Abstract

Triple collocation (TC) is a popular technique for determining the data quality of three products that estimate the same geophysical variable using mutually independent methods. When TC is applied to a triplet of one point-scale in situ and two coarse-scale datasets that have the similar spatial resolution, the TC-derived performance metric for the point-scale dataset can be used to assess its spatial representativeness. In this study, the spatial representativeness of in situ snow depth measurements from the meteorological stations in northeast China was assessed using an unbiased correlation metric ρt,X12 estimated with TC. Stations are considered representative if ρt,X120.5; that is, in situ measurements explain no less than 50% of the variations in the “ground truth” of the snow depth averaged at the coarse scale (0.25°). The results confirmed that TC can be used to reliably exploit existing sparse snow depth networks. The main findings are as follows. 1) Among all the 98 stations in the study region, 86 stations have valid ρt,X12 values, of which 57 stations are representative for the entire snow season (October–December, January–April). 2) Seasonal variations in ρt,X12 are large: 63 stations are representative during the snow accumulation period (December–February), whereas only 25 stations are representative during the snow ablation period (October–November, March–April). 3) The ρt,X12 is positively correlated with mean snow depth, which largely determines the global decreasing trend in ρt,X12 from north to south. After removing this trend, residuals in ρt,X12 can be explained by heterogeneity features concerning elevation and conditional probability of snow presence near the stations.

1. Introduction

Snow cover is a key component in the global water cycle and directly impacts the Earth’s energy balance and climate dynamics (Cohen 1994). Remote sensing is the most efficient way to regularly measure snow cover and depth on global and regional scales (Armstrong and Brodzik 2002; Foster et al. 2011). The Scanning Multichannel Microwave Radiometer (SMMR), Special Sensor Microwave Imager (SSM/I), and Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E) have been routinely used to retrieve snow depth and snow water equivalent (SWE) since the 1970s (Che et al. 2016). Satellite snow products are increasingly used for modeling and monitoring in various fields such as hydrology (Berezowski et al. 2015), climate research (Bormann et al. 2012), glaciology (Stroeve et al. 2005), and numerical weather prediction (Brasnett 1999).

Validation of microwave snow depth products with ground truth data is key to improving inversion algorithms. However, owing to the high spatial variability of snow depth, the validation process can be quite challenging. An in situ snow depth measurement can only be representative over a very small spatial scale (Clark et al. 2011; Trujillo et al. 2007), whereas satellite-derived snow depth represents the mean value of a microwave footprint with a size of 25 km × 25 km or larger (Vander Jagt et al. 2013). If satellite-derived snow depth is directly compared with point measurements, the obtained errors are likely dominated by representativeness errors due to the variability of the snow depth field on subgrid scales as opposed to snow depth inversion model errors (Brasnett 1999; Tustison et al. 2001; Chang et al. 2005; Liston 1999, 2004).

To evaluate the spatial representativeness of the point-scale snow depth, most studies attempted to obtain the difference between the point measurement and the area average, and argued that a point measurement is representative if its value deviates less than 10% from the area average (Neumann et al. 2006; Molotch and Bales 2005, 2006; Rice and Bales 2010; Meromy et al. 2013; Grünewald and Lehning 2013; Grünewald et al. 2013). This method requires a dense sampling network, based on which upscaling to the coarse scale can be achieved by using spatial modeling methods. Although this method has been successfully applied at the watershed scale, it is of limited use for estimating the spatial representativeness of sparse meteorological stations that provide only one in situ observation for a satellite footprint. Since it is logistically prohibitive to carry out extensive snow surveys or set up dense networks over hundreds of operational meteorological stations, the limitations of point measurements at these stations in adequately representing snow depth for the surrounding area have been questioned but not explored in detail (Blöschl 1999; Neumann et al. 2006; Derksen et al. 2003; Chang et al. 2005; Grünewald and Lehning 2013; Meromy et al. 2013).

A promising way to evaluate the representativeness of a point-scale dataset is the triple collocation (TC) technique, which estimates the data quality of three mutually independent datasets without treating any dataset as perfectly observed “truth” (Stoffelen 1998). TC has now become a standard procedure in comprehensive satellite validation processes, especially in soil moisture research (Scipal et al. 2008; Dorigo et al. 2010, 2015; Chen et al. 2017; Gruber et al. 2016a,b, 2017). When TC is applied to a triplet containing one point-scale and two coarse-scale datasets that have the similar spatial resolution, performance metrics associated with the point-scale dataset indicate its spatial representativeness, assuming that the instrumental random error can be neglected (Gruber et al. 2013, 2016a; Chen et al. 2017). The most prominent feature of using TC to assess the spatial representativeness is that it is data-driven and does not need field surveys or dense sampling networks. The credibility of using the TC-derived correlation metric or random error variances in representing the closeness of the point-scale data to the coarse-scale ground truth has been confirmed at densely instrumented validation sites by Miralles et al. (2010) and Chen et al. (2017).

