# Improvement of Spatial Interpolation of Precipitation Distribution Using Cokriging Incorporating Rain-Gauge and Satellite (SMOS) Soil Moisture Data

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}). Quarterly precipitation data collected during a 5-year period (2010–2014) from 113–116 rain-gauge stations located in the study area were used. Additionally, monthly precipitations in the years 2014–2017 from over 400 rain-gauge stations located in Poland were used. The spatiotemporal data on soil moisture (SM) from the Soil Moisture and Ocean Salinity (SMOS) global satellite (launched in 2009) were used as an auxiliary variable in addition to precipitation for the OCK method. The predictive performance of the spatial distribution of precipitations was the best for OCK for all quarters, as indicated by the coefficient of determination (R

^{2}= 0.944–0.992), and was less efficient (R

^{2}= 0.039–0.634) for the OK and IDW methods. As for monthly precipitation, the performance of OCK was considerably higher than that of IDW and OK, similarly as with quarterly precipitation. The performance of all interpolation methods was better for monthly than for quarterly precipitations. The study indicates that SMOS data can be a valuable source of auxiliary data in the cokriging and/or other multivariate methods for better estimation of the spatial distribution of precipitations in various regions of the world.

## 1. Introduction

^{3}m

^{−3}) [43,46,47,48,49], and thus show good global performance [43,45]. Recent studies have demonstrated that assimilation of remotely sensed soil moisture has the potential to improve the quality of the near-real-time SMOS-based rainfall product [50,51] and to determine soil water resources [52].

## 2. Study Area and Data Used

^{2}. The study area is dominated by soils with texture of loamy sands and loams. In the case of the examined area of Poland, it was approximately 325,000 km

^{2}. Quarterly precipitation data for a 5-year period (2010–2014) from 113–116 rain gauges retrieved from Tutiempo Network, S.L., Copyright 2018, located in a variety of networks were used (Figure 1A). Monthly precipitation data were obtained from over 400 ground meteorological stations of the Institute of Meteorology and Water Management—National Research Institute (IMGW) (Figure 1B). These free-access data are in ASCII format [57]. The original 1-day precipitation values were summed to monthly blocks in order to be consistent with the time scale of the SM from SMOS. It was assumed that a given weather station is representative of the entire SMOS pixel in which it is located. The rain-gauge stations were located at different altitudes varying from 1 to 1988 m a.s.l. The average altitude was 311 m with standard deviation as high as 320 m and, consequently, a coefficient of variation (average divided by standard deviation and multiplied by 100%) of 103%. The distribution of the altitudes exhibited a positive asymmetry and high slenderness. Skewness and kurtosis were 2.658 and 8.886, respectively. For this study, the SMOS L2 v. 650 datasets provided by the European Space Agency were examined. Based on the SMOS mission data processing algorithm, for each record assigned to a single DGG (Discrete Global Grid) node number, L2 retrieval has been carried out, under the condition that such pixel is not masked by no measurement value (−999 is assigned as an indicator of no measurements). The procedure for masking is defined by the series of quality flags defined in the mission Algorithm Theoretical Basis Document (ATBD) [58]. Additionally, variety of flags is applied to define the scene for which retrieval is conducted (including complex urban areas). The largest complications originate from external sources, such as radiofrequency interference (RFI) [58]. RFI contaminates the original signal, leading to exaggerated values of brightness temperature and, in consequence, unphysical values of SM. Thus, such pixels are masked by −999 (no measurement) value, before the L2 processing algorithm is applied. The datasets were downloaded from ftp://smos-ds-02.eo.esa.int/SMOS/L2SM/MIR_SMUDP2/. Next, further products were built from these data. The soil moisture contents in the topsoil (less than ~10 cm) were extracted from the SMOS satellite data for about 5000 (for Poland and neighbouring countries) and 2000 (for Poland) points on the Discrete Global Grid (15 km grid) using the Icosahedral Snyder Equal Area (ISEA) map 4H9 projection. SMOS soil moisture was averaged quarterly in 2010–2014 [59] and monthly in 2014–2017. For the purpose of geostatistical analysis, a regularization procedure was performed (i.e., it was assumed that each pixel is represented by one point located in its centre).

