TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-1249-2015Theoretical analysis of errors when estimating snow
distribution through point measurementsTrujilloE.ernesto.trujillo@epfl.chLehningM.https://orcid.org/0000-0002-8442-0875School of Architecture, Civil and Environmental Engineering,
École Polytechnique Fédérale de Lausanne, Lausanne, SwitzerlandWSL Institute for Snow and Avalanche Research SLF, Davos,
SwitzerlandE. Trujillo (ernesto.trujillo@epfl.ch)19June2015931249126428November20145January201519May201525May2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/9/1249/2015/tc-9-1249-2015.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/9/1249/2015/tc-9-1249-2015.pdf
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 (∼ 1 m) LIDAR snow depths
at two locations in Colorado, we show that the sample errors follow the
theoretical behavior. Furthermore, we show how the determination of the
spatial location of the measurements can be reduced to an optimization
problem for the case of the predefined number of measurements, or to the
designation of an acceptable uncertainty level to determine the total number
of regularly spaced measurements required to achieve such an error. On this
basis, a series of figures are presented as an aid for snow survey design
under the conditions described, and under the assumption of prior knowledge
of the spatial covariance/correlation properties. With this methodology,
better objective survey designs can be accomplished that are tailored to the
specific applications for which the measurements are going to be used. The
theoretical framework can be extended to other spatially distributed snow
variables (e.g., SWE – snow water equivalent) whose statistical properties
are comparable to those of snow depth.
Introduction
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 km2 domains to within 2 % error. Despite of
these insights, these types of empirical approaches can be site-dependent,
they do not provide a theoretical quantitative framework for the assessment
of uncertainties associated with a particular sampling design, they do not
allow for an optimal sampling strategy (e.g., selecting the number of points
and locations for a desired accuracy level), and they do not take advantage
of the increased knowledge of the characteristics of the heterogeneity of
snow cover properties.
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.
Background
Let Z(x)
denote a random field function of the coordinates x in the
n-dimensional space R. Bold-italic letters represent a
location vector from hereon. In our case, the
field can represent, e.g., snow depth or snow water equivalent (SWE) at a
given time of the year. The mean of the process over a domain A (e.g., a
profile section or an area) is defined as follows:
μz(A)=1A∫Az(x)dx.
In practice, the mean is often obtained from the arithmetic average of
measurements at a finite number of locations, N, within the domain as follows:
Z‾=1N∑i=1Nz(xi).
The performance of the estimator Z‾
can be evaluated by calculating the expected value of the square difference
between the estimator Z‾ and the true mean μz(A)σZ‾2A=E1N∑i=1Nzxi-1A∫Azxdx2.
For a 1st order stationary process (i.e., the mean independent of location;
e.g., Cressie (1993), Sect. 2; and Journel and Huijbregts (1978), Sect. 2),
(Eq. 3) can be expressed as
σZ‾2A=1N2∑i=1NVARzxi+2N2∑i=1N-1∑j=i+1NCOVzxizxj-2N⋅A∑i=1N∫ACOVzxizxjdxj+1A2∫A∫ACOVzxizxjdxidxj,
where VAR[ ] and COV[ ] are the variance and the covariance, respectively.
If we further assume that the process is second order stationary (e.g.,
Cressie (1993), Sect. 2; and Journel and Huijbregts (1978), Sect. 2),
that is, if the mean and the variance are independent of the location,
and the covariance function depends only on the vector difference
xi-xj. Equation (3) can be expressed as
σZ‾2A=σp21N+2N2∑i=1N-1∑j=i+1NCORRxi-xj-2NA∑i=1N∫ACORRxi-xjdxj+1A2∫A∫ACORRxi-xjdxidxj,
where CORR[ ] is the correlation function, and σp2 is the
variance of the point process.
The first two terms in Eq. (5) are the total sum of the covariances (or
correlation as σp2 has been factored out) between all point
locations i=1,…,N (e.g., measurement locations). The first of the two
terms is only a function of the number of points, while the second is a
function of the number of points, N, and the correlations between the
locations. Such correlations are themselves a function of the separation
vectors (both in magnitude and direction), and the parameters of the
correlation function. These two terms are independent of the size of the area
A, and can be thought of as the portion of the error caused by the
correlation between the point processes at the locations i=1,…,N
(e.g., measurement locations). Term 3 accounts for the correlation between
the measurement locations and the continuous process over the domain A.
