# Upscaling In Situ Soil Moisture Observations to Pixel Averages with Spatio-Temporal Geostatistics

^{1}

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^{3}

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## Abstract

**:**

^{3}·m

^{−3}and can be used as a high-resolution soil moisture product for watershed-scale soil moisture monitoring.

## 1. Introduction

^{2}cm

^{2}), whereas satellite sensors provide soil moisture estimates for a much larger spatial “block” support (typically > 10

^{2}km

^{2}). Considering the large within-block spatial variation of point-support soil moisture [6], a large number of ground-based measurements are needed to upscale soil moisture to a satellite-based footprint for validation purposes. In addition, soil moisture varies over time and ground-based measurements are not all taken simultaneously nor at the same time as the satellite-based measurement.

## 2. Study Area and Data Description

^{3}·m

^{−3}accuracy levels for soil moisture retrievals [17]. In addition to remote sensing data, a ground-based Ecological and Hydrological Wireless Sensor Network (EHWSN) was also installed to monitor farmland soil moisture during the HiWATER campaign [18]. Specifically, fifty EHWSN nodes named WATERNET were deployed to measure soil moisture at 0–4 cm depth for validation of the remote sensing product. The spatial distribution of WATERNET is shown in Figure 1. It covers approximately 36 (6 × 6) PLMR pixels. Soil moisture was measured using the frequency-domain reflectometry method and a Hydro Probe II (HP-II) sensor. The laboratory calibration shows that the WATERNET soil moisture instrument error is 0.010 m

^{3}·m

^{−3}[19]. WATERNET observes soil moisture at 1-minute temporal resolution. The high measurement accuracy and synchronization with satellite and airborne remote sensing overpasses makes WATERNET suitable for validation of the PLMR soil moisture products. In addition, a 1 km × 1 km pixel (the black square in Figure 1), which contains 50 EHWSN nodes jointly named SoilNET, was used as an intensive observation zone to capture small-scale soil moisture variation [20]. SoilNET is an ad hoc network designed by the Jülich Research Center [21]. It measures soil moisture at 0–4 cm depth every 5 min. Laboratory calibration showed that the instrument error of SoilNET is 0.015 m

^{3}·m

^{−3}, which is larger than that of WATERNET.

**Figure 1.**Soil moisture monitoring network in the Yingke–Daman irrigation district. The triangles and circles show the measurement locations of the WATERNET and SoilNET wireless sensor network, respectively. The background is a 3 m resolution Thermal Airborne Spectrographic Imager (TASI) for land surface Temperature (LST) retrieval.

## 3. Methods

#### 3.1. Upscaling with Spatio-Temporal Regression Block Kriging

#### 3.1.1. Spatio-Temporal Random Field Model

**s**

_{i}, t

_{i}), i = 1, …, n. The objective of spatio-temporal upscaling is to make a prediction of Z(B), i.e., the average of Z for an unobserved spatio-temporal “block” B, which is defined as a subset of the space-time domain: B ⊂ S × T. For instance, B might be a 1 × 1 square kilometer area integrated over a 24 h period. In order to predict Z(B) we must first define a model for Z.

**s**, t) is a deterministic trend that represents variation that can be explained by external environmental “covariates”. R(

**s**, t) is the spatio-temporal correlated stochastic residual, typically representing small-scale, “noisy” variation.

#### 3.1.2. The Trend Component

**s**, t) may be written as [23,25]:

_{i}are unknown regression coefficients, the f

_{i}are covariates that must be exhaustively known over the spatio-temporal domain, and p is the number of covariates. Covariate f

_{0}is taken as unity, resulting in β

_{0}representing the intercept.

**s**,t) has only spatial components. For this, a linear multiple regression model is established:

**X**

_{k}is a covariates matrix with dimensions $n\times \left(p+1\right)$, and the β

_{k}are parameters to be estimated. LST, NDVI, and FVC are auxiliary environmental variables extracted by taking the block average of the k-th ASTER fly-over.

#### 3.1.3. The Residual Component

**s**,t) at the n observation points. Since the trend component in Equation (1) cannot explain all variation in soil moisture, the residuals of the regression model will have spatio-temporal variation that in addition might be correlated in space and time. This indicates that a spatio-temporal variogram may be estimated from the residuals at the observation locations and used to predict the residuals using kriging.

**s**

_{i}, t

_{i}) and R(

**s**

_{j}, t

_{j}) with the variogram, assuming space-time second-order stationarity:

**h**

_{ij}=

**s**

_{i}−

**s**

_{j}and τ

_{ij}= t

_{i}− t

_{j}the distance in space and time, respectively, and E denotes the mathematical expectation. In Equation (4), we assume that the semivariance of R at points (

**s**

_{i}, t

_{i}) and (

**s**

_{j}, t

_{j}) only depends on the separation distance (

**h**

_{ij}, τ

_{ij}). This assumption might be difficult to satisfy for the variable Z but are more plausible for the residual variable R. Note that keeping both spatial and temporal distances separate implies that zonal and geometric space-time anisotropies can be accommodated.