Validations of microwave snow depth and soil moisture share a high degree of similarity. The success of TC applications in soil moisture studies has prompted us to adopt this technique for snow depth studies. To the best of our knowledge, this study is the first attempt to apply TC to evaluate the spatial representativeness of point-scale snow depth measurements from meteorological stations. Besides assessing the spatial representativeness, we investigated the answers to two questions, which have not been fully explored in previous TC studies.

  1. How representativeness varies with season? Representativeness of a station is not a constant (Bohnenstengel et al. 2011); it can change considerably from the snow accumulation period to the snow ablation period owing to the variations in the spatial heterogeneity of snow depth (Molotch and Bales 2005; Winstral and Marks 2014). Some researchers argued that the observations need to be selected with the specific objective of representing either the accumulation or the ablation season process (Molotch and Bales 2005). Understanding the seasonal variations in representativeness can help us choose the most representative stations according to the time of the snow depth product and hence make full use of the existing networks.

  2. What factors in the vicinity of stations play a dominant role in determining representativeness? Understanding the dominant factors has two advantages. First, it provides an indirect approach to validate representativeness assessments. Strong heterogeneity usually results in low representativeness; thus, the representativeness assessments are generally reasonable if they are strongly correlated with heterogeneity features. Second, dominant factors can be used to predict representativeness, which is potentially useful in choosing the representative locations for new sites.

The remainder of this paper is organized as follows. Section 2 introduces the TC technique and how TC is used to evaluate station representativeness. Section 3 describes the study region, datasets, TC implementation process, and the method of extracting heterogeneity features. Results and discussion are presented in sections 4 and 5, respectively.

2. Introduction of the TC technique

a. TC approaches

The most commonly used error model for TC analysis is the following model (Gruber et al. 2016a):

 
Xi=αi+βit+εi,
(1)

where Xi (i ∈ {1, 2, 3}) are three collocated and independent datasets of the same geophysical variable linearly related to the true underlying value t with additive zero-mean random errors εi. The terms Xi, t, εi are all random variables; αi and βi are the intercepts and slopes, respectively, representing systematic additive and multiplicative biases of dataset Xi with respect to the true signal t.

There are four main underlying assumptions for the error model of TC (Zwieback et al. 2012; Gruber et al. 2016a,b): (i) linearity between the true signal and the observations; (ii) signal and error stationarity; (iii) error orthogonality: independence between the errors and the true signal, that is, Cov(t, εi) = 0; and (iv) zero error cross correlation: independence between the errors of Xi and Xj, that is, Cov(εi, εj) = 0, for ij.

Following McColl et al. (2014), the covariances between the different datasets are calculated as follows:

 
Cov(Xi,Xj)=E(XiXj)E(Xi)E(Xj)
 
=βiβjσt2+βiCov(t,εj)+βjCov(t,εi)+Cov(εi,εj),
(2)

where σt2=var(t). Using the assumptions of error orthogonality and zero error cross correlation, the equation is reduced to (3):

 
QijCov(Xi,Xj)={βiβjσt2,forijβiβjσt2+σεi2,fori=j,
(3)

where σεi2=var(εi), representing the variance of random error in dataset Xi. Since there are six equations (Q11, Q12, Q13, Q22, Q23, Q33) but seven unknowns (β1,β2,β3,σε1,σε2,σε3,σt), the system is underdetermined. It can be solved by defining a new variable θi = βiσt. Then, the equations can be rewritten as in (4):

 
Qij={θiθj,forijθi2+σεi2,fori=j.
(4)

Now there are six equations and six unknowns, and the system can be solved. Variable θi2, which provides estimates of the sensitivity of datasets Xi to ground truth changes (Gruber et al. 2016a), can be written as follows:

 
{θ12=β12σt2=Q12Q13Q23θ22=β22σt2=Q12Q23Q13θ32=β32σt2=Q13Q23Q12.
(5)

The estimation equation for error variances can be written as follows:

 
{σε12=Q11Q12Q13Q23σε22=Q22Q12Q23Q13σε32=Q33Q13Q23Q12.
(6)

Since σεi2 is the absolute random error variance affected by the dynamic range of the data, Draper et al. (2013) proposed relative error variance (fMSEi), which is calculated by normalizing the error variances with the corresponding dataset variances:

 
fMSEi=σεi2Qii.
(7)