This term can be seen as a negative contribution to the total error assuming
that the sum of the integrals is positive. The term is a function of the
number of points, N, the domain area, A, the location of the points and
the correlation structure, characterized using the parameters of the
correlation function. Lastly, term 4 is the contribution to the error caused
by the intrinsic correlation structure of the continuous process over the
domain. This term is a function of the domain (e.g., size and shape of A)
and the correlation structure (e.g., parameters of the correlation function).
Data
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 × 1 km
areas (intensive study areas – ISAs) in the Colorado Rocky Mountains close
to maximum snow accumulation in April 2003. The data were processed from
snow-on (8–9 April 2013) and snow-off (18–19 September 2013) LIDAR
elevation returns with an average horizontal spacing of 1.5 m and vertical
tolerance of 0.05 m. The final CLPX snow depth contour product (0.10 m
vertical spacing) was generated from these returns. This product was used to
generate gridded snow depth surfaces with 1024 × 1024 elements over
the ISAs, for a grid resolution of 0.977 m. For this study two areas will be
used: the Fraser–St Louis Creek ISA (FS) and the Rabbit Ears–Walton
Creek ISA (RW) (Fig. 1). The FS ISA is covered by a moderate density
coniferous (lodgepole pine) forest on a flat aspect with low relief. The RW
ISA is characterized by a broad meadow interspersed with small, dense stands
of coniferous forest and with low rolling topography. The snow depth
distributions in these ISAs show differences that are relevant for the
analysis of the methodology introduced here. At the FS ISA, the snow depth
distribution is relatively isotropic (Fig. 1b), with short spatial
correlation memory and little variation in the spatial scaling properties
(i.e., power-spectral exponents and scaling breaks) with direction (Trujillo
et al., 2007). On the other hand, the spatial distribution of snow depth in
the RW ISA is more anisotropic (Fig. 1c), with longer spatial correlation
memory along a principal direction aligned with the predominant wind
direction vs. shorter memory along the perpendicular direction, and with
variations in the power-spectral exponents and scaling breaks according to
the predominant wind directions (Trujillo et al., 2007).
(a) Location of the Fraser and Rabbit Ears study areas in
the state of Colorado.
(b) LIDAR Snow depth distributions on 8 April 2003, at the Saint
Louis Creek intensive study area (ISA) and (c) on 9 April at the
Rabbit Ears ISA.
(a) Sample normalized snow depth profile (mean = 0,
standard deviation = 1) in a forested environment from LIDAR (1 m
resolution) at the Fraser–St. Louis Creek (FS) intensive study area (ISA)
of the Cold Land Processes Experiment (CLPX) (Trujillo et al., 2007; Cline et
al., 2009). The profile is sampled with regular separations (spacing) of 5,
10, 25, 50, and 100 m (from top to bottom, respectively). (b)
Average values within sampling intervals (same as in a) vs. point
samples for normalized snow depth profiles in the FS ISA. The red line is a
linear regression fit, with slope β and r2 as indicated in each
plot. (c) Histograms of the difference between the point and average
values for each of the sampling intervals. The vertical red line marks the
mean difference.
One-dimensional process
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 “d”. Each
of these point measurements is meant to represent the mean snow depth over a
particular distance surrounding the measurement. The question is, over what
distance is this assumption valid? In this case, the intrinsic assumption is
that the measurement is representative over the distance “d”, but at this
point the validity of such an assumption is not proven.
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 (μ=0, σ=1)
separated every 30 m in both the x (east) and y (north) directions and
for an area of 500 m × 500 m. The 30 m separation between
profiles is chosen to reduce the spatial correlation between them.
