In recent years, marked improvements in our knowledge of the statistical
properties of the spatial distribution of snow properties have been achieved
thanks to improvements in measuring technologies (e.g., LIDAR, terrestrial
laser scanning (TLS), and ground-penetrating radar (GPR)). Despite this,
objective and quantitative frameworks for the evaluation of errors in snow
measurements have been lacking. Here, we present a theoretical framework for
quantitative evaluations of the uncertainty in average snow depth derived
from point measurements over a profile section or an area. The error is
defined as the expected value of the squared difference between the real mean
of the profile/field and the sample mean from a limited number of
measurements. The model is tested for one- and two-dimensional survey designs
that range from a single measurement to an increasing number of regularly
spaced measurements. Using high-resolution (

The assessment of uncertainties of snow measurements remains a challenging problem in snow science. Snow cover properties are highly heterogeneous over space and time and the representativeness of measurements of snow stage variables (e.g., snow depth, snow density, and snow water equivalent (SWE)) is often overlooked due to difficulties associated with the assessment of such uncertainties. This has been, at least in part, due to the limited knowledge of the characteristics of the spatial statistical properties of variables such as snow depth and SWE, particularly at the small scale (sub-meter to tens of meters). However, recent improvements in remote sensing of snow (e.g., light detection and ranging (LIDAR) and radar technologies) have allowed significant progress in the quantitative understanding of the small-scale heterogeneity of snow covers in different environments (e.g., Trujillo et al., 2007, 2009; Mott et al., 2011).

Point or local measurements of snow properties will continue to be necessary for purposes ranging from inexpensive evaluation of the amount of snow over a particular area, to validation of models and remote sensing measurements. Such measurements have a footprint representative of a very small area surrounding the measurement location (i.e., support, following the nomenclature proposed by Blöschl, 1999), and the integration of several measurements is necessary for a better representation of the snow variable in question over a given area. Because of this, tools for quantitative evaluations of the representativeness and uncertainty of measurements need to be introduced, and the uncertainty of such measurements should be more widely discussed in the field of snow sciences.

Currently, efforts to assess the reliability and uncertainty of snow
measurements have focused on statistical analyses using point measurements
(e.g., Pomeroy and Gray, 1995; Yang and Woo, 1999; Watson et al., 2006;
Rice and Bales, 2010; Lopez-Moreno et al., 2011; Meromy et al., 2013) or
synthetically generated fields in a Monte Carlo framework (e.g., Kronholm
and Birkeland, 2007; Shea and Jamieson, 2010), comparisons between remotely
sensed and ground data (e.g., Chang et al., 2005; Grünewald and
Lehning, 2014), and analyses of subsets drawn from spatially distributed
remotely sensed data (e.g., McCreight et al., 2014). These studies have
been useful to empirically quantify uncertainties associated with point
measurements. For example, Pomeroy and Gray (1995) present an
equation for determining the minimum number of surveys points required to be
confident that the mean falls within a certain envelop around the sample
mean based on the CV of SWE or snow depth. McCreight et al. (2014) use
the NASA's Cold Land Processes Experiment (CLPX) LIDAR snow depth data set
(also used in this study) to empirically address questions regarding the
inference of larger-scale snow depths from sparse observations. They
evaluate estimation uncertainty from random sampling for varying sample
size. Their conclusions indicate that adding observations to a randomly
distributed survey pattern leads to a reduction in both percent-error in
snow volume over the study areas, as well as its uncertainty. They also add
that with a few hundred observations, one can expect to infer the true mean
snow depth over the 1 km

Another possible approach is one in which the expected error in the estimation of a particular statistical moment of a field over a defined domain (e.g., areal mean or standard deviation from a finite number of measurements) is determined on the basis of known statistical properties of the field in question. Such an approach uses geostatistical principles that have been proposed by Matheron (1955, 1970) and others, and that have been applied in mining geostatistics (Journel and Huijbregts, 1978), the analysis of uncertainties in measuring precipitation (Rodríguez-Iturbe and Mejía, 1974), and for a more general analysis of the effects of sampling of random fields as examples of environmental variables (e.g., Skøien and Blöschl, 2006). Implementation of these types of approaches appear to be lacking in the numerous studies using point measurements to represent snow distribution. Often in these studies, the spatial snow distribution derived from point measurements is addressed as the “true” distribution, which is then used for evaluating the performance of interpolation methodologies, regressions trees, and hydrological models. These comparisons ignore the intrinsic error incurred when extrapolating the original point measurements, leaving a proportion of uncertainty unaccounted for that can be significant. The principal motivation of the present study is to encourage the use of more objective and quantitative methodologies for error evaluation in snow sciences. The approach presented below can be used for objective survey design to estimate snow distribution from point measurements. We do not intend to present our approach as novel in the general geostatistical sense; instead, we present the derivation with the specific application for snow sciences in mind. However, because of the general nature of the random fields' theory the development is based on, similar developments can indeed be applied to other environmental variables that can be described as a random field.