Combining (7), (4), and (3), fMSEi can be written as follows:

 
fMSEi=σεi2θi2+σεi2=σεi2βi2σt2+σεi2,
(8)

where βi2σt2 represents the signal and σεi2 represents the noise (Gruber et al. 2016a; McColl et al. 2014); thus, fMSEi is not only a measure of relative error, but also a measure of signal-to-noise ratio (SNR). Furthermore, fMSEi is related to the linear correlation coefficient of Xi with the underlying true signal t (denoted by ρt,Xi). According to McColl et al. (2014), the relationship between ρt,Xi and the ordinary least squares (OLS) slope βi can be written as in (9):

 
ρt,Xi=βiσtQii.
(9)

Combining (7), (8), and (9), we obtain (10):

 
ρt,Xi2=βi2σt2Qii=Qiiσεi2Qii=1σεi2Qii=1fMSEi.
(10)

Equation (10) indicates that ρt,Xi2 and fMSEi are complementary. When fMSEi is 0.5, the coefficient of determination ρt,Xi2 for the linear error model is 0.5, and the correlation coefficient of Xi with t is 0.5 (≈0.71).

b. Representativeness analysis of point-scale data with TC

While TC is a powerful tool for estimating random errors and removing systematic differences between the signal variance component of observations, it is affected by representativeness errors (Yilmaz and Crow 2014). TC assumes that the three datasets represent the same signal, which is very unlikely given that the three datasets can have very different spatial measurement support (McColl et al. 2014; Gruber et al. 2016a). When a triplet consists of one point-scale in situ dataset and two coarse-scale datasets that have the similar spatial resolution, the high-resolution signal in the point-scale dataset cannot be detectable for coarse-scale datasets and therefore be regarded as error (Gruber et al. 2016a). In other words, TC will penalize the point-scale dataset for its limited representativeness at the coarse scale, whereas no representativeness error is assigned to the error estimates of the coarse-scale datasets (Gruber et al. 2016a; Yilmaz and Crow 2014). This characteristic of TC opens an opportunity for evaluating the spatial representativeness of point-scale data efficiently, which has been proved feasible by recent studies on soil moisture validation networks (Gruber et al. 2013; Chen et al. 2017; Miralles et al. 2010).

Although most previous works used random error variances σεi2 to evaluate representativeness (Gruber et al. 2013), which is an absolute error metric and sensitive to data range and variability, more recent studies recommend using the correlation metric ρt,Xi2 (Chen et al. 2017; Gruber et al. 2017). As a unitless measure, ρt,Xi2 can be fairly compared across space and time. Moreover, ρt,Xi2 is interpretable, making it easier to choose a threshold value. In this study, we assume that the value of ρt,Xi2 for a representative station should be equal to or larger than 0.5. This criterion has been used in other studies (Chen et a1. 2017) and can be interpreted as follows: a station is representative if its measurements can explain no less than 50% of the variance in the ground truth of the area average, or from the SNR perspective, the signal level contained in the in situ measurements is no less than the noise level. Since ρt,Xi represents the correlation between Xi and t, the criterion can also be interpreted as follows: a station is representative if the correlation coefficient between in situ measurements and the ground truth of the area average is equal to or larger than 0.5 (≈0.71).

Note that TC assumes stationarity of random error variance, thus, the representativeness estimates represent the averaged condition of the entire period (Loew and Schlenz 2011). Considering that error is often time variant, the result for the entire period can be inaccurate for a particular subset of the time period. If representativeness of a particular time period is of interest, data inputs should be confined within the corresponding time period.

3. Study region, data, and methods

a. Study region

The study region is northeast China (38°–56°N, 120°–135°E), which is one of the three primary snow-covered regions in China (Li et al. 2008) and is characterized by taiga snow (Sturm et al. 1995). The region with a total area of 1.26 × 106 km2 encompasses the provinces of Heilongjiang, Jilin, Liaoning, and the eastern part of Inner Mongolia. The regional climate includes warm temperate, medium temperate, and subarctic zones. Annual precipitation is approximately 430–680 mm, of which 5%–10% is snowfall (He et al. 2013; Zhang et al. 2016). There are three mountain ranges (Daxinganling, Xiaoxinganling, and Changbaishan Mountains) and two large plains (Songnen and Sanjiang) in the region. Primary land cover types are forest (40%), farmland (30%), and grassland (20%). Figure 1 shows the spatial pattern of tree cover (%) and elevation (m) in the study region.

Fig. 1.

Tree cover (provided by MOD44B), elevation (provided by GTOPO30), and distribution of meteorological stations in the study region.

Fig. 1.

Tree cover (provided by MOD44B), elevation (provided by GTOPO30), and distribution of meteorological stations in the study region.

b. Data

1) Meteorological

There are 98 operational meteorological stations located within the study region. Figure 1 shows that in general, these stations are evenly distributed, although more stations are located on plains than in mountains. Snow depth and snow density are measured manually using a snow tube with a cross-sectional area of 100 cm2 at the fixed snow experiment field at each station. Measurement frequency is one every day for snow depth and every five days for snow density, and each record is obtained from the average of three measurements (Dai and Che 2011).