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
(∼ 1 m), and a second one with a moving average (MA) of the first one
with an averaging window equal to the sampling spacing. Ultimately, the
variance/standard deviation of the first profile (∼ 1 m) is larger
than that of the MA, with a distribution that reflects these differences. The
samples drawn from the first profile will reflect a larger variance than that
of the samples from the MA profile as they are drawn from these
distributions, and this is what is reflected in Figs. 2 and 3. The degree of
overestimation can be quantified through the slope of the regression line (in
red in Figs. 2b and 3b). In the forested environment (Fig. 2b), the slopes
range between 0.8 and 0.13, with decreasing slopes with increasing spacing.
These slopes indicate that, on average, the magnitudes of the mean values are 0.8
times the magnitudes of the point values for the 5 m spacing and 0.1 times for the
100 m spacing. In the open and wind-dominated environment, the slopes are
higher and range between 0.97 and 0.23 from 5 m spacing and 100 m spacing,
respectively. A clear difference emerges: forested environments require
shorter separation between single measurements if the snow depth profile is
to be accurately captured by the measurements. The variability within the
size of the interval determines the degree of uncertainty associated with the
point measurements, as the sub-interval variability is related to the degree
of overestimation of the mean value within the interval.
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 (μ=0, σ=1), allowing us to
focus on the rescaled spatial variations. What is highlighted is the
relevance of the spatial structure of the profile rather than the absolute
variance. This spatial structure can be quantified by, for example, the
spatial covariance/correlation function.
(a) As Fig. 2 but for an open and wind influenced
environment at the Rabbit Ears–Walton Creek (RW) ISA of the CLPX (Trujillo
et al., 2007; Cline et al., 2009). (b) Average values within
sampling intervals (same as in a) vs. point samples for
normalized snow depth profiles in the RW ISA. The red line is a linear
regression fit, with slope β and r2 as indicated in each plot.
(c) Histograms of the difference between the point and average
values for each of the sampling intervals. The vertical red line marks the
mean difference.
Sub-interval standard deviation (a) and range
(b) for varying interval lengths for profiles of snow depth in a
forested environment (FS) and an open and wind-influenced environment (RW)
in the Colorado Rocky Mountains (same regions as those in Figs. 2 and 3). The
mean standard deviation and mean range for the study areas are shown by the
solid lines, while the shaded areas cover the quantiles between 25 and 75 %
of the values for all the intervals in these areas.
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 (∼ 100 m). The standard deviation
increases with interval size in both environments, with the steepest increase
at the lower interval sizes. Furthermore, the standard deviation tends to
stabilize more rapidly in the forested environments, with an increase of only
1.8 cm between 30 and 100 m. On the other hand, the standard deviation
continues to increase in the open environment at RW, with less of an
asymptotical behavior for the scales analyzed. Complementary, the shaded
areas (25 to 75 % quantiles) give an idea of the variability of
standard deviation values, with a much wider range in RW vs. FS, and an
increase in the range between quantiles with interval size in RW.
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
(σZ‾2σp2) for the case of a single
measurement along a profile section of length L, as in Fig. 5a. The survey
case applied to profiles in FS and RW along the x and y directions. Solid
lines are the theoretical error using exponential decay exponents derived
from the functions fitted to the sampled correlation functions of the two
surfaces in the x and y directions.
(a) Theoretical normalized squared error for a single
measurement in the middle of a section of length, L, and for an exponential
correlation function with a decay exponent, ν. (b) and
(c) Comparison of the theoretical and sampled normalized squared
error for the same survey case applied to profiles in FS and RW along the x
and y directions. Dashed lines are the theoretical error from Eq. (7) using
exponential decay exponents derived from the functions fitted to the sampled
correlation functions of the two surfaces in the x and y directions.
Case 1: single measurement along a profile section
Equation (2) can be used to evaluate the uncertainty of a single measurement
along a profile section of length L. For this case, as well as for the
following cases in this article, an exponential covariance with a decay
exponent ν (ν > 0) will be assumed:
COVh,σ,ν=σ2exp-νhforσ2>0,andν>0,,
where σ2 is the variance, and h is
the length of the vector h. For this one-dimensional case and
combining Eqs. (6) and (5), the following expression is obtained:
σZ‾2x,L,νσp2=1-2Lν2-exp-νx-exp-ν⋅L-x+1L2ν2L+2νexp-νL-2ν,
where x is the distance from one extreme of the section to the location of
the measurement (Fig. 5a). The normalized squared error
σZ‾2x,L,vσp2 is
minimized at x equal to half of the section length, L/ 2, regardless
of ν. The existence of a correlation in the profile leads to this
solution, as the middle location contains more information about its
surroundings. Also, this solution is different from the solution for an
uncorrelated profile (e.g., white noise), for which the squared error would
be equal to the variance, independent of the location of the measurement.
