# Geostatistical analysis of soil moisture distribution in a part of Solani River catchment

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## Abstract

The aim of this paper is to estimate soil moisture at spatial level by applying geostatistical techniques on the point observations of soil moisture in parts of Solani River catchment in Haridwar district of India. Undisturbed soil samples were collected at 69 locations with soil core sampler at a depth of 0–10 cm from the soil surface. Out of these, discrete soil moisture observations at 49 locations were used to generate a spatial soil moisture distribution map of the region. Two geostatistical techniques, namely, moving average and kriging, were adopted. Root mean square error (RMSE) between observed and estimated soil moisture at remaining 20 locations was determined to assess the accuracy of the estimated soil moisture. Both techniques resulted in low RMSE at small limiting distance, which increased with the increase in the limiting distance. The root mean square error varied from 7.42 to 9.77 in moving average method, while in case of kriging it varied from 7.33 to 9.99 indicating similar performance of the two techniques.

## Keywords

Soil moisture Geostatistics Moving average Limiting distance Lag distance Kriging Root mean square error## Introduction

Soil moisture plays a key role in various hydrological, environmental and agricultural applications. It governs infiltration and surface runoff, since the hydraulic conductivity of soil depends upon the available water content in the soil. It also influences the plant evapotranspiration thereby making it an important component of water balance equation. Knowledge of spatial distribution of soil moisture is essential for predicting runoff at the catchment scale (Fitzjohn et al. 1998; Western et al. 1999, 2001) and in the design of irrigation scheduling (Blonquist et al. 2006; Vellidis et al. 2008). For optimum irrigation practices, it becomes essential to accurately estimate the soil moisture. Moreover, to model overland flow from a precipitation event over a catchment, the estimation of antecedent soil moisture is a prerequisite.

Various studies (Oevelen 1998; Feng et al. 2004; De Lannoy et al. 2006) have shown that hydrometeorological conditions, soil properties and land cover govern the presence of moisture in the unsaturated zone of soil. These studies (Anderson and Burt 1977, 1978; Beven and Kirkby 1979; Chorley 1980; Moore et al. 1988; Wood et al. 1990; 1993; Bárdossy and Lehmann 1998) have shown that soil moisture is highly variable both in space and time. The characteristics of spatial variability of soil moisture depend on the scale of observation (Lakhankar et al. 2010).

For estimation of soil moisture at catchment level using observed point observations of soil moisture data, various interpolation methods such as weighted average, inverse distance interpolation, spline interpolation and kriging can be employed (Bárdossy and Lehmann 1998; Thattai and Islam 2000). For the estimation of soil moisture at spatial scale using observed point observations of soil moisture data, various interpolation techniques such as weighted average, inverse distance interpolation, spline interpolation and kriging can be employed. Dynamic multiple linear regression technique was adopted by Wilson et al. (2005) to compute soil moisture using topographic attributes. Downer and Ogden (2003) estimated soil moisture using gridded surface subsurface hydrologic analysis (GSSHA) hydrologic model with a root mean square error of 0.1. Pandey and Pandey (2010) mapped soil moisture in Udaipur, India, applying ordinary kriging. The study concluded that the krigged values were consistent and true representative of soil moisture values. Said et al. (2008) have experimented with ANN method to estimate soil moisture in the Solani River catchment. However, application of geostatistical interpolation technique is new in the study area.

The aim of this paper is to evaluate the efficacy of geostatistical interpolation techniques and to estimate soil moisture at spatial level from the in situ observed soil moisture data at point locations. The selection of an estimation technique may be site specific. Therefore, an evaluation of the techniques may be necessary to identify the appropriate one to be adopted in the prediction model. In the present study, two interpolation techniques, namely, moving average and ordinary kriging, have been evaluated to estimate the soil moisture at spatial scale.

## Study area

^{2}. The geographical coordinates of the sample points were recorded with the help of hand held global positioning system (GPS). Of the measured 69 locations, 52 samples were collected from the vegetated land covered with wheat, sugar cane, mustard and berseem crop. The wheat was in early growing stage, whereas the rest of the crops were at matured level. The remaining 17 locations correspond to bare soil fields. Out of 69 locations, 49 locations were considered for the model development whereas remaining 20 locations were kept for model validation. A due representation for vegetated and bare soil samples has been given in model development and its validation. Figure 2 depicts the geographical representation of the soil sampled locations.

The undisturbed soil samples were collected with the help of piston sampler from both bare soil and vegetated surfaces from the upper 0–10 cm thick soil layer. The samples were weighed and then oven-dried at 105 °C for 24 h to compute the volumetric soil moisture content. The observed volumetric soil moisture in the area varied from a minimum 3 % to a maximum of 43 %.