2) Satellite snow depth from GlobSnow product

The daily GlobSnow snow water equivalent (SWE) product (version 2.0) from 2000 to 2013 was downloaded at http://www.globsnow.info/. The product was retrieved by using a data-assimilation-based approach which combines spaceborne passive radiometer data (SMMR, SSM/I, and SSMIS) with data from ground-based synoptic weather stations (Pulliainen 2006; Takala et al. 2011). According to Takala et al. (2011), constant snow density (0.24 g cm−3) can be used to convert SWE to snow depth. GlobSnow SWE information is provided for terrestrial nonmountainous regions of the Northern Hemisphere, excluding glaciers and Greenland. The product has been validated using independent SWE reference data from Russia, the former Soviet Union, Finland, and Canada, and the results indicate overall strong retrieval performance with root-mean-squared error (RMSE) below 40 mm for cases when SWE is below 150 mm. Retrieval uncertainty increases when SWE is above this threshold (Takala et al. 2011). The GlobSnow results over China are based on passive microwave information because no Chinese stations are available for data assimilation, which ensures the independence between the GlobSnow data and meteorological observations in this study.

3) Snow depth analysis data from CMC

Canadian Meteorological Centre (CMC) daily snow depth analysis data (version 1) from 2000 to 2013 were downloaded at https://nsidc.org/data/NSIDC-0447 (Brown and Brasnett 2010). The snow depth production includes two steps (Brasnett 1999). The first step is to calculate the background field of snow depth based on the forecasts of precipitation and the analysis of screen-level temperature. Specifically, precipitation is considered snow if the analyzed screen-level temperature is zero or less. The snow melting algorithm is applied when the analyzed temperature is greater than zero. When there is neither snowfall nor melting, it is assumed that the mass of the snowpack is conserved. The next step is to incorporate the snow depth observations wherever they are available by performing statistical interpolation every 6 h on a 1/3° grid. Compared to the microwave and optical snow cover results, the CMC snow depth data show more skill than climatology (Brasnett 1999). Since snow depth observations in China are not available in the CMC snow depth analysis system, the snow depth results for China are model derived, which are the initial guess field of the simplified assumptions regarding snowfall, melting, and aging.

4) Satellite snow cover

Representativeness of in situ snow depth measurements is largely determined by the spatial heterogeneity of snow depth in the surrounding area. Due to the unavailability of subgrid scale snow depth distribution near a station, optical satellite snow cover data with a high spatial resolution are often used as a substitute (Molotch and Bales 2005, 2006). In this study, the “binary” (i.e., snow or not snow) snow cover map provided by the MODIS daily snow cover product (MOD10A1, from 2000 to 2013) with a spatial resolution of 500 m was used. The MODIS snow mapping algorithm is based on a normalized difference snow index (NDSI) approach. Normalized difference vegetation index (NDVI) criteria are also applied to enable snow detection in areas with dense vegetation (Dong and Menzel 2016). Many studies have confirmed high accuracy and consistency of MODIS snow cover images by comparing them with other high spatial resolution satellite-derived snow products or ground-based point snow depth measurements (Maurer et al. 2003; Hall and Riggs 2007).

5) Other ancillary data

Forest coverage is a potentially important factor that influences station representativeness and satellite snow depth product quality (Vander Jagt et al. 2013). Snow depth tends to be underestimated because forest attenuation and its upwelling radiation can decrease the passive microwave brightness temperature signal from snow cover underneath (Chang 1996). Meanwhile, operational meteorological stations measure snow depth in clearings where snow dynamics differ from those in the forest (Raleigh et al. 2013). Therefore, stations located near dense forests tend to have problems representing the surrounding environment. In this study, the MODIS percent tree cover product (MOD44B, with a spatial resolution of 500 m) was used to extract the mean forest coverage for the surrounding area of each station (Hansen et al. 2002).

Elevation is another important factor that needs to be considered (Jost et al. 2007; Grünewald et al. 2013). Complex topography within a microwave footprint makes it difficult to extract the snow signature. Different viewing angles of mountains by the ascending and descending orbits further complicate the problem (Chang and Rango 2000). Grids with high standard deviations of elevation are often excluded from interpolation or assimilation analysis (Takala et al. 2011). In this study, the global DEM data (GTOPO30, with a spatial resolution of approximately 1 km) were used to extract standard deviations of elevation in the surrounding area of each station.

c. Methods

1) TC analysis

Daily snow depth measurements at each meteorological station in the snow season (October–December, January–April) from 2000 to 2013 were used. The data were passed through temporal consistency and range checks. Each record has a quality control flag and only high-quality data were used in the analysis. According to the longitude and latitude of a station, we located the microwave grid (25 km × 25 km) whose center is the closest to the station and extracted the snow depth value from both GlobSnow and CMC products. Samples with all three collocated data exhibiting zero values were removed from the analysis.