(a) and (b) Theoretical normalized squared error
for the three-point pattern along a profile section in Fig. 5b, and for
profile section lengths (L) of 1 (a) and 25 (b). Each of
the colored lines corresponds to a specific decay exponent, ν, and the
black line marks the theoretical solution for aoptimal. (c)
Theoretical normalized error and aoptimal for isolines of profile section
lengths (L) and exponential decay exponents (ν) for the three-point
pattern along a profile section of length L in Fig. 5b.
Theoretical and sampled normalized squared error
(σZ‾2σp2) for the three-point
pattern along a profile section in Fig. 5b, and for profile section lengths
(L) between 10 and 80 m in FS and RW. The solid lines are the theoretical
error from Eq. (8) using exponential decay exponents derived from the
functions fitted to the sampled correlation functions of the two surfaces in
the x and y directions, while the dots correspond to the sampled error
for profiles in FS (a–d) and RW (e–h).
Theoretical normalized squared error (σZ‾2σp2) for the N-point pattern along a profile section
in Fig. 5c, and for profile section lengths (L) between 10 and 80 obtained
from Eq. (10).
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 x (Fig. 5a) between 0 and L
along the profile sections. Each sample section of length L will provide a
single difference for each of the x values. These sample differences are then
used to calculate the mean normalized squared error for each x, and the same
is repeated for each section length L. The results indicate that the real
snow depth profiles behave as predicted by the model of the error, with a
minimum error at x equal to half of the section length. Another difference
highlighted by these results is the difference between the sample errors in
the forested environment (FS) vs. the open environment (RW) for the
larger interval sizes (e.g., 50 m). The sampled normalized squared error in
the forested environment shows only a mild decrease in the square error to
around 0.7–0.8 towards the inside of the section length. However, this
decrease is achieved for the measurement along most of the interval length
with the exception of the extremes. This can be explained by the
relationship between the spatial memory of snow depth (e.g., the correlation
function) and the section length. Densely forested environments exhibit
correlation lengths that are shorter than those in open and wind influenced
environments (e.g., Trujillo et al., 2007, 2009). As the
section length increases beyond such correlation lengths, a measurement
location towards the middle of the interval contains less information of the
surrounding snow depths in a forested environment (e.g., FS) vs. an open
and wind influenced environment (e.g., RW). This is observed in Fig. 6c
vs. Fig. 6f, with the results in RW showing a more clear minimum
towards the center of the profile section. The results also show a poorer
performance of the model in RW vs. FS, as the exponential correlation
model has a poorer fit in RW at the shorter-lag range; However, model
performance is improved for longer section lengths (e.g., Fig. 6c and f)
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,
ν, and the length of the section, L (Fig. 7a). The normalized squared
error increases with interval length, with a steeper increase for larger
exponential decay exponents, for which the squared error approaches that of
an uncorrelated field more rapidly. The theoretical model is tested on the
snow depth fields at FS and RW. The test consists of calculating the sampled
normalized squared error as the average of all squared differences between
the mid-section snow depth and the mean from all LIDAR grid points within
each interval of length L. This is done for profiles separated every 30 m,
similar to the analysis above, and for profiles along the x and y directions.
