# Analysis of Large Scale Spatial Variability of Soil Moisture Using a Geostatistical Method

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## Abstract

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## 1. Introduction

^{2}) catchment areas [13–18]. However, areal extent of these studies is too small for robust soil moisture analysis at precipitation scales as well as spatial scales (20–50 Km) of soil moisture retrieval from passive microwave satellite data. The spatial extent (or footprint) of AGRMET and satellite based passive microwave radiometers is comparable to average distance between two Oklahoma Mesonet soil moisture sensors. This distance is approximately equal to precipitation storm-scales, which drive the soil moisture spatial structures [19], therefore the knowledge of the spatial structure at these relatively crude resolutions is needed with the increase in availability large scale remote sensing satellite radiometer (WindSat, AMSR-E, SMOS, NPOESS) data for soil moisture retrieval. Figure 1 illustrates typical application variogram and kriging analysis to bring new spatial dynamic of soil moisture characteristics into the forecast models to improve hydrological modeling and assimilation results.

## 2. Study Area and Data Sets

#### 2.1. In Situ Oklahoma Mesonet

^{9}) weather and soil observations [21]. The instruments used in Mesonet Network for soil moisture measurement, are a Campbell Scientific 229-L heat dissipation sensor [23]. This sensor consists of a heating element and thermocouple emplaced in epoxy in a hypodermic needle, which is encased in a porous ceramic matrix. This sensor is typically used to measure soil metric potential by determining the temperature difference of the sensor before and after a heat pulse is introduced. The temperature difference is then converted into Fractional Water Index (FWI) and volumetric soil moisture. For physically-based land surface models, the more quantitative volumetric soil moisture measure is preferable due to mass transport of water within the soil column.

#### 2.2. AGRMET Model

## 3. Methodology

#### 3.1. Variogram and Kriging

_{i}= measured sample value at point i, Z

_{i+h}= measured sample value at point i + h, and N (h) = total number of sample couples for the separation interval h.

_{0}), sill (C

_{0}+ C), and decorrelation length or range of spatial dependence (A

_{0}) (Figure 3) are determined. These parameters of the variogram model describe the characteristics of spatial variation. The nugget is the y-intercept of the variogram indicating the semivariance between the two closest points separated in the spatial field. The sill of the variogram model represents the spatially dependent variance. Theoretically, the sill is equivalent to the maximum semivariance when the variogram model is bounded. The decorrelation length or range measures the limit of dependence of a given variable and is the distance at which the variogram reaches its sill. This is the limit of spatial dependence. If the decorrelation length is large then long-range variations dominate; if it is small, then the major variation occurs over short distances [26].

- γ(h) = semivariance for interval distance class h;
- h = the separation distance interval
- C
_{0}= nugget variance ≥ 0; - C = structural variance ≥ C
_{0}; and - A
_{0}= decorrelation length or range parameter.

_{0}, pairs of points will no longer be autocorrelated and the variogram reaches an asymptote (spherical model effective range A = A

_{0}). However, in the case of Gaussian or hyperbolic models the sill never meets the asymptote. In such a condition, the effective range (A = √3A

_{0}) is the distance at which the sill (C

_{0}+ C) is within 5% of the asymptote. The utility of correlated variables become less useful at lengths beyond their horizontal decorrelation length scales. These measures are typically represented by an exponential length scale decay in their correlations such that the covariance is proportional to exp (–1/A), where A is the decorrelation length.

#### 3.2. Geostatistical Spatial Analysis

^{2}of the fitting procedure were determined. The model with the higher value of R

^{2}was selected as an appropriate model to represent the sample variogram. The theoretical variogram model (Gaussian, spherical, exponential, or linear) that best fits the experimental variogram of AGRMET and Oklahoma Mesonet data was selected for soil moisture mapping using the block Kriging technique [26].

#### 3.3. Oklahoma Mesonet Data Screening and Quality Control

## 4. Results and Discussion

#### 4.1. In Situ and Model Soil Moisture Comparison (Oklahoma Mesonet vs. AGRMET)

#### 4.2. Variogram Analysis of Soil Moisture

_{0}). The decorrelation length was higher for dry periods before precipitation and decreases with increasing soil moisture during and after precipitation events. Soil moisture decorrelation length is found to be higher than the precipitation decorrelation length, and is discussed in Haberlandt [35].

_{0})] throughout the month for AGRMET and Oklahoma Mesonet data (Figure 7) were higher at wet soil conditions. This is because the precipitation-forced soil moisture patterns are at their strongest during wet events and have yet to be damped during the dry-down phase. The dry-down phase tends to diminish spatial patterns of soil moisture since each value slowly converges to lower soil moisture values that tend to be clustered at similar low soil moisture levels. However, if precipitation occurs in only part of the study area, higher values of MSH may not be observed.

