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Spatial Representativeness Analysis for Snow Depth Measurements of
Meteorological Stations in Northeast China

YUANYUAN WANG AND ZHAOJUN ZHENG

National Satellite Meteorological Center, China Meteorological Administration, Beijing, China

(Manuscript received 14 June 2019, in final form 20 February 2020)
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 r²_t,X estimated with TC. Stations are

1

considered representative if r²_t,X $ 0:5; that is, in situ measurements explain no less than 50% of the variations

1

in the ‘‘ground truth’’ of the snow depth averaged at the coarse scale (0.258). 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 r²_t,X values, of which 57 stations are

1

representative for the entire snow season (October–December, January–April). 2) Seasonal variations in r_t²_,X

1

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 r²_t,X is positively correlated with mean snow depth, which largely determines the global decreasing

1

trend in r²_t,X from north to south. After removing this trend, residuals in r²_t,X can be explained by heterogeneity features concerning elevation and conditional probability of snow presence near the stations.

• 1
• 1

1. Introduction

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 3 25 km or larger (Vander Jagt et al. 2013). If satellitederived 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).

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).

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

Corresponding author: Yuanyuan Wang, wangyuany@ cma.gov.cn

DOI: 10.1175/JHM-D-19-0134.1

Ó 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright

Policy (www.ametsoc.org/PUBSReuseLicenses).

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10% from the area average (Neumann et al. 2006; questions, which have not been fully explored in previ-

Molotch and Bales 2005, 2006; Rice and Bales 2010; ous TC studies. Meromy et al. 2013; Grünewald and Lehning 2013;

1) How representativeness varies with season?
Grünewald et al. 2013). This method requires a dense
Representativeness of a station is not a constant sampling network, based on which upscaling to the

(Bohnenstengel et al. 2011); it can change con-

coarse scale can be achieved by using spatial modeling siderably from the snow accumulation period to methods. Although this method has been successthe snow ablation period owing to the variafully applied at the watershed scale, it is of limited tions in the spatial heterogeneity of snow depth use for estimating the spatial representativeness of

(Molotch and Bales 2005; Winstral and Marks

sparse meteorological stations that provide only one
2014). Some researchers argued that the obserin situ observation for a satellite footprint. Since it vations need to be selected with the specific is logistically prohibitive to carry out extensive snow objective of representing either the accumulation surveys or set up dense networks over hundreds of opor the ablation season process (Molotch and erational meteorological stations, the limitations of
Bales 2005). Understanding the seasonal variapoint measurements at these stations in adequately tions in representativeness can help us choose the representing snow depth for the surrounding area have most representative stations according to the been questioned but not explored in detail (Blöschl time of the snow depth product and hence make

1999; Neumann et al. 2006; Derksen et al. 2003; Chang

full use of the existing networks.

et al. 2005; Grünewald and Lehning 2013; Meromy

2) What factors in the vicinity of stations play a

et al. 2013).

dominant role in determining representativeness?
A promising way to evaluate the representativeness
Understanding the dominant factors has two advanof a point-scale dataset is the triple collocation (TC) tages. First, it provides an indirect approach to validate technique, which estimates the data quality of three representativeness assessments. Strong heterogeneity mutually independent datasets without treating any usually results in low representativeness; thus, the dataset as perfectly observed ‘‘truth’’ (Stoffelen 1998). representativeness assessments are generally reason-
TC has now become a standard procedure in compreable if they are strongly correlated with heterohensive satellite validation processes, especially in soil geneity features. Second, dominant factors can be

moisture research (Scipal et al. 2008; Dorigo et al. 2010,

used to predict representativeness, which is poten-

2015; Chen et al. 2017; Gruber et al. 2016a,b, 2017).

tially useful in choosing the representative locations
When TC is applied to a triplet containing one pointfor new sites. scale and two coarse-scale datasets that have the similar

• spatial resolution, performance metrics associated with
• The remainder of this paper is organized as follows.

the point-scale dataset indicate its spatial representa- Section 2 introduces the TC technique and how TC is tiveness, assuming that the instrumental random error used to evaluate station representativeness. Section 3 can be neglected (Gruber et al. 2013, 2016a; Chen et al. describes the study region, datasets, TC implementation 2017). The most prominent feature of using TC to assess process, and the method of extracting heterogeneity feathe spatial representativeness is that it is data-driven and tures. Results and discussion are presented in sections 4 does not need field surveys or dense sampling net- and 5, respectively. works. 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

2. Introduction of the TC technique

a. TC approaches

ground truth has been confirmed at densely instrumented validation sites by Miralles et al. (2010)

and Chen et al. (2017).

The most commonly used error model for TC analysis is the following model (Gruber et al. 2016a):
Validations of microwave snow depth and soil mois-

ture share a high degree of similarity. The success of TC applications in soil moisture studies has prompted

X_i 5 a_i 1 b_it 1 «_i ,

(1) us to adopt this technique for snow depth studies. where X_i (i 2 {1, 2, 3}) are three collocated and indeTo the best of our knowledge, this study is the first at- pendent datasets of the same geophysical variable linetempt to apply TC to evaluate the spatial representa- arly related to the true underlying value t with additive tiveness of point-scale snow depth measurements from zero-mean random errors «_i. The terms X_i, t, «_i are meteorological stations. Besides assessing the spatial all random variables; a_i and b_i are the intercepts and representativeness, we investigated the answers to two slopes, respectively, representing systematic additive

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(6)

8

Q₁₂Q₁₃ Q₂₃

and multiplicative biases of dataset X_i with respect to the true signal t.

2

«

>>>

s

5 Q₁₁

222

1

>>>>

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, «_i) 5 0; and (iv) zero error cross correlation: independence between the errors of X_i and X_j, that is, Cov(«_i, «_j) 5 0, for i ¼ j.