The theoretical normalized squared error is estimated from Eq. (7) using the
exponential decay exponent from the model fitted to the sampled correlation
function. The results show that the theoretical model reproduces the sampled
squared error remarkably well, even reproducing the anisotropic properties
of the correlograms, represented by the different exponents of the
exponential model along x and y directions (Fig. 7b and c). The model also
reproduces the different behavior of the squared error between both fields
(i.e., FS and RW), showing that the normalized squared error increases more
rapidly and is larger in the forested environment (Fig. 7b) vs. the
open environment (Fig. 7c). However, it should be noted here that as the
error is normalized and as the variance of the field in the open environment
is larger (Fig. 4a), the absolute squared error could reach higher values
in the open environment (RW). In this regard, one feature to discuss here is
the assumption that the point variance of snow depth in these environments
has been estimated as the spatial variance over the entire study area, as it
is generally practiced in time series analysis and geostatistics. In
practice, this is the only possible approach because there is limited
information to estimate the point variance from multiple realizations of the
process at each spatial location, as inter-annual and intra-annual snow depth
fields are not available, not only for these areas, but for almost any area
where this methodology may be applied.
Case 2: three measurements along a profile section
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 “a”, with the middle measurement centered in the section of
length L (Fig. 5b), and for an exponential covariance function with
parameter ν, Eq. (5) leads to the following expression:
σZ‾2a,L,νσp2=13+292exp-νa-exp-2νa-43Lν3-exp-νL21+exp-νa+expνa+1L2ν2L+2νexp-νL-2ν.
Equation (8) can be minimized to determine the optimal separation distance
between points, a, as a function of L and ν:
aoptimal=-1νlnt,
where
t=B+B2-4AB2AA=4ν9andB=-43Lexp-νL2.
The combination of Eqs. (8) and (9) can be used to determine the normalized
squared error, σZ‾2σp2, and the
optimal distance, aoptimal, for the measurement pattern in
Fig. 5b. The model predicts that the normalized squared error is minimized at
an intermediate location between 0 and L/ 2 (black lines in Fig. 8a
and b). The results show an increase in the error with interval size, L, as
well as little sensitivity of aoptimal to ν. This latter
feature can be seen as an advantage since small biases in the estimation of
ν will not result in significant biases in the estimation of
aoptimal. One could almost assume a value of aoptimal
without prior knowledge of the exponential decay exponent, selecting
aoptimal within the range of values indicated by the model for a
rage of possible exponential decay exponents. Note that aoptimal
is located close to the 60 % distance from the center towards the outer
boundary of the profile section for all section lengths (Fig. 8a and b). On
the other hand, the measurement error displays a higher sensitivity to ν
around aoptimal, indicating that biases in the estimation of
ν would have a more noticeable effect on the estimation of the
measurement error. This is further clarified in Fig. 8c, in which the
normalized error (not squared) and aoptimal can be obtained for
corresponding profile section lengths (L) and exponential decay exponents
(ν) based on the isolines shown. For example, for a profile section of
30 m, and an exponential decay exponent of 0.2 m-1, the normalized
error is 0.32 and aoptimal is 9.63 m (see intersect of the two
isolines in Fig. 8c). The normalized error in Fig. 8c is not squared,
highlighting the sensitivity of the measurement error to ν, which
represents the degree of spatial correlation of the profile in this case
(e.g., lower values indicate stronger spatial memory/correlation, hence lower
measurement errors).
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 a being varied between 0 and L/ 2
(Fig. 9). For each value of a, the normalized squared error is estimated
based on the means obtained using the three snow depth samples for each
section. All squared differences are then averaged to obtain the values
presented in the figure. Sampled and modeled errors follow the same trend
across all a values and for the different L values in Fig. 9. The minimum
error is also reproduced by the model proving the applicability of the model
for estimating the optimal separation between measurements. The model does
perform better in the forested environment of FS vs. RW, particularly for
lower a values. This can be justified as the exponential covariance model
displays a better fit in FS over RW, particularly over the lower range of lag
values. Also, note that both the modeled and sampled normalized squared
errors are lower for the snow depth profiles at RW because of the longer
spatial memory of the snow depth distribution in this environment (higher
spatial correlations) when compared to that in FS.
Case 3: N measurements along a profile section
As stated above, the measurement error can be reduced by increasing the
number of measurements taken over a given section of length L. Let us focus
on the case of stratified sampling where N regularly spaced measurements are
taken over the interval (Fig. 5c), and to quantify this reduction we can
use Eq. (5) and the exponential covariance model. Equation (5) can then be
reduced to the following:
σZ‾2N,L,νσp2=1N+2N2∑k=1N-1kexp-νL-kLN-4Lν1-1N∑k=1Nexp-νLNN-k+12-2L2ν21-Lν-exp-νL.
