We present a simple method that allows snow depth measurements to
be converted to snow water equivalent (SWE) estimates. These estimates are
useful to individuals interested in water resources, ecological function,
and avalanche forecasting. They can also be assimilated into models to help
improve predictions of total water volumes over large regions. The
conversion of depth to SWE is particularly valuable since snow depth
measurements are far more numerous than costlier and more complex SWE
measurements. Our model regresses SWE against snow depth (

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.

Conceptual sketch of the evolution of snow water equivalent (SWE) over the course of a water year (black line). Also shown is the evolution of SWE with snowpack depth over a water year (red line). Note the hysteresis loop due to the densification of the snowpack.

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

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

Given the relative ease in obtaining depth measurements, it is common to use

Many studies express

Other approaches choose to parameterize

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

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

In this section, we list sources of 1970–present snow data utilized for this study (Table 1). With regards to snow coring devices, we refer to them using the terminology preferred in the references describing the data sets.

SNOTEL (Serreze et al., 1999; Dressler et al., 2006) and SCAN (Schaefer et
al., 2007) stations in the contiguous United States (CONUS) and Alaska
typically record sub-daily observations of

Distribution of measurement locations used in this study.

Goodison et al. (1987) note that Canada has no national digital archive of
snow observations from the many independent agencies that collect snow data
and that snow data are instead managed provincially. The quantity and
availability of the data vary considerably among the provinces. The Water
Management Branch of the British Columbia (BC) Ministry of the Environment
manages a comparatively dense network of Automated Snow Weather Stations
(ASWSs) that measure SWE,

The snow survey program (USDA, 2008) dates to the 1930s and includes a large number of snow course and aerial marker sites (Fig. 2c) in western North America. While the measurement frequency is variable, it is most commonly monthly. To generate a data set for this study, data were extracted using the National Water and Climate Center Report Generator 2.0. This allows filtering by time period, elevation band, and other elements. All sites with data between 1980 and 2018 were included (Fig. 2c).

In addition to the data from the SCAN sites, snow data for this project from
the northeastern United States come from two networks and three research sites (Fig. 2b). The Maine Cooperative Snow Survey (MCSS, 2018) network includes

The Sleepers River, Vermont Research Watershed in Danville, Vermont (Shanley
and Chalmers, 1999), is a USGS site that includes 15 stations with long-term
weekly records of

In the spring of 2018, we conducted 3 weeks of fieldwork in the Chugach
Mountains in coastal Alaska, near the city of Valdez (Fig. 2d–e). We
measured

Figure 3 demonstrates that it is not uncommon for automated snow pillow
measurements to become noisy or nonphysical, at times reporting large
depths when there is no SWE reported. This is different from instances when
physically plausible but very low densities might be reported; say in
response to early season dry, light snowfalls. It was therefore desirable to
apply some objective, uniform procedure to each station's data set in order
to remove clear outlier points, while minimizing the removal of valid data
points. We recognize that there is no accepted standardized method for
cleaning bivariate SWE–

Sample time series of SWE and

The Mahalanobis distance (MD; De Maesschalck et al., 2000) quantifies how far a point lies from the mean of a bivariate distribution. The distances are in terms of the number of standard deviations along the respective principal component axes of the distribution. For highly correlated bivariate data, the MD can be qualitatively thought of as a measure of how far a given point deviates from an ellipse enclosing the bulk of the data. One problem is that the MD is based on the statistical properties of the bivariate data (mean, covariance) and these properties can be adversely affected by outlier values. Therefore, it has been suggested (e.g., Leys et al., 2018) that a “robust” MD (RMD) be calculated. The RMD is essentially the MD calculated based on statistical properties of the distribution unaffected by the outliers. This can be done using the minimum covariance determinant (MCD) method as first introduced by Rousseeuw (1984).

Once RMDs have been calculated for a bivariate data set, there is the
question of how large an RMD must be in order for the data point to be
considered an outlier. For bivariate normal data, the distribution of the
square of the RMD is

A scatter plot of SWE vs.