Previous research demonstrated that raw values with varying dynamic ranges and climatology tend to violate TC assumptions (Gruber et al. 2016a; Dorigo et al. 2010; Chen et al. 2017). Therefore, we calculated the snow depth anomaly by removing climatological signals from the raw time series of each product. The climatology was obtained by first averaging multiannual data for each day of year (DOY) and then smoothed by a moving window of 7 days.

Using the triplet data consisting of in situ observed snow depth anomaly (represented by X1), satellite-estimated snow depth anomaly (GlobSnow product represented by X2), and snowpack model-estimated snow depth anomaly (CMC product represented by X3), we calculated the correlation coefficients between any two of the three anomaly time series for each station. Those stations that failed to show significant positive correlation (p value > 0.05) or had less than 100 data points in the collocated triple time series were considered unqualified and masked from the TC analysis. To analyze the seasonal variation in representativeness, we divided the collocated time series into two periods: 1) stable or accumulation period (December–February) when temperature is very low and snow often accumulates steadily and 2) unstable or ablation period (October, November, March, April) when temperature is higher and fast snowmelt may result in patchy and heterogeneous snow distribution. For each period, unqualified stations (no significant positive intercorrelation or sample size is less than 100) were excluded from the TC analysis.

2) Extraction of explanatory variables

All explanatory variables were extracted from a window region centered around each station. The window region has a spatial extent of 51 pixels × 51 pixels (a pixel is 500 m), approximately the same size of a microwave grid. Mean forest coverage data (MFC) were extracted from the MOD44B product. The terrain index (TI), which is defined as the standard deviation of elevation within the window region, was extracted from the GTOPO30 product. With regard to the snow cover climatology features, eight variables were designed and extracted from the MOD10A1 “binary” data over 2000–13.

Regional mean and standard deviation of snow cover frequency in the surrounding area of a station is calculated as follows:

 
{mFsnow=mean(Fsnowi,j)vFsnow=std(Fsnowi,j),
(11)

where Fsnowi,j is the accumulated snow cover frequency for pixel (i, j) in the window region over time span T (i.e., 14 snow seasons over 2000–13).

 
Fsnowi,j=n=1Tsnowi,j,nT,
(12)

where snowi,j,n = 1, if pixel (i, j) is covered by snow at time n; otherwise, snowi,j,n = 0.

The presence of snow-free land in the cold season means no snow or snow has melted, which provides somewhat different information from snow cover frequency. Using the similar approach, regional mean and standard deviation of nonsnow cover frequency in the surrounding area of a station is calculated as follows:

 
{mFland=mean(Flandi,j)vFland=std(Flandi,j),
(13)

where Flandi,j is the accumulated non–snow cover frequency for pixel (i, j) in a window region over time span T:

 
Flandi,j=n=1Tlandi,j,nT,
(14)

where landi,j,n = 1, if pixel (i, j) is not covered by snow at time n; otherwise landi,j,n = 0.

If a station is representative, then when it is covered by snow, its surrounding area should also be covered by snow with high probability. This consistency information cannot be extracted from mFsnow or vFsnow. Therefore, we calculated the regional mean and standard deviation of conditional probability of snow presence as follows:

 
{mPsnow=mean(Psnowi,j)vPsnow=std(Psnowi,j),
(15)

where Psnowi,j is the probability of snow presence at pixel (i, j) when the station (center pixel) is covered by snow (Dong and Menzel 2016):

 
Psnowi,j=n=1TSi,j,n/n=1TScenter,n,
(16)

where n=1TScenter,n is the number of days the station is snow covered; n=1TSi,j,n is the number of days both pixel (i, j) and the station are snow covered.

Using the similar formula, regional mean and standard deviation of conditional probability of nonsnow presence are calculated as follows:

 
{mPland=mean(Plandi,j)vPland=std(Plandi,j),
(17)

where Plandi,j is the probability of nonsnow presence at pixel (i, j) when the station (center pixel) is not covered by snow:

 
Plandi,j=n=1TLi,j,n/n=1TLcenter,n,
(18)

where n=1TLcenter,n is the number of days the station is not covered by snow; n=1TLi,j,n is the number of days neither the pixel (i, j) nor the station is covered by snow.

Note that the snow climatology features were extracted from the cloud-free data; thus, they are biased to the clear-sky condition. Considering that snow depth has a time-integrative nature and will not change quickly with sky conditions, we assume that such a bias will not have a significant impact on climatological results.