The normalized squared error
(σZ‾2σp2) obtained with Eq. (10) for
profiles sections of lengths between 10 and 80 shows a steep decrease with
N (Fig. 10), with a steeper decrease for higher exponential decay
exponents. For the longer profile sections (e.g., 80, Fig. 10d), small
reductions in the squared error are achieved beyond only a few measurements
(e.g., N=16). Equation (10) and the results in Fig. 10 can be used to
determine the number of measurements necessary to achieve a desired accuracy
level. One could, for example, design a survey to sample a snow depth profile
with a mean value every 10 m. The number of measurements required to achieve
a desired level of accuracy can be obtained from Fig. 10a, based on previous
knowledge of the sample estimate of the exponential decay exponent. This can
be achieved thanks to the intra-annual and inter-annual persistence of the
spatial patterns, and hence, the spatial statistical properties of snow depth
fields in mountain environments, as shown in previous studies using both
manual surveys and LIDAR measurements (e.g., Deems et al., 2008; Sturm and
Wagner, 2010; Schirmer et al., 2011; Melvold and Skaugen, 2013; Helfrich et
al., 2014). A detailed spatial survey (e.g., dense manual measurements or
TLS), sampling different portions of an area can be used to determine the
covariance/correlation characteristics of the snow depth distribution, with
which the model for the error can be applied. An a priori estimate of the
exponential decay exponent may also be possible and will be tested in future
applications of the framework, given the relative insensitivity of the error
with respect to ν.
Theoretical and sampled normalized squared error
(σZ‾2σp2) for the N-point pattern
along a profile section in Fig. 5c, and for profile section lengths (L)
between 10 and 80 m in FS and RW. The solid point markers are the
theoretical error from (10) using exponential decay exponents derived from
the functions fitted to the sampled correlograms of the two surfaces in the
x and y directions, while the circle markers with the dotted lines
correspond to the sampled error for profiles in FS (a–d) and RW
(e–h).
Sample survey designs with (a) a five-point pattern centered
in the area, and (b) a regularly spaced pattern. For the five-point
pattern, a can vary between 0 and L/ 2, while for the N×N
points pattern, the separation between the measurements is determined by the
number of points.
(a) Theoretical normalized squared error
(σZ‾2σp2) for the two-dimensional
case with a single measurement in the middle of a square area with side
dimension L. (b) Theoretical and sampled normalized squared error
for the same two-dimensional survey applied to the snow depth field in FS.
The dashed line is the theoretical error derived for an exponential decay
exponent of 0.17 derived from the sampled correlation function of snow depth
in FS, while the solid line is the sampled normalized squared error for the
snow cover in FS.
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 N measurements distributed
along profile sections of length L. As the snow depth fields are
gridded at ∼ 1 m resolution, the location of the measurements is
approximated to the closest coordinate in the profile section following the
pattern in Fig. 5c. Once again, sampled and modeled errors follow closely the
same trend for the different L values in both FS and RW (Fig. 11). The
error decreases with N, with a rapid decay at the lower N values,
illustrating that the error can be drastically reduced by simply increasing
the number of measurements by a small amount. The normalized squared error
across all N values is lower for RW than for FS, consistent with the higher
spatial correlations observed in the snow depth fields of RW vs. FS. Once
again, there are some differences between the sampled and modeled normalized
squared error in RW for the shorter profile lengths and for small N values:
a consequence of the poorer fit of the exponential model for the shorter lag
range in RW. However, the model is still able to reproduce the error in both
fields, and the applicability of the model is illustrated even when the fit
of the correlation model can be improved.
Two-dimensional process
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 x-y plane for
three cases in a square area: (a) one single measurement in the center of the
area, (b) five measurements radiating out from the center (Fig. 12a), and (c)
N by N measurements regularly spaced in the x and y directions
(Fig. 12b).
Theoretical normalized squared error (σZ‾2σp2) as a function of the distance a from the center of
the area for square areas of side dimensions (L) between 10 and 80. Each
curve corresponds to an exponential decay (ν) between 0.1 and 5.