Scatter plot of SWE vs.

Summary of information about the data sets used in this study. Data sets in bold font were used to construct the regression model. The numbers of stations and data points reflect the post-processed data.

The 30-year climate normals at 1 km resolution for North America were obtained from the ClimateNA project (Wang et al., 2016). This project provides grids for minimum, maximum, and mean temperature, and total precipitation for a given month. These grids are based on the PRISM normals (Daly et al., 1994) and are available for the periods 1961–1990 and 1981–2010. For this study, the more recent climatology was used. The ClimateNA project also provides a wide array of derived bioclimatic variables, such as precipitation as snow (PAS), frost-free period (FFP), mean annual relative humidity (RH), and others. Wang et al. (2012) summarize these additional variables and how they are derived. Figure 5 shows gridded maps of winter (sum of December, January, February) precipitation (PPTWT) and the temperature difference (TD) between the mean temperature of the warmest month and the mean temperature of the coldest month. The latter variable (TD) is a measure of continentality.

Gridded maps of winter (December, January, February) precipitation (PPTWT) and temperature difference (TD) between the mean of the warmest month and the mean of the coldest month) for North America. Maps are for the 1981–2010 climatological period.

In order to demonstrate the varying degrees of influence of explanatory
variables, several regression models were constructed. In each case, the
model was built by randomly selecting 50 % of the paired SWE–

The simplest equation, and one that is supported by the strong correlation
seen in the portions of Fig. 3 when SWE is present, is one that expresses
SWE as a function of

Recall from Figs. 1 and 4 that there is a hysteresis loop in the SWE–

As these two equations are discontinuous at DOY

A final model was constructed by incorporating climatological variables.
Again, the emphasis in this study is on methods that can be implemented at
locations lacking the time series of weather variables that might be
available at a weather or SNOTEL station. Climatological normals are unable
to account for interannual variability, but they do preserve the high
spatial gradients in climate that can lead to spatial gradients in snowpack
characteristics. Stepwise linear regression was used to determine which
variables to include in the regression. The initial list of potential
variables included was

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

Two-dimensional histograms (heat maps;

Summary of performance metrics for the three regression models presented in Sect. 2.2.

Heat map of SWE residuals as a function of DOY for the application of the multivariable two-equation model to the western North America snow pillow validation data set.

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.

Probability density function of snow pillow station root-mean-square error (RMSE) normalized by station winter precipitation (PPTWT) for the application of the multivariable two-equation model to the western North America snow pillow validation data set.

Spatial distribution of snow pillow station root-mean-square error (RMSE) normalized by station winter precipitation (PPTWT) for the application of the multivariable two-equation model to the western North America snow pillow validation data set.

A key objective of this study is to regress climatological information in a
continuous rather than a discrete way. The work by Sturm et al. (2010)
therefore provides a valuable point of comparison. In that study, the
authors developed the following equation for density

Model parameters by snow class for Sturm et al. (2010).

To make a comparison, the snow class for each SNOTEL and British Columbia
snow survey (rows 1 and 3 of Table 1) site was determined using a 1 km snow
class grid (Sturm et al., 2010). The aggregated data set from these stations
was made up of 27 % alpine, 14 % maritime, 10 % prairie, 11 %
tundra, and 38 % taiga data points. Equation (11) was then used to
estimate snow density (and then SWE) for every point in the validation
data set described in Sect. 2.2. Figure 10 compares the SWE estimates from
the Sturm model and from the current multivariable, two-equation model
(Eqs. 5, 7–8). The upper left panel of Fig. 10 shows all of the data,
and the remaining panels show the results for each snow class. In all cases,
the current model provides better estimates (narrower
cloud of points; closer
to the

Comparison of the multivariable, two-equation model of the current study with the model of Sturm et al. (2010), applied to the western North America snow pillow validation data set. The subpanels show modeled SWE vs. observed SWE for all of the data binned together, as well as for the data broken out by the snow classes identified by Sturm et al. (1995). The gray symbols show the Sturm result and the transparent heat maps (warmer colors indicate greater density of points) show the current result.