4. Results

a. Representativeness assessments for the entire period

Among 86 stations that can output valid TC results for the entire period, 57 stations have ρt,X12 values larger than 0.50, indicating that 66% of the stations are spatially representative. Among the remaining 29 stations, 26 have moderate representativeness (0.25ρt,X12<0.5), and 3 have low representativeness (0<ρt,X12<0.25). Table 1 shows that the average value and standard deviation of ρt,X12 are 0.59 and 0.19, respectively. The spatial distribution of ρt,X12 is shown in Fig. 2a. Stations with low or no representativeness are generally located in the southern region, whereas the representative stations are in the northern region. Stations with moderate representativeness are mainly located in the southern region, but some of them are mixed with representative stations near the northern mountainous region.

Table 1.

Statistics of ρt,X12 for different periods. The number in the parentheses represents the number of stations with valid TC results.

Statistics of ρt,X12 for different periods. The number in the parentheses represents the number of stations with valid TC results.
Statistics of ρt,X12 for different periods. The number in the parentheses represents the number of stations with valid TC results.
Fig. 2.

Spatial distribution of ρt,X12 for the (a) entire period, (b) stable period, and (c) unstable period.

Fig. 2.

Spatial distribution of ρt,X12 for the (a) entire period, (b) stable period, and (c) unstable period.

b. Seasonal variations in representativeness

Statistical parameters of ρt,X12 are shown in Table 1, where ρt,X12 for the unstable period is noticeably lower than ρt,X12 for the stable period. If the same threshold (ρt,X120.5) is used, the number of representative stations for stable and unstable periods is 63 and 25, respectively. Therefore, more stations can be used for coarse-scale data validation during the stable period (Fig. 2b). By contrast, only a small number of stations can be reliably used during the unstable period, and these stations are mainly located in the northern cold region or in the middle region with flat terrain (Fig. 2c).

Owing to a lower sampling density and weaker positive correlations among the three snow depth products, the number of stations that can output valid representativeness results for the unstable period is only 62, while it is 90 for the stable period. For a fair comparison, the following analysis is based on 61 stations that output valid results for both periods. The scatterplot of ρt,X12 (Fig. 3a) shows that most points are under the 1:1 line, indicating that stations are generally less representative during the unstable period. Some points deviate from the 1:1 line dramatically, suggesting that they suffer large reductions in representativeness when the period shifts. Twenty-nine stations that are representative during the stable period become nonrepresentative during the unstable period. However, there are four stations showing slightly higher representativeness values during the unstable period. With regard to the standard deviation of random errors σε1, a less pronounced seasonal variation is observed (Fig. 3b). Most points are scattered around the 1:1 line. Several stations show higher values of σε1 during the stable period, most likely due to a larger magnitude of snow depth.

Fig. 3.

Comparison for (a) representativeness measure ρt,X12 and (b) random error standard deviations σε1 between stable and unstable period.

Fig. 3.

Comparison for (a) representativeness measure ρt,X12 and (b) random error standard deviations σε1 between stable and unstable period.

c. Explaining representativeness

Prior to the representativeness modeling, we analyzed the correlation coefficients among the explanatory variables (Table 2). There are several pairs of variables showing very strong correlations, such as mPsnow and vPsnow, mFland and mean_SD, mFsnow and mFland, vPsnow and TI, and vFsnow and MFC. Specifically, two variables related to the conditional probability of snow presence (mPsnow, vPsnow) are strongly related to two physiographic factors (TI and MFC). Three variables related to the mean frequency of snow or land occurrence (mFsnow, vFsnow, mFland) are strongly correlated with the mean snow depth of the stations (mean_SD). The results indicate that information overlap exists between snow climatology features and physiographic characteristics in the surrounding area of the stations.

Table 2 also shows the correlation coefficients of ρt,X12 with explanatory variables for both stable and unstable periods. Only mFsnow, mFland, and mean_SD are highly correlated with ρt,X12, regardless of the season analyzed. The correlation during the unstable period is noticeably stronger. Since mFsnow and mFland are strongly correlated with mean_SD, we infer that stations with deeper snow tend to have better representativeness. Since snow depth in the northern region is larger, the northern stations are generally more representative than the southern stations, and this is partly attributed to the definition of ρt,X12. Being a relative error metric, ρt,X12 tends to favor data with larger variations, which is the case for northern stations with deeper snow.

Table 2.

Correlation coefficients among variables. Strong correlations (|r| > 0.4) are shown in bold.

Correlation coefficients among variables. Strong correlations (|r| > 0.4) are shown in bold.
Correlation coefficients among variables. Strong correlations (|r| > 0.4) are shown in bold.