## Methodology

A number of interpolation techniques have been used for the analysis of distribution of the soil moisture at spatial scale (Feng et al. 2004; Wilson et al. 2005; Lakhankar et al. 2010). The most common interpolation techniques are moving average, trend surface and kriging (Kratze et al. 2006). The applicability of these techniques depends upon various factors such as distribution of sampled data in the space, the type of surfaces to be generated and tolerance of estimation errors. Hence, an evaluation of these geostatistical techniques may help in arriving at the most appropriate technique for estimation of soil moisture estimation at spatial level in Indian conditions. In the present study, the performance of moving average and kriging geostatistical techniques has been evaluated in estimating soil moisture distribution in a part of Solani River catchment.

## Moving average technique

*n*is the number of observed soil moisture locations within the search radius, \(W_{i}\) and \(M_{\text{voi}}\) are the weights assigned and the volumetric moisture content in percentage, respectively, at the

*i*th location. The weight \(W_{i}\) for each observed soil moisture location may be computed as,

*i*th observed soil moisture location from the estimated soil moisture location. The \(h_{\text{ri}}\) is computed as,

*i*th observed soil moisture location from the estimated soil moisture location, and \(h_{l}\) is the search radius. The accuracy of estimation depends upon the search radius and the weight exponent. Therefore, several trials are conducted before arriving at the acceptable values for the search radius and weight exponents.

## Kriging

Kriging is a geostatistical data interpolation technique based on the assumption that the data are spatially correlated. Ordinary kriging works on selected theoretical semi-variogram model used for computing semi-variance values at the points where soil moisture is to be estimated. A plot of the calculated semi-variance values against the distance (lags) is known as a semi-variogram. Increasing the lag distance of the semi-variance values consider average over more points, thus decreasing the fluctuations of the experimental semi-variogram. Several theoretical semi-variogram models are possible that include linear, spherical, circular, exponential and Gaussian (Teegavarapu and Chandramouli 2005) to fit over the experimentally constructed semi-variogram. The most suitable semi-variogram model may be found based on the RMS error between the semi-variance values obtained from experimentally observed data and the theoretical model predicted semi-variance values. Often the experimental semi-variogram values do not approach to zero at the origin and intersect the positive *y*-axis. This is due to the residual or spatially uncorrelated noise, which is also known as the nugget (Kitanidis 1997). The stabilized semi-variogram value is known as the sill, and the distance at which the semi-variogram values approach the sill is called the range.

The parameters, \(C_{0}\) and \(a\) denote nugget and range. The summation of \(C_{0}\) and \(C_{1}\) is referred to as sill and the sill value at range, \(a\), is the desired semi-variance.

*i*th location, and

*n*is the number of locations within the limiting distance. The \(W_{i}\) are computed by solving the following equations,

*p*where soil moisture is to be estimated, and the location \(i\) where soil moisture is observed. \(\gamma_{(hik)}\) is semi-variance value for the distance between the locations

*i*and

*k*and \(W_{i}\) is the weight for point \(i\). \(\lambda\) is a Lagrange multiplier that is used to minimize the possible estimation error in kriging interpolation.

The limiting distance governs the selection of observed locations around a location considered to estimate the soil moisture. The limiting distance is usually taken as smaller than the range of the selected semi-variogram. Observed locations that lie beyond specified limiting distance are not considered in the interpolation of soil moisture.

## Results and discussion

It can be seen from Fig. 3a that the method has failed to predict soil moisture at few places in case of small limiting distance. This is due to non-availability of observed sufficient soil moisture locations within the selected limiting distance. However, by increasing the limiting distance to 2,000 m and beyond, the moving average method is able to estimate the soil moisture over the entire study area. It can thus be concluded that for the effective application of moving average method, several experimental trials on limiting distance and weight exponent may therefore be necessary to obtain the optimal limiting distance and the weight exponent, which may also be case dependent.