^{3}m

^{−3}, with the highest and the lowest values in 2010 (0.171 m

^{3}m

^{−3}) and in 2012 (0.128 m

^{3}m

^{−3}), respectively (Figure 2B). The coefficients of variation were similar in all study years (26.6–33.3%). The minimum and maximum soil moisture values were close to zero and 0.5 m

^{3}m

^{−3}. The skewness (0.20–0.57) and kurtosis (0.46–1.79) values indicate that soil moisture distribution was positively skewed and slightly narrow. The mean soil moisture in the individual quarters varied from 0.12 to 0.2 m

^{3}m

^{−3}. The variability of soil moisture was, in general, highest in quarter I (to 40%) and lowest in quarter II (around 23%). The yearly standard deviations ranged from 0.04 to 0.051 m

^{3}m

^{−3}. The standard variation was approximately 6.3% in quarter I and lower (up to ca. twofold) in the other quarters. The coefficient of variation was greater than 33% in each year and approximately 40% in quarter I of 2012 and 2013, with similar and lower variability in quarters II and III and increased variability in quarter IV. Histograms of soil moisture distributions indicate that the asymmetries were negative in quarter III and mostly positive in the other seasons (see Figure 3 in [59]). To obtain the normal distribution required in geostatistics, the variables with the highest asymmetry were transformed using the square root.

^{3}m

^{−3}with the highest and the lowest values of monthly soil moisture in January 2017 (0.085 m

^{3}m

^{−3}) and in February 2017 (0.254 m

^{3}m

^{−3}), respectively (Figure 2D). The coefficients of variation were generally similar in all study years (23.9%–48.0%) except January 2017 (97%). The minimum and maximum of monthly soil moisture values were respectively close to zero and 0.748 m

^{3}m

^{−3}(in wetlands). The values of skewness (−0.425 to 2.008) indicate that soil moisture distribution was slightly negatively and positively skewed, and those of kurtosis (−0.07 to 5.8) that it had a normal or narrow shape. Both the skewness and kurtosis values indicate that the monthly soil moisture distributions were close to the normal distribution.

## 3. Methodology

#### 3.1. Semivariograms and Cross-Semivariograms

_{1}) and soil moisture (z

_{2}) was performed using geostatistical methods. The normality of precipitation was obtained after square root transformation. After that, the soil moisture distribution was close to normal and thereby met the required condition of a stationary or quasi-stationary process. The experimental semivariogram γ(h) and cross-semivariogram between precipitation (z

_{1}) and soil moisture (z

_{2}) − γ

_{12}(h) for the distance h (°) were calculated from the following equations [60]:

_{1}(x

_{i}), z

_{1}(x

_{i}+h)], [z

_{2}(x

_{i}), z

_{2}(x

_{i}+h)], distant by h, and x

_{i}is the spatial coordinate. For semivariograms and cross-semivariograms determined empirically, the following three mathematical models were selected:

- −
- spherical model:

- −
- exponential model:

- −
- Gaussian model:

_{0}is the nugget variance ≥0, C is the structural variance ≥C

_{0}, A

_{0}is the range parameter, and C

_{0}+ C is the sill. In the case of the spherical isotropic model, the effective range A = A

_{0}. In the case of the exponential isotropic model, the effective range A = 3A

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

_{0}) is within 5% of the asymptote. In the case of the Gaussian model, the effective range A = 3

^{0.5}A

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

_{0}) is within 5% of the asymptote. In the case of the anisotropic model, the effective range $A=\sqrt{{A}_{1}^{2}\left[{\mathit{cos}}^{2}\left(\theta -\phi \right)\right]+{A}_{2}^{2}\left[{\mathit{sin}}^{2}\left(\theta -\phi \right)\right]}$, where A

_{1}is the range parameter for the major axis (ϕ) and A

_{2}is the range parameter for the minor axis (ϕ + 90). In the case of the exponential anisotropic model, the range (or effective range) is 3A

_{1}for the major axis and 3A

_{2}for the minor axis, ϕ is the angle of maximum variation, and θ is the angle between pairs. In the case of the Gaussian anisotropic model, the range (or effective range) is 3

^{0.5}A

_{1}for the major axis and 3

^{0.5}A

_{2}for the minor axis, ϕ is the angle of maximum variation, and θ is the angle between pairs. To evaluate anisotropy, the azimuth direction A

_{z}with the lowest semivariance values defined by smaller in the major direction (lower average semivariance) and largest in the minor (90°–offset) direction was used.

^{2}values were selected.

#### 3.2. Interpolation Methods

#### 3.2.1. Inverse Distance Weighting

_{j}*(h) is the estimated precipitation value at desired location j, z

_{i}is the measured sample value at point i, h

_{j}is the distance between z

_{j}*(h) and z

_{i}, s is the smoothing factor, and p is the weighting power.