>><

Q₁₂Q₂₃ Q₁₃ s²_« 5 Q₂₂

.

2

>>>>>>>>>:

Q₁₃Q₂₃ Q₁₂

2

s_« 5 Q₃₃

3

Since s²_« is the absolute random error variance af-

i

fected by the dynamic range of the data, Draper et al. (2013) proposed relative error variance (fMSE_i), which is calculated by normalizing the error variances with the corresponding dataset variances:
Following McColl et al. (2014), the covariances between the different datasets are calculated as follows:

Cov(X_i, X_j) 5 E(X_iX_j) 2 E(X_i)E(X_j)

s_«²

i

5 b_ib_js²_t 1 b_iCov(t, «_j) 1 b_jCov(t, «_i)

• fMSE_i 5
• .
• (7)

Q_ii

1 Cov(«_i, «_j),
(2)
Combining (7), (4), and (3), fMSE_i can be written as

follows: where s²_t 5 var(t). Using the assumptions of error or-

thogonality and zero error cross correlation, the equation is reduced to (3):

• s_«²
• s_«²

• i
• i

fMSE_i 5

5

• ,
• (8)

u²_i 1 s²_« b²_i s_t² 1 s²_«

• i
• i

(

b_ib_js²_t ,

b_ib_js²_t 1 s²_« , for i 5 j for i ¼ j

• where b²_i s_t² represents the signal and s_«² represents the
• Q_ij [ Cov(X_i, X_j) 5
• ,
• (3)

i

noise (Gruber et al. 2016a; McColl et al. 2014); thus,

fMSE_i is not only a measure of relative error, but also a measure of signal-to-noise ratio (SNR). Furthermore, fMSE_i is related to the linear correlation coefficient of

i

where s²_« 5 var(«_i), representing the variance of

i

random error in dataset X_i. Since there are six equations (Q₁₁, Q₁₂, Q₁₃, Q₂₂, Q₂₃, Q₃₃) but seven unknowns (b₁, b₂, b₃, s_« , s_« , s_« , s_t), the system is
X_i with the underlying true signal t (denoted by r_t,X ). According to McColl et al. (2014), the relationship be-

i

• 1
• 2
• 3

underdetermined. It can be solved by defining a new variable u_i 5 b_is_t. Then, the equations can be rewritten as in (4): tween r_t,X and the ordinary least squares (OLS) slope b_i can be written as in (9):

i

(

b_is_t

pffiffiffiffiffiffi

u_iu_j,

for i ¼ j

r_t,X

5

• .
• (9)

i

Q_ii
Q_ij 5

• .
• (4)

u²_i 1 s²_« , for i 5 j

i

Combining (7), (8), and (9), we obtain (10):
Now there are six equations and six unknowns, and the system can be solved. Variable u²_i , which provides estimates of the sensitivity of datasets X_i to ground truth changes (Gruber et al. 2016a), can be written as follows:

Q_ii 2 s²_«

s_«²

b²_i s²_t r²_t,X

• 5
• 5

ⁱ 5 1 2 ⁱ 5 1 2 fMSE_i . (10)

i

• Q_ii
• Q_ii
• Q_ii

Equation (10) indicates that r²_t,X and fMSE_i are

8

Q₁₂Q₁₃

i

21
21
2

t

>>>

u 5 b s 5

complementary. When fMSE_i is 0.5, the coefficient of determination r²_t,X for the linear error model is 0.5, and

Q₂₃

>>>>

i

• pffiffiffiffiffiffi
• >

><

the correlation coefficient of X_i with t is 0:5 (’0.71).

Q₁₂Q₂₃ Q₁₃ u²₂ 5 b²₂s²_t 5

• .
• (5)

>>

b. Representativeness analysis of point-scale data with TC

>>>>>>>:

Q₁₃Q₂₃ Q₁₂

• 2
• 2
• 2

u₃ 5 b₃s_t 5

While TC is a powerful tool for estimating random errors and removing systematic differences between the
The estimation equation for error variances can be signal variance component of observations, it is affected

• written as follows:
• by representativeness errors (Yilmaz and Crow 2014).

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TC assumes that the three datasets represent the same regions in China (Li et al. 2008) and is characterized by signal, which is very unlikely given that the three datasets taiga snow (Sturm et al. 1995). The region with a total can have very different spatial measurement support area of 1.26 3 10⁶ km² encompasses the provinces of (McColl et al. 2014; Gruber et al. 2016a). When a triplet Heilongjiang, Jilin, Liaoning, and the eastern part of consists of one point-scale in situ dataset and two coarse- Inner Mongolia. The regional climate includes warm scale datasets that have the similar spatial resolution, the temperate, medium temperate, and subarctic zones. high-resolution signal in the point-scale dataset cannot be Annual precipitation is approximately 430–680 mm, of detectable for coarse-scale datasets and therefore be re- which 5%–10% is snowfall (He et al. 2013; Zhang et al. garded as error (Gruber et al. 2016a). In other words, TC 2016). There are three mountain ranges (Daxinganling, will penalize the point-scale dataset for its limited rep- Xiaoxinganling, and Changbaishan Mountains) and two resentativeness at the coarse scale, whereas no repre- large plains (Songnen and Sanjiang) in the region. sentativeness error is assigned to the error estimates of Primary land cover types are forest (40%), farmland the coarse-scale datasets (Gruber et al. 2016a; Yilmaz and (30%), and grassland (20%). Figure 1 shows the spatial Crow 2014). This characteristic of TC opens an oppor- pattern of tree cover (%) and elevation (m) in the tunity for evaluating the spatial representativeness of study region. point-scale data efficiently, which has been proved feasi-