Theoretical and sampled normalized squared error
(σZ‾2σp2) for the 5-point pattern
in Fig. 12a over square areas of side dimensions (L) between 10.7 and
79.1 m. The separation distance (a) is varied from the center
outwards. The solid line is the theoretical error derived for an exponential
decay exponent of 0.17 derived from the sampled correlation function of snow
depth in FS, while the solid red point markers are the sampled normalized
squared error for the snow cover in FS.
Theoretical normalized squared error (σZ‾2σp2) for the N by N
point pattern in Fig. 12b, and for areas of side dimension (L)
between 10 and 80. The exponential exponent is
varied between 0.1 and 5.
Theoretical and sampled normalized squared error
(σZ‾2σp2) for the N by N point
pattern in Fig. 12b, and over square areas of side dimensions (L) between
10.7 and 79.1 m. The solid black point markers are the theoretical error for
an exponential decay exponent of 0.17 derived from the sampled correlogram of
snow depth in FS. The dotted red lines with circle markers are the sampled
normalized squared error for the snow cover in FS.
For the isotropic case, the covariance/correlation function is only dependent
on the magnitude of the lag vector,
hi,j=xi-xj,
and, consequently, the error is represented by,
σZ‾2A=σp21N+2N2∑i=1N-1∑j=i+1NCORRhi,j-2NA∑i=1N∫ACORRhi,jdxj+1A2∫A∫ACORRhi,jdxidxj.
The exponential correlation function for the isotropic case takes the
following form:
CORRh,ν=exp-νh,
where h is the magnitude of the lag vector. Replacing the correlation function in the expression
for σZ‾2, we obtain,
σZ‾2=σp21N+2N2∑i=1N-1∑j=i+1Nexp-νxi-xj-2NA∑i=1N∫Aexp-νxi-xjdxj+1A2∫A∫Aexp-νxi-xjdxjdxi.
For the case of a rectangular area of side dimension Lx and Ly in
the corresponding x and y directions, the equation becomes,
σZ‾2=σp21N+2N2∑i=1N-1∑j=i+1Nexp-νxi-xj2+yi-yj212-2NA∑i=1N∫0Ly∫0Lxexp-νxi-x2+yi-y212dxdy+1A2∫0Ly∫0Lx∫0Ly∫0Lxexp-νx′-x2+y′-y212dxdydx′dy′.
The limits of the integrals can be changed depending on the desired location
of the origin. In this case, the origin is located at the lower-left corner.
As discussed earlier, the first term is only a function of N, such that the
base error is the variance of the point process divided by the number of
points. The second term is a function of N, the location of the points, and
the decay rate ν. The third term is a function of N, A, the location
of the points, and the decay rate ν. The fourth term is a function of
A and ν, but is independent of the location of the points and N
(i.e., independent of the survey design, and only a function of the
correlation structure of the continuous process).
Case 1: single measurement in the center of the area
In this case, we focus on a single measurement in the middle of a square area
of side dimension L. Numerical solution of Eq. (15) shows that the
normalized squared error increases rapidly with L, with a steeper increase
for higher exponential decay exponents (Fig. 13a), which approach a
normalized squared error of 1 for L values less than 10 (e.g.,
1 ≤ν≤ 5). The theoretical results in Fig. 13a can be used
to determine the discrepancy between a single measurement in the middle of an
area and the areal mean for a second order stationary and anisotropic process
with an exponential covariance/correlation function. Comparison of the
modeled and sampled normalized square errors for the FS snow depth field
indicate very good agreement between modeled and sample errors (Fig. 13b).
The sample error is estimated following the same procedure explained for the
one-dimensional cases, although in the two-dimensional space. Both sampled
and modeled errors show the same behavior across L values between 1 and
100 m, although the scatter in the sampled error increases for larger L
values. This can be explained by the smaller number of samples to estimate
the mean normalized squared error and the fact that the correlation structure
decays rapidly and a single sample becomes less correlated to the surrounding
area for these larger areas. The model introduced here can then be used to
assess the representativeness of a single measurement over an area
objectively and accurately, and it can be extended for other
covariance/correlation functions as needed.