On this basis, the error in the estimation of spatial means from point measurements over a particular domain (e.g., a profile, or an area) can be quantified as the expected value of the squared difference between the real mean and the sample mean obtained from a limited number of point measurements. Such an approach, as it will be shown here, uses spatial statistical properties of snow depth fields in a way that allows for an objective evaluation of the estimation error for snow depth measurements. The sections below illustrate the use of such methodology for optimal design of sample strategies in the specific context of snow depth. However, the methodology can also be implemented for other snow variables such as snow water equivalent.

Let

The first two terms in Eq. (5) are the total sum of the covariances (or
correlation as

For the analyses and tests of the methodology presented here, light detection
and ranging (LIDAR) snow depths obtained as part of the NASA's Cold Land
Processes Experiment (CLPX) will be used (Cline et al., 2009). The data set
consists of spatially distributed snow depths for 1 km

The spatial representation of the snow cover requires a basic assumption on the scale or resolution at which a field or profile is going to be represented. This relies on the spatial support of the measurements. For the case of snow depths, point measurements from local surveys using a snow depth probe are frequently used for this representation. Generally, there are additional sources of uncertainty associated with these types of measurements, such as the accuracy of the position of the measurement in space or deviations in the vertical angle of penetration of the probe through the snow pack. These uncertainties are additional to any of the uncertainties estimated using the methodology discussed here.

The one-dimensional case provides a good opportunity to illustrate the
limitations of point measurements. Consider the case of a snow depth profile
that is measured using a snow depth probe at a regular spacing “

The answer to this question is conditioned to how variable the profile is and
over what distances. To address this, let us look at two snow depth profiles,
one in a forested environment (FS) and another in an open environment (RW) in
the Colorado Rocky Mountains (Figs. 2a and 3a, respectively). The variability
in the profiles is markedly different, with variations over shorter distances
in the forested area, and a smoother profile in the open and wind influenced
environment. This is reflected in the spatial correlation structure of these
snow depth profiles, with stronger correlations over longer distances in open
and wind-influenced environments with respect to that in forested
environments (Trujillo et al., 2007, 2009). These differences should be
considered when selecting the sampling frequency required to capture the
variability and accurately represent the mean conditions within a particular
sampling spacing. This is illustrated by comparing the mean snow depth for a
particular resolution to the point value at the center of the interval
(Fig. 2b in a forested environment and Fig. 3b in an open and wind-influenced
environment). In the figures, average vs. point values at several sampling
intervals are compared for normalized profiles (

Firstly, the resulting comparison shows that the point values generally
overestimate the variability in mean snow depths if we replace the mean snow
depth distribution by its point sample. To clarify this, let us consider here
two snow depth profiles, one with the snow depths at the nominal scale
(

Secondly, the differences between average and point values for each spacing
distance are generally more scattered in the forested environment than in the
open environment, and in both environments the degree of scattering increases
with spacing (Figs. 2c and 3c). However, it is important to note here that we
are comparing normalized profiles (

Sub-interval standard deviation

In addition to differences in correlation structure, there are also
differences in the absolute variability in snow depth in these environments
(Fig. 4). Contrary to the normalized snow depth discussed above, the
subinterval standard deviation as a function of interval size along the
profiles is higher in the open and wind-influenced environment at RW vs.
the forested environment at FS (Fig. 4a). Mean standard deviation values in
the open environment are twice as large as those at the forested environment
towards the larger interval sizes (

Consistent with the standard deviation, the sub-interval mean range (range defined as the difference between the maximum and minimum snow depths within an interval) increases with interval size in both FS and RW (Fig. 4b). However, the mean range is larger in the open environment at RW and the rate of increase with interval size is also steeper. Similarly, the shaded areas indicate wider distribution of range values in the open environment at RW, while they are relatively uniformly distributed around the mean across interval sizes in the forested environment at FS. The results in Figs. 2–4 illustrate this contrasting behavior between the snow covers in these environments and their influence on measurement strategies: that is, the forested environments requires shorter separation between measurements for accurate representation of the snow cover; however, in the wind-influence and open environment, the subinterval variability is higher indicating wider variations around any sampled measurement within the interval.

Ultimately, the number and distance between measurements and the specific arrangement of the measurements are all conditioned to what the measurements are needed for. Hydrologic applications may not require a highly detail representation of a snow depth profile (or a field), and representing the average conditions over a given distance (or area) is sufficient, but small-scale process-based studies may require a more detailed characterization over shorter distances (or smaller areas). This implies that the decision depends on the particular usage that the measurements will support. In the following sections, the equations presented in the background (Sect. 2) will be applied to evaluate the uncertainty associated with multiple measurement designs for profiles and fields of snow depth.

Survey designs for the sampling of a snow profile.