**s**, t) consists of three stationary and independent components: a purely spatial process (with constant realizations over time), a purely temporal process (realizations are constant in space), and a spatio-temporal process for which distance in space is made comparable to distance in time by introducing a space-time anisotropy ratio. We also assume spatial isotropy, so that the spatial distance

**h**becomes a scalar h. Thus, the sum-metric covariance structure can be represented by:

#### 3.1.4. Spatio-Temporal Regression Block Kriging

**C**is the variance-covariance matrix of the stochastic residuals at observation points as derived from Equation (6), ${c}_{B}={\left[{c}_{1}\left(B\right),{c}_{2}\left(B\right),\dots ,{c}_{n}\left(B\right)\right]}^{\text{'}}$ with ${c}_{i}\left(B\right)={{\displaystyle \int}}_{B}C\left(s-{s}_{i},t-{t}_{i}\right)dsdt/\left|B\right|$ for i = 1, 2, ..., n, ${x}_{B}={\left[1,{x}_{1}\left(B\right),{x}_{2}\left(B\right),\dots ,{x}_{p}\left(B\right)\right]}^{\text{'}}$, with ${x}_{j}\left(B\right)={\int}_{B}{x}_{j}\left(s,t\right)dsdt/\left|B\right|$ for j = 1, 2, ..., p, and

**μ**are Lagrange multipliers introduced to satisfy the unbiasedness constraint [26]. The solution of Equation (9) for

**λ**and

**μ**yields the best linear unbiased prediction (BLUP) $\widehat{Z}\left(B\right)$, which is given by:

**s**,t) is constant), STRBK reduces to spatio-temporal ordinary block kriging (STOBK). The predictor and the corresponding prediction error variance of STOBK can be easily obtained from Equations (10) and (11) as:

**1**is an n-dimensional row vector whose elements are set to unity.

#### 3.2. Accuracy Assessment

## 4. Results

#### 4.1. Data Summary

**Figure 2.**Multiple parallel time series plot of 50 WATERNET soil moisture observations. The top-left panel shows the parallel multiple time series of soil moisture. The yellow to red palette represents variation in surface soil moisture. White represents no data. The right panel presents summary statistics of soil moisture for each node. The black dots denote the median while the horizontal lines represent the lower and upper quartiles. The lower panel shows the average moisture time series of the 50 nodes.

Date | Time Duration | Min | Max | Mean | SD | |
---|---|---|---|---|---|---|

10 July 2012 | 9:00–15:00 | WATERNET | 0.198 | 0.339 | 0.256 | 0.033 |

SoilNET | 0.221 | 0.289 | 0.252 | 0.027 | ||

2 August 2012 | 9:00–15:00 | WATERNET | 0.160 | 0.369 | 0.274 | 0.042 |

SoilNET | 0.237 | 0.312 | 0.278 | 0.034 |

**Figure 3.**ASTER-derived LST, NDVI, and FVC on 10 July 2012 (

**top**) and 2 August 2012 (

**bottom**). The 6 × 6 grids represent the PLMR pixels coverage.

#### 4.2. Regression Modeling of Spatial Trends

^{3}·m

^{−3}.

#### 4.3. Variogram Analysis of the Residuals

**Figure 4.**Point sample variogram (

**left**) of residuals from multiple linear regression and the fitted sum-metric model (

**right**) for 10 July (

**upper**), and 2 August 2012 (

**bottom**). Units are (m

^{3}·m

^{−3})

^{2}.

**Table 2.**Parameters of the fitted sum-metric variogram model for soil moisture regression residuals.

Date | Component | Model | Nugget | Sill | Range | Anisotropy |
---|---|---|---|---|---|---|

10 July | Spatial | Exponential | 0 | 0.00098 | 201.1 m | |

Temporal | Nugget | 0 | 0 | 0 min | ||

Spatio-temporal | Exponential | 0 | 0.00029 | 35.0 m | 1.56 m/min | |

2 August | Spatial | Exponential | 0 | 0.00079 | 265.3 m | |

Temporal | Nugget | 0 | 0.000012 | 10.0 min | ||

Spatio-temporal | Exponential | 0 | 0.00068 | 24.0 m | 1.79 m/min |

#### 4.4. Accuracy Assessment

Upscaling Error (m^{3}·m^{−3}) | |||
---|---|---|---|

BK | STOBK | STRBK | |

10 July | 0.0016 | 0.0014 | 0.0013 |

2 August | 0.0017 | 0.0017 | 0.0016 |

^{3}·m

^{−3}and the ME is 0.016 m

^{3}·m

^{−3}; 2 August yields RMSE = 0.027 m

^{3}·m

^{−3}with an ME of 0.005 m

^{3}·m

^{−3}. The PLMR product at 10 July is, thus, more accurate than the product of 2 August. The main reason is that the soil moisture is relatively homogeneous during 10 July.

**Figure 5.**Comparison of the STRBK upscaled predictions with the values of PLMR retrievals on 10 July (

**left**), and 2 August 2012 (

**right**). The blue line is the fitted linear regression, the dashed line is the 1:1 line. The error bars show the 95% prediction intervals, as derived from the block kriging variances.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, J.; Ge, Y.; Heuvelink, G.B.M.; Zhou, C.
Upscaling *In Situ* Soil Moisture Observations to Pixel Averages with Spatio-Temporal Geostatistics. *Remote Sens.* **2015**, *7*, 11372-11388.
https://doi.org/10.3390/rs70911372

**AMA Style**

Wang J, Ge Y, Heuvelink GBM, Zhou C.
Upscaling *In Situ* Soil Moisture Observations to Pixel Averages with Spatio-Temporal Geostatistics. *Remote Sensing*. 2015; 7(9):11372-11388.
https://doi.org/10.3390/rs70911372

**Chicago/Turabian Style**

Wang, Jianghao, Yong Ge, Gerard B. M. Heuvelink, and Chenghu Zhou.
2015. "Upscaling *In Situ* Soil Moisture Observations to Pixel Averages with Spatio-Temporal Geostatistics" *Remote Sensing* 7, no. 9: 11372-11388.
https://doi.org/10.3390/rs70911372