Comparison of model performance by Sturm et al. (2010) and the current study.

Comparison of the multivariable, two-equation model of the current study with the model of Sturm et al. (2010), applied to the western North America snow pillow validation data set. The panels show probability density functions of the residuals of the model fits for all of the data binned together, as well as for the data broken out by the snow classes identified by Sturm et al. (1995). The gray lines show the Sturm result and the colored lines show the current result. The vertical lines show the mean error, or the model bias, for both the Sturm and current results.

In order to provide an additional comparison, the simple model of Pistocchi
(2016) was also applied to the validation data set. His model calculates the
bulk density as

A final point of comparison can be provided by the model of Jonas et al. (2009). The full version of that model contains region-specific offset
parameters that are not relevant to North America, so the following partial
version of the model is used (their Eq. 4):

Model coefficients

The regression equations in this study were developed using a large
collection of snow pillow sites in CONUS, AK, and BC. The snow pillow sites
are limited to locations west of approximately 105

Figure 12 graphically summarizes the data sets and the performance of the
multivariable two-equation model of the current study. The RMSE values are
comparable to those found for the western stations, but, given the
comparatively thinner snowpacks in the northeast, represent a larger
relative error (Table 6). The bias of the model is consistently positive, in
contrast to the western stations where the bias was negligible. Note that
Table 6 also includes results from the application of the other three models
discussed. Sturm et al. (2010) cannot be applied to several of the data sets
since their available 1 km snow class data set cuts off at

Results from application of the multivariable, two-equation
model to numerous northeastern US data sets. The left column shows the SWE–

Performance metrics for various models applied to the northeastern US data sets. Bold font is used to highlight the model with the best performance for each data set.

The NRCS snow course and aerial marker data were also left out of the model building process so they provide an additional and completely independent comparison of the various models considered. Recall that these data come from snow course (coring measurements) and aerial surveys, which are different measurement methods than the snow pillows, which provided the data for construction of the current regression model. Figure 13 shows the aggregated snow course/aerial marker data set, along with the performance of the multivariable two-equation model of the current study. Table 7 summarizes the results and demonstrates that the current model has the best performance.

Performance metrics for various models applied to the NRCS snow course and aerial marker data set. Bold font is used to highlight the model with the best performance.

Results from application of the multivariable, two-equation model to the NRCS snow course/aerial marker data set. Panel

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

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–

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

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–

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.

Numerous online data sets were used for this project and were obtained from
the following locations:

NRCS Snow Telemetry,

NRCS Soil Climate Analysis Network,

British Columbia Automated Snow Weather Stations,

Maine Cooperative Snow Survey,

New York Snow Survey,

Sleepers River Research Watershed, snow data not available online; request
data from contact at

Hubbard Brook Experimental Forest,

Climatological Data,

NRCS snow course/aerial marker data,

A MATLAB function for calculating SWE based on the results is this paper has
been made publicly available at GitHub (

JK contributed (acquisition, formatting, preliminary analysis) the Alaska and British Columbia data, JMH contributed the SNOTEL data, EAB and JK contributed the northeastern USA data, DFH contributed the western USA snow course/aerial marker data, RLC contributed the Chugach data. All authors contributed to the conception and direction of this study and to writing and editing of the manuscript. DFH conducted the regression analysis and was the principal author of the manuscript.

The authors declare that they have no conflict of interest.

We thank Matthew Sturm, Adam Winstral, and the third anonymous referee for their careful and thoughtful reviews of this paper.

This research has been supported by NASA (grant no. NNX17AG67A), CUAHSI (Pathfinder Fellowship grant), and the NSF (grant no. MSB-ECA 1802726).

This paper was edited by Jürg Schweizer and reviewed by Matthew Sturm, Adam Winstral, and one anonymous referee.