Besides the smooth latitudinal trend in ρt,X12, local variations in representativeness are also evident (Fig. 2). It is important to identify the factors that affect local variations because the task of choosing a representative location for a new site is often spatially confined within a region where climatic variations can be neglected. We tried all possible combinations of two variables (except mean_SD) to model the residuals in ρt,X12 that cannot be explained by mean_SD. The model incorporating mPsnow and TI as independent variables showed the highest accuracy. The linear regression model with three variables (mPsnow, TI, mean_SD) achieved satisfactory performance (r = 0.70) in estimating ρt,X12 for the unstable period, but the model accuracy was less satisfactory (r = 0.54) for the stable period due to the weaker correlations between ρt,X12 and explanatory variables.

5. Discussion

a. Definition of representativeness

Previous studies have argued that a point measurement is representative if its value deviates less than 10% from an area average. This requirement may be applicable when choosing representative locations for new sites, but for existing networks, it is too stringent. Finding stations that fulfill such a requirement can be very difficult or even impossible (Molotch and Bales 2005). Due to the stable physiographic features and similar meteorological forcing conditions from year to year, a station’s measurements can be consistently lower or higher than the area average of its surroundings (Liston 2004; Deems et al. 2008; Schirmer et al. 2011; Nitta et al. 2014), suggesting that a large fraction of the difference has a systematic nature. If this systematic difference can be corrected and the in situ measurements match the area average value after the correction, then the station should be considered representative. Therefore, we prefer the definition in Vanderlinden et al. (2012) that representative locations are the locations where measurements are either close to the average values or can be easily transformed to obtain the average values.

The largest obstacle to evaluating representativeness is the lack of area averages (i.e., ground truth at the footprint scale) against which a station can be compared and corrected. The TC technique overcomes this obstacle by including three collocated independent datasets [Xi (i ∈ {1, 2, 3})]. Without the knowledge of the ground truth t, TC can estimate the standard deviation of random errors σεi and coefficients of determination ρt,Xi2 of the linear regression model that predicts t with Xi. When X1 represents the point-scale dataset, σε1 and ρt,X12 represent the accuracy of predicting the ground truth using point-scale dataset. If X1 is of limited spatial representativeness for the coarse-scale, which is the spatial measurement support for both X2 and X3, then the prediction accuracy of X1 will be low because TC favors majority (signal shared by X2 and X3) and punishes minority (signal only from X1). Therefore, σεi and ρt,X12 can be used to assess the spatial representativeness of the point-scale dataset. Note that σεi is not the root-mean-square difference (RMSD) between X1 and t, but the RMSD between linear-corrected (or upscaled) X1 and t because the systematic difference is removed. Nonetheless, being a measure of absolute error, σε1 is sensitive to data variability and cannot be compared fairly across space and time; thus, we used ρt,X12 to assess representativeness. Note that ρt,X12 reflects the joint impacts of instrumental performance and spatial representativeness. Since snow depth observations of meteorological stations are made manually, we can assume that the instrumental error is negligible. However, for automatic sensors, the instrumental errors can be high due to miscalibration or bad deployment (Gruber et al. 2013); therefore, data quality of the automatic sensors should be checked carefully before using ρt,X12 to assess spatial representativeness.

b. Relating representativeness to spatially explicit variables

Spatial representativeness of in situ snow depth measurements is determined by the spatial variability of snow depth in the surrounding area. Previous studies that evaluated the representativeness spatially modeled the snow depth. This study omits spatial modeling, which is infeasible for sparse networks, and directly assesses the representativeness of meteorological stations. Similar to the spatial pattern of snow depth shaped by a range of different processes that occur across a hierarchy of spatial scales (Trujillo et al. 2007; Jost et al. 2007; Clark et al. 2011; Melvold and Skaugen 2013), representativeness distribution also reveals a smooth climate pattern with additional local variations. After removing the climate pattern, local variations in representativeness can be best explained by mPsnow (inconsistency in snow cover presence between a station and its surroundings) and TI (terrain heterogeneity in the surrounding area). This result agrees with the common knowledge that a point-scale dataset is less likely to be representative in heterogeneous regions; thus, it provides an indirect proof that the TC-derived representativeness assessment in this study is reasonable.