#### 3.2.2. Ordinary Kriging

_{i}) is the value measured at point x

_{i}, z*(x

_{o}) is the estimated value at the point of estimation x

_{o}, and λ

_{i}is the weights. The weights are determined from a system of equations after inclusion of the condition of estimator nonbias and its effectiveness:

#### 3.2.3. Ordinary Cokriging (OCK)

_{1}) and soil moisture (z

_{2})). The main advantage of the method is the possibility of indirect reconstruction of the spatial variability of precipitation, the measurement of which is difficult and expensive, through field analysis of soil moisture, which is easier to determine with standard measuring equipment or available from satellite observations.

_{o,}can be made with the help of the estimation method known as the cokriging approach. The mathematical basis for cokriging is the theorem on the linear relationship of the unknown estimator z

_{2}

^{*}(x

_{o}) expressed by the following formula [60]:

_{1i}and λ

_{2j}are weights associated with z

_{1}and z

_{2}. N

_{1}and N

_{2}are the numbers of neighbours of z

_{1}and z

_{2}included in the estimation at point x

_{o}. Cokriging weights are determined from a system of equations with the inclusion of the condition of estimator nonbias and its effectiveness:

_{1}and μ

_{2}are Lagrangian factors, and C

_{11}, C

_{12}, C

_{21}, and C

_{22}represent covariance between the variables. The relationships between the semivariance γ(h) and the covariance C are expressed by the equation γ(h) = C(0) − C(h). In our study, the cokriging approach was used to enhance the estimation of spatial precipitation distribution using sparse data from rain-gauge stations and more densely sampled SMOS SM (as auxiliary variable) complementing the former.

## 4. Results

#### 4.1. Statistics of Rain-Gauge Data

#### 4.2. Correlation Analysis

#### 4.3. Semivariogram and Cross-Variogram Models

^{2}> 0.8) mostly to spherical, exponential, and Gaussian models. For quarterly precipitations and soil moisture, the best model to describe the spatial relationship was the exponential model. The ranges (A) of spatial dependence in semivariograms for quarterly precipitations were greater (from 1.26° for III 2010 to 6.47° for II 2010) with predominant values below 3 than for quarterly soil moisture contents (from 1.00° for I 2010 to 4.08° for IV 2013) with predominating values below 2° (Table 3). In the case of cross-semivariograms, the ranges of spatial dependence between rainfall and soil moisture appreciably increased, reaching a maximum of 8.98° with predominant values below 5°. Anisotropy (A

_{z}) varied for precipitations from 50° to 120° with predominant values above 100°, for SMC from 0° to 172° with predominance below 120°, and for P_SM from 12° to 122° with predominance above 100°. The ratios C

_{0}/(C

_{0}+ C) were in most cases (50 out of 60) <0.25, indicating strong spatial dependence; in the other cases, it was moderate [62] (0.25–0.75). It is worth noting that the spatial dependences (nugget/sill) were more frequently stronger in OCK (0.00–0.179) (except one, 0.403) than OK (0.018–0.25).

_{z}) varied for precipitations and soil moisture from 58° to 130° with predominant values from 58° to 95° and P_SM from 100° to 120°. The ratios C

_{0}/(C

_{0}+ C) were in cases (16 out of 32) <0.25, indicating strong spatial dependence; in the other cases, it was moderate (0.25–0.75). It is worth noting that the spatial dependences (nugget/sill) were strong in OCK (0.00–0.068).

#### 4.4. Comparison of the Interpolation Methods and Cross-Validation

^{2}—coefficient of determination, and SE Pre.—standard error prediction were calculated to compare the accuracy of each interpolator (Table 4). In the case of quarterly precipitation in the years 2010–2014, the ranges of regression coefficients (a), SE, SE Pre. (predicted), and R

^{2}for IDW and OK were 0.552–1.428, 0.082–0.927, 29.1–108.9, 0.039–0.599 and 0.610–1.517, 0.072–0.339, 28.7–110.1, 0.040–0.634, respectively (Table 4). These values indicate a slightly better accuracy of OK than IDW. In turn, OCK indicates that the directional coefficients (a) of regression equations are above 1, up to 1.44, with small SEs (<0.032) and SE Pre. (<23) and with high values of R

^{2}(>0.94). The R

^{2}values for IDW and OK, which are the lowest in quarter I in most years (in 3 of 5 years), correspond with the lowest precipitation, but are similar in all the quarters in the case of OCK (Table 4). The appreciably lower SE and SE Pre. values along with the higher R

^{2}for OCK than IDW and OK clearly indicate better performance of the former.