Case 2: five measurements radiating out from the center of the area
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 a
from the center leads to a similar optimization problem as illustrated in
case 2 of the one-dimensional examples (Sect. 4.2). In the two-dimensional
case, Eq. (15) does not have an explicit solution for a, and numerical
implementation is required. The equation can be solved by simply replacing
the point coordinates and the correlation function parameters. Following this
approach, the normalized squared error can be obtained for areas of varying
sizes (Fig. 14). Similar to the one-dimensional example (case 2, Sect. 4.2),
σZ‾2σp2 decreases with a, reaching
a minimum at an intermediate distance from the middle point outwards. The
decay in σZ‾2σp2 is more rapid for
the least correlated processes (i.e., higher decay exponents) reaching a
value close to the base normalized square error that is a function of the
number of points (i.e., 1/N=1/5 in this case). An extended analysis of
the effect of each of the terms in the equation is included in the
Supplement. The error, as shown in Fig. 14, is minimized as a consequence of
two balancing terms that lead to this intermediate solution. The optimal
solution is a balance between reducing the correlation between the individual
measurements (e.g., increasing the separation between the location of the
measurements) but increasing the correlation between the measurements and the
surrounding area (e.g., locating the measurements closer to the middle of the
area). These two competing effects lead to an optimization problem based on
the location of the point measurements. For the least correlated processes,
the error resembles the behavior of an uncorrelated field once the
measurements become effectively decorrelated (e.g., a > 1 in
Fig. 14b for ν=5). Figure 14 exemplifies how Eq. (14) can be used to
determine the optimal measurement location for areas of different sizes, and
to determine the associated error with configurations other than the optimal.
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 (L) between 10
and 79 m, with a being varied between 0 and L/ 2 (Fig. 15). For
each value of a, the normalized squared error is estimated based on the
means obtained using the five snow depth samples for each section. All
squared differences are then averaged to obtain the values presented in the
figure. Once again, the sampled and modeled errors follow the same trend
across all a values and for the different L values. The minimum error and
aoptimal are also reproduced closely by the model, and as the
area size increases, the sampled and modeled error approach the error for an
uncorrelated field at larger separations (i.e., 0.2). These results
illustrate that the performance of the model in the two-dimensional space is
remarkable, similar to what was observed in the one-dimensional case.
Case 3: N by N measurements regularly spaced in the x and y directions
Similarly to the one-dimensional case, the two-dimensional case of N by N
regularly spaced measurements (Fig. 12b) leads to a decreasing normalized
squared error with N (Fig. 16). There is a sharp decrease in the error by
just increasing the number of measurements in the lower range of N. The
analysis illustrates that stratified sampling, as shown here, is an excellent
approach for minimizing the error. For a 10 by 10 area for example,
increasing N to 4 (N2=16) reduces the normalized squared error to
less than 0.05. It is also worth noting here that for this two-dimensional
case, the error is less sensitive to the value of the exponential decay
exponent (ν) for the higher N values as the mean is accurately captured
regardless of the correlation of the field. Beyond a certain number of
measurements regularly distributed in the area, the measurements gather
enough information such that there are only very minor improvements with the
addition of new measurements, regardless of the exponent value. Figure 16
serves as an example of how the methodology can be used for objective
selection of the number of measurements necessary to achieve a desired
accuracy level using prior knowledge of the spatial covariance function.
The performance of the model is tested again for a square area with side
length (L) between 10 and 79 m using the snow depth field in FS, and for
an increasing number of rows/columns of measurements leading to a total
number of measurements of N2 (Fig. 17). The results illustrate again the
accurate performance of the theoretical model, with sampled and model errors
following closely the same squared errors. Both sampled and modeled errors
increase as the size of the area increases, as expected. These results
complete the model performance tests for the two-dimensional isotropic case.
Summary and conclusions
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-1). Hence, the biases in the estimated
error and the survey design remain small.
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.
The Supplement related to this article is available online at doi:10.5194/tc-9-1249-2015-supplement.
Acknowledgements
Data for this article were obtained from NASA's Cold Land Processes
experiment (CLPX), available at
http://nsidc.org/data/docs/daac/nsidc0157_clpx_lidar.
Edited by: R. Brown
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