#### 4.3. Kriging Performance Assessment

^{2}). No specific trends in bias and RMSE, specific to the wet and dry periods were observed at the locations (Figure 9a). During the month, mostly positive biases were observed at KING, MARE, MAYR, and MEDI. The negative bias observed at KETC, MINC, NOWA, and OKEM. LAHO and OKMU sites were small. The average RMSE of 10 jackknifed sites was found to be 3.2% through September 2003 (Table 1). Larger RMSE (∼5.5%) were observed at KETC, MEDI and MIAM; and lower RMSE (about 1–1.5%) were observed at LAHO, MAYR and OKMU Mesonet sites. The RMSE values could potentially be lowered through use of a co-kriging analysis [26] by including precipitation as an additional variable.

## 5. Summary and Conclusions

## Acknowledgments

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**Figure 1.**Application of variogram and kriging analysis in calibration and validation of soil moisture information for data assimilation process.

**Figure 2.**The distribution of AGRMET grid points and Oklahoma Mesonet sites used in the geostatistical analysis.

**Figure 3.**A generalized variogram model shows the essential components: nugget is the y-intercept that represents the semi-variance between two closest points; Sill signifies maximum semi-variance; and decorrelation length measures spatial continuity.

**Figure 4.**Soil moisture and precipitation comparisons between in situ Oklahoma Mesonet and AGRMET data for BUFF (a, d), PAUL (b, e), and SHAW (c, f) during September 2003.

**Figure 5.**Mean and standard deviation of precipitation (a) observed by Oklahoma Mesonet, (b) AGRMET precipitation (c) soil moisture measured at Oklahoma Mesonet sites and derived from AGRMET model.

**Figure 6.**Soil moisture variograms of AGRMET (a-b) and Oklahoma Mesonet (c-d), where symbols are the experimental semi-variances and the solid lines show the fitted model. Variograms (a) and (c) are before precipitation event (253 day), and (b) and (d) after a precipitation event (254 day). Gaussian and Spherical models best fit the AGRMET and Oklahoma Mesonet soil moisture data, respectively.

**Figure 7.**(a) Decorrelation lengths are higher for AGRMET compared to Oklahoma Mesonet soil moisture data, (b) Magnitude of spatial heterogeneity (MSH) is the ratio of Sill and Nugget for AGRMET and Oklahoma Mesonet soil moisture data. The MSH represents magnitude of spatial dependence which is higher during wet soil conditions, (c) Variogram elements (Sill and Nugget) show higher Sill (variance) and Nugget for Oklahoma Mesonet data compared to for AGRMET soil moisture data.

**Figure 8.**Kriged map of soil moisture for AGRMET data (a and b) and Mesonet data (c and d) generated using semi-variograms shown in Figure 6. Figures (a) and (c) are before precipitation event (253 day), and (b) and (d) for after precipitation event (253 day).

**Figure 9.**Kriging performance assessment (a) Bias and RMSE between kriged soil moisture map and ten Jackknifed site locations (b) Area average whole network, Root mean square difference (RMSD) and bias between kriged Mesonet and AGRMET soil moisture maps shows higher RMSD during dry period.

**Table 1.**This table shows the performance of Kriging in terms of volumetric soil moisture at each jack-knifed Mesonet site for September 2003.

Site Name | Latitude | Longitude | Bias | RMSE | Correlation Coefficient |
---|---|---|---|---|---|

KETC | 34.529 | −97.765 | −0.059 | 0.060 | 0.81 |

KING | 35.881 | −97.911 | +0.027 | 0.030 | 0.97 |

LAHO | 36.384 | −98.111 | −0.006 | 0.008 | 0.94 |

MARE | 36.064 | −97.213 | +0.047 | 0.048 | 0.86 |

MAYR | 36.987 | −99.011 | +0.012 | 0.013 | 0.80 |

MEDI | 34.729 | −98.567 | +0.060 | 0.061 | 0.94 |

MINC | 35.272 | −97.956 | −0.023 | 0.026 | 0.86 |

NOWA | 36.744 | −95.608 | −0.027 | 0.032 | 0.46 |

OKEM | 35.432 | −96.263 | −0.025 | 0.026 | 0.86 |

OKMU | 35.581 | −95.915 | −0.011 | 0.013 | 0.91 |

All Sites | -- | -- | -- | 0.032 | 0.84 |

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**MDPI and ACS Style**

Lakhankar, T.; Jones, A.S.; Combs, C.L.; Sengupta, M.; Vonder Haar, T.H.; Khanbilvardi, R.
Analysis of Large Scale Spatial Variability of Soil Moisture Using a Geostatistical Method. *Sensors* **2010**, *10*, 913-932.
https://doi.org/10.3390/s100100913

**AMA Style**

Lakhankar T, Jones AS, Combs CL, Sengupta M, Vonder Haar TH, Khanbilvardi R.
Analysis of Large Scale Spatial Variability of Soil Moisture Using a Geostatistical Method. *Sensors*. 2010; 10(1):913-932.
https://doi.org/10.3390/s100100913

**Chicago/Turabian Style**

Lakhankar, Tarendra, Andrew S. Jones, Cynthia L. Combs, Manajit Sengupta, Thomas H. Vonder Haar, and Reza Khanbilvardi.
2010. "Analysis of Large Scale Spatial Variability of Soil Moisture Using a Geostatistical Method" *Sensors* 10, no. 1: 913-932.
https://doi.org/10.3390/s100100913