Effect of limiting distance and weight exponent on estimated soil moisture using moving average technique

Trial no. | Limiting distance (m) | Weight exponent | RMSE (%) |
---|---|---|---|

1 | 1,000 | 0.5 | 7.42 |

1.0 | 7.43 | ||

1.5 | 7.44 | ||

2.0 | 7.45 | ||

2.5 | 7.46 | ||

2 | 2,000 | 0.5 | 10.27 |

1.0 | 10.31 | ||

1.5 | 10.39 | ||

2.0 | 10.43 | ||

2.5 | 10.47 | ||

3 | 3,000 | 0.5 | 9.70 |

1.0 | 9.71 | ||

1.5 | 9.80 | ||

2.0 | 9.91 | ||

2.5 | 9.99 | ||

4 | 4,000 | 0.5 | 9.44 |

1.0 | 9.40 | ||

1.5 | 9.51 | ||

2.0 | 9.65 | ||

2.5 | 9.77 |

In summary, it can be reiterated that the moving average method has limitation in selecting appropriate limiting distance as the user may have to determine it experimentally. To overcome this limitation, Kitanidis (1997) has proposed the use of ordinary kriging under the domain of semi-variogram model wherein the limiting distance may be taken as less than or equal to the range.

Experimental semi-variance values at different lag distances

Lag distance (m) | Number of location pairs | Semi-variance |
---|---|---|

1,000 | 10 | 50.75 |

2,000 | 55 | 121.69 |

3,000 | 101 | 122.96 |

4,000 | 116 | 93.05 |

5,000 | 139 | 103.13 |

6,000 | 146 | 91.3 |

7,000 | 138 | 97.48 |

8,000 | 101 | 124.18 |

9,000 | 79 | 88.06 |

10,000 | 87 | 110.39 |

11,000 | 68 | 101.49 |

12,000 | 51 | 122.31 |

13,000 | 34 | 122.49 |

14,000 | 19 | 121.44 |

15,000 | 10 | 130.17 |

16,000 | 10 | 151.67 |

17,000 | 9 | 209.82 |

18,000 | 3 | 99.66 |

The accuracy of ordinary kriging depends primarily upon the theoretical semi-variogram model employed to fit the experimental semi-variogram. Therefore, three theoretical semi-variogram models, namely, spherical, Gaussian and exponential models have been used in this study.

RMSE between actual and model computed semi-variance values

Semi-variogram model | Lag distance (m) | Nugget | Sill | Range (m) | RMSE (%) |
---|---|---|---|---|---|

Spherical | 1,000 | 48 | 101 | 2,000 | 10.32 |

2,000 | 50 | 105 | 2,800 | 9.12 | |

3,000 | 102 | 114 | 3,000 | 11.19 | |

4,000 | 98 | 114 | 4,000 | 12.54 | |

Gaussian | 1,000 | 48 | 101 | 2,000 | 10.43 |

2,000 | 50 | 105 | 2,800 | 10.24 | |

3,000 | 102 | 114 | 3,000 | 11.19 | |

4,000 | 98 | 114 | 4,000 | 12.51 | |

Exponential | 1,000 | 48 | 101 | 2,000 | 10.32 |

2,000 | 50 | 105 | 2,800 | 10.23 | |

3,000 | 102 | 114 | 3,000 | 12.19 | |

4,000 | 98 | 114 | 4,000 | 12.51 |

It can also be seen that as the lag distance increases, the RMSE between the semi-variance values increases. The optimum lag distance has been found to be 2,000 m in case of spherical model.

It can be seen from Fig. 5a that kriging method also is unable to estimate soil moisture in regions where observed locations for interpolation are at a greater distance than the limiting distance. However, with the increase in the limiting distance to 2,000 m and beyond, the kriging method is able to estimate the soil moisture distribution entire area.

RMSE between the observed and the estimated soil moisture using ordinary kriging

Limiting distance (m) | RMSE (%) |
---|---|

1,000 | 7.33 |

2,000 | 9.57 |

2,700 | 9.99 |

## Conclusions

Spatial and temporal knowledge of soil moisture is important to effectively model the surface runoff from a river catchment. The moving average and kriging methods were employed to estimate soil moisture at spatial scale in a part of Solani River catchment. Observed soil moisture data at 49 sample locations points were used to estimate soil moisture spatial scale, whereas the soil moisture data at 20 locations were used to validate the results by computing RMSE between the observed and estimated soil moisture using the two methods.

For the dataset used, for small limiting distances, due to non-availability of sufficient observed soil moisture locations within the adopted limiting distance, the moving average method was not able to estimate the soil moisture in the region. The effect of variation in weight exponent was also found insignificant in this method. The kriging method was also unable to estimate soil moisture in regions where observed locations for interpolation were at a distance greater than the limiting distance. However, with the increase in the limiting distance to 2,000 m and beyond, the kriging method was able to estimate the soil moisture distribution entire area. Increase in the limiting distance beyond 1,000 m resulted in increase RMSE in both the cases. From the comparison of the two methods, the kriging appears to be a more practical method due to its dependency on data-constructed semi-variogram.

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