Comparison of the theoretical and sampled normalized squared error
(

Equation (2) can be used to evaluate the uncertainty of a single measurement
along a profile section of length

Theoretical and sampled normalized squared error
(

Theoretical normalized squared error (

The results here are confirmed with an analysis of LIDAR snow depths
profiles in FS and RW (Fig. 6). The analysis consists of calculating the
difference between the mean and the point value for sections of a given
length (varied between 10–50 m) and for

Model and sampled results thus support that the measurement location can be
fixed in the middle of the interval, and the normalized squared error can
then be described as a function of both the exponential decay exponent,

From Eq. (5) it is also evident that increasing the number of measurements
will reduce the squared error. In the case of three measurements separated by
a distance “

The performance of the model is tested against the normalized squared error
obtained from the same snow depth profiles in FS and RW. The test consists of
estimating the normalized squared error for profiles sections of length
between 10 and 80 m, with

As stated above, the measurement error can be reduced by increasing the
number of measurements taken over a given section of length

Theoretical and sampled normalized squared error
(

Sample survey designs with

Following the method described in the previous section, we test the
performance of the model against the normalized squared error obtained from
the same snow depth profiles in FS and RW. In this case, the sampled squared
error is estimated for

Similar to the one-dimensional process, Eq. (5) can be formulated to
calculate the squared error in the two-dimensional space. To exemplify this,
we apply the methodology to an isotropic process over the

Theoretical normalized squared error (

Theoretical and sampled normalized squared error
(

Theoretical normalized squared error (

Theoretical and sampled normalized squared error
(

For the isotropic case, the covariance/correlation function is only dependent
on the magnitude of the lag vector,

As discussed earlier, the first term is only a function of

In this case, we focus on a single measurement in the middle of a square area
of side dimension

The case of five measurements radiating out from the center (Fig. 12a), with
a point in the middle of the area and four points separated by a distance

The performance of the model is tested against the normalized squared error
obtained from the snow depth field in FS. The test consists of estimating the
normalized squared error for a square area with side length (

Similarly to the one-dimensional case, the two-dimensional case of

The performance of the model is tested again for a square area with side
length (

A methodology for an objective evaluation of the error in capturing mean snow depths from point measurements is presented based on the expected value of the squared difference between the real average snow depth and the mean of a finite number of snow depth samples within a defined domain (e.g., a profile section or an area). The model can be used for assisting the design of survey strategies such that the error is minimized in the case of a limited and predetermined number of measurements, or such that the desired number of measurements is determined based on a predefined acceptable uncertainty level. The model is applied to one- and two-dimensional survey examples using LIDAR snow depths collected in the Colorado Rockies. The results confirm that the model is capable of reproducing the estimation error of the mean from a finite number of samples for real snow depth fields.

Here, we should highlight some of the implications of the assumptions made in the model. In simplified terms, the second-order stationarity assumption implies that the mean and the variance of the process/variable (e.g., snow depth) are independent of the spatial location, and that the covariance is dependent only on the separation vector (i.e., lag). Although these assumptions may be less valid over larger scales (e.g., greater than 100 m), in the context of the model application to snow depth the assumption should be valid at smaller scales. We present these examples to show how the error can be quantified with good accuracy at such smaller scales. Application of these types of approaches at larger scales will require additional evaluation with particular attention as to what the specific demands of the application are. Also, the methodology presented here is not suitable for discontinuous snow cover if both snow-covered and snow-free areas are considered in the error estimation. This case has not been considered in the development here.

Implementation of the model in practice requires prior assumption of a
correlation/covariance model and estimates of the model parameters (e.g., the
decay exponent for the exponential case). In the examples presented here we
use LIDAR data for the parameter estimation to illustrate the applicability
of the model and its ability to estimate the error using real snow depth
data. Snow distributions in mountain environments have been shown to be
consistent intra- and inter-annually because the controlling processes are
relatively consistent during the season and from season to season. Such
consistency suggests that the correlation/covariance model should also be
consistent, as well as the parameters of the model. These parameters can be
estimated via a dense survey either manually or with TLS of one or more small
plots of a size similar to the size that is aimed to be represented. These
surveys would not necessarily have to be repeated as the parameters and
covariance models should be preserved. Detailed surveys can be conducted
under different conditions to characterize the range of the correlation
models and parameters (e.g., after a snow storm, or close to peak
accumulation). Also here, we should point out that although we show results
for a wide range of the exponential decay exponent values, we are finding
that most of the values that we have observed are in the lower range of those
presented (e.g., 0.1–0.2 m

Currently, remote sensing technologies (e.g., TLS, Airborne LIDAR, and ground penetrating radar) are allowing for the characterization of snow cover properties at increasing resolutions in both space and time. Such improvements can be utilized in the context presented here providing information about the range of best fitting covariance/correlation models and parameters for different conditions, supporting the application of methodologies such as the one presented here. With such improvements, survey designs can be optimized such that estimation errors can be explicitly addressed and accounted for, particularly when extrapolating a limited number of measurements to estimate the spatial distribution of snow. Such applications will continue to be relevant despite of the aforementioned improvements, as access to these technologies is limited by their cost and the expertise that is required for their application.

Data for this article were obtained from NASA's Cold Land Processes
experiment (CLPX), available at