The relationship uncovered also highlights the necessity of designing variables more pertinent to the heterogeneity of snow depth in explaining representativeness. Specifically, long time series snow cover data should be explored fully to describe the snow cover pattern, which is indicative of the snow depth pattern to some extent. In future work, different sizes of window region can be selected to explore the scaling effect of snow cover heterogeneity around stations because this study only extracted explanatory variables from a window region with a fixed size. Furthermore, future work could use gap filling and smoothing techniques to improve the snow cover data quality so that snow climatology features are more accurate.

c. Understanding seasonal variations in representativeness

Our results confirm that representativeness of a station shows large seasonal variations. Since ρt,X12 is positively correlated with snow depth, almost all stations have better representativeness during the stable period when snow is deeper. Although most stations suffer reductions in ρt,X12 when the time changes from the stable to the unstable period, the magnitudes are different. To understand the factors that affect the magnitudes, we calculated the correlation coefficients between explanatory variables and the reduction in ρt,X12. We found that mean_SD has the largest impact on seasonal variations (r = −0.48). The smaller the annual mean snow depth is, the larger the seasonal variations are. Thus, when the time changes from the stable to the unstable period, many representative stations located in the southern region, where snow is usually shallower, become nonrepresentative (Figs. 2b,c). We also found that stations with low forest cover in the surrounding area tend to exhibit a large reduction in representativeness (r = −0.34), and terrain has a similar but weaker effect (r = −0.26). For regions where forest cover is low or terrain is flat, high wind speed or strong radiation often occurs during the snow ablation season, which can easily lead to patchy or heterogeneous snow distributions. By contrast, snow distribution can be quite homogeneous during the snow accumulation period for these regions. Therefore, when using stations in less forested regions or in flat regions for validation, their time-variant representativeness should be considered.

d. Comparison of two coarse-scale snow depth products

The TC-derived representativeness assessments are impacted by the quality of two coarse-scale snow depth products (GlobSnow, CMC) used in this study. If there is no significant positive correlation between the two products, TC analysis cannot be carried out. If the two products are both strongly correlated with the underlying ground truth, the representativeness results will be more credible. To better understand the coarse-scale data quality, we calculated pixel-wise Pearson correlation coefficients Ri,j between GlobSnow and CMC based on the snow depth anomaly time series. The correlation coefficient Ri,j depicts the consistency in temporal variations between the two datasets. Under the framework of TC, Ri,j can also be written as follows (Gruber et al. 2016a):

 
Ri,j=QijQiiQjj=βiσtQiiβjσtQjj=ρt,Xiρt,Xj.
(19)

Equation (19) shows that Ri,j is determined by the multiplication of the correlation coefficients of the two products (Xi, Xj) with the “ground truth” (t). High Ri,j values mean that both products are of high quality, while low Ri,j values indicate that at least one product is not of high quality. Figure 4 shows the Ri,j maps for three periods (stable, unstable, and entire period). During the stable period, Ri,j values are high for most areas, although there is a belt of low values in the southern area (near the coastal region). During the unstable period, Ri,j values are much lower in the northern and eastern mountainous areas, and the southern area still exhibits the lowest values across the region. For the entire time period, Ri,j shows the medium values, and its spatial pattern reflects the combined effect in both stable and unstable periods. These results indicate that the quality of the two coarse-scale snow depth products varies with space and time. For regions with lower Ri,j values, the representativeness results may have higher uncertainties because at least one of the coarse-scale snow depth products is not strongly correlated with the ground truth. Future studies should focus on these regions and attempt to decrease the uncertainties in the representativeness assessments by introducing new datasets or carrying out snow surveys.

Fig. 4.

Correlation coefficients between GlobSnow and CMC snow depth anomaly time series for the (a) stable period, (b) unstable period, and (c) entire period.

Fig. 4.

Correlation coefficients between GlobSnow and CMC snow depth anomaly time series for the (a) stable period, (b) unstable period, and (c) entire period.

6. Conclusions

This study demonstrates that the TC technique is capable of efficiently and effectively evaluating the spatial representativeness of point-scale snow depth measurements with respect to satellite gridscale data for hundreds of meteorological stations. By choosing the spatial representative stations (ρt,X120.5), validation work can be extended from a few core validation sites to widespread sparse networks. The correlation metric should be adopted to validate new coarse-scale snow depth data because large systematic biases may still exist for those representative stations. The seasonality in the TC-derived representativeness highlights the necessity of choosing the best stations according to the time period of the products under consideration. Representativeness assessments were indirectly proved to be reasonable by the close relationship between the two variables depicting heterogeneity features (mPsnow and TI) and the representativeness metric ρt,X12. Despite its advantages, TC cannot output the correct results when the three collocated datasets are not accurate enough or have high levels of error correlations. In the future, more mutually independent snow depth products of high quality should be used to extend TC to an arbitrary number of datasets and to increase the robustness and accuracy of the representativeness evaluation. Snow surveys around the stations should also be performed to provide a benchmark for directly evaluating the TC approach.

Acknowledgments

This research is funded by the National Key Research and Development Program of China (2018YFC1506605; 2018YFC1506501) and Science and Technology Basic Resources Investigation Program of China (2017FY100500).

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