpack depth that can exist over short (meter-scale) distances. The variability of the Chugach avalanche probe measurements was assessed by taking the standard deviation of eight depth measurements per site. The average of this standard deviation over the sites was 22 cm and the average coefficient of variation (standard deviation normalized by the mean) over the sites was 15 %. This variability is a function of the surface roughness of the underlying terrain, and also a function of wind redistribution of snow. Propagating this uncertainty through the regression equations yields a slightly higher (16 %) uncertainty in the SWE estimates. CSO participants can do three things to ensure that their recorded depth measurements are as representative as possible. First, avoid measurements in areas of significant wind scour or deposition. Second, avoid measurements in terrain likely to have significant surface roughness (rocks, fallen logs, etc.). Third, take several measurements and average them.

Expansion of CSO measurements in areas lacking SWE measurements can increase our understanding of the extreme spatial variability in snow distribution and the inherent uncertainties associated with modeling SWE in these regions. It could also prove useful for estimating watershed-scale SWE in regions like the northeastern United States, which is currently limited to five automated SCAN sites with historical SWE measurements for only the past 2 decades. Additionally, historical snow depth measurements are more widely available in the Global Historical Climatology Network (GHCN-Daily; Menne et al., 2012), with several records extending back to the late 1800s. While many of the GHCN stations are confined to lower elevations with shallower snow depths, the broader network of quality-controlled snow depth data paired with daily GHCN temperature and precipitation measurements could potentially be used to reconstruct SWE in the eastern United States given additional model development and refinement.

We have developed a new, easy-to-use method for converting snow depth measurements to snow water equivalent estimates. The key difference between our approach and previous approaches is that we directly regress in climatological variables in a continuous fashion, rather than a discrete one. Given the abundance of freely available climatological norms, a depth measurement tagged with coordinates (latitude and longitude) and a time stamp is easily and immediately converted into SWE.

We developed this model with data from paired SWE–h measurements from the western United States and British Columbia. The model was tested against entirely independent data (primarily snow course, some snow pillow) from the northeastern United States and was found to perform well, albeit with larger biases and root-mean-squared errors. The model was tested against other well-known regression equations and was found to perform better. The model was also tested against a large data set of independent snow course and aerial marker measurements from western North America. For this second independent test, the current model outperformed the other models considered.

This model is not a replacement for more sophisticated snow models that evolve the snowpack based on high-frequency (e.g., daily or sub-daily) weather data inputs. The intended purpose of this model is to constrain SWE estimates in circumstances where snow depth is known, but weather variables are not, a common issue in sparsely instrumented areas in North America.