^{2}for IDW and OK were 0.989–1.129, 0.022–0.067, 6.2–29.5, 0.352–0.844 and 0.927–1.086, 0.020–0.064, 5.6–29.5, 0.348–0.865, respectively (Table 4). As for OCK, the values of regression coefficients (a) (1.018–1.283) were similar to those in IDW and OK, whereas those of SE (<0.031) and SE Pre. (<16) were lower, and R

^{2}(0.810–0.995) was in most cases considerably larger. The above data indicate better performance of OCK than IDW and OK, similarly as with quarterly precipitation.

^{2}were larger, and those of SE and SE Pre. were lower for monthly than for quarterly precipitations (Table 4).

#### 4.5. Maps of Precipitations

## 5. Discussion

^{2}= 0.944–0.992 between the measured and estimated precipitations), compared with both ordinary kriging (OK) (R

^{2}= 0.040–0.634) and inverse distance weighting (IDW) (R

^{2}= 0.039–0.599) using only the rain-gauge data. The greater suitability of the OCK method was supported by the smaller values of nugget and standard error prediction (SE Pre.) and the larger range of influence of the cross-semivariogram model than that of the direct (single) semivariogram for the above variables. As for monthly precipitation in the years 2014–2017, the performance of OCK (R

^{2}= 0.810–0.995) was considerably higher than that of IDW (R

^{2}= 0.352–0.844) and OK (R

^{2}= 0.348–0.865), similarly as with quarterly precipitation. The performance of all interpolation methods was better for monthly than for quarterly precipitations as shown by larger R

^{2}and lower SE and SE Pre. values.

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Spatial distribution of rain-gauge stations in Poland and neighbouring countries and SMOS pixel (

**A**). Map was created using Google Earth (v. 7.3.2.5776). Google, proprietary software, https://www.google.com/earth/. The background maps from Google Maps (https://www.google.com/maps/@51.1367607,20.6545385,5.5z), accessed 12 April 2017, and https://pl.wikipedia.org/wiki/Plik:Poland_location_map_white.svg (

**B**). Elevation map of Poland, n.p.m.—above sea level (

**C**), https://pl.wikipedia.org/wiki/Mapa_hipsometryczna#/media/Plik:Poland-hipsometric_map.jpg. The background maps were modified using Microsoft Office PowerPoint 2019.

**Figure 2.**Mean, minimal (Min), and maximal (Max) values for annual rainfall (

**A**) and soil moisture data (

**B**) with standard deviations in Poland and neighbouring countries for the study period 2010–2014 and for monthly rainfall (

**C**) and soil moisture data (

**D**) in Poland for the study period 2014–2017.

**Figure 3.**Spatial distribution of monthly rainfall (2D maps) in Poland estimated by the ordinary cokriging interpolation method for selected months in the years 2014 and 2016 (1° = approximately 100 km). Maps were created using Gamma Design Software GS+10 [60].

Quarter of the Year | I 2010 | II 2010 | III 2010 | IV 2010 | I 2011 | II 2011 | III 2011 | IV 2011 | I 2012 | II 2012 | III 2012 | IV 2012 | I 2013 | II 2013 | III 2013 | IV 2013 | I 2014 | II 2014 | III 2014 | IV 2014 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

n | 115 | 115 | 116 | 116 | 116 | 116 | 113 | 116 | 115 | 116 | 116 | 116 | 114 | 114 | 114 | 114 | 114 | 116 | 116 | 110 |

Mean | 104.0 | 277.1 | 344.0 | 148.3 | 80.6 | 163.3 | 261.1 | 91.7 | 113.8 | 185.5 | 219.4 | 145.5 | 141.0 | 265.5 | 190.2 | 106.5 | 93.8 | 207.9 | 275.5 | 114.1 |

SD | 36.3 | 140.2 | 120.6 | 50.6 | 29.7 | 74.9 | 88.3 | 53.2 | 71.1 | 62.5 | 86.4 | 44.1 | 63.4 | 82.6 | 73.6 | 50.2 | 31.7 | 99.1 | 120.5 | 42.7 |

CV (%) | 34.9 | 50.6 | 35.0 | 34.1 | 36.9 | 45.9 | 33.8 | 58.0 | 62.4 | 33.7 | 39.4 | 30.3 | 45.0 | 31.1 | 38.7 | 47.1 | 33.8 | 47.7 | 43.7 | 37.4 |

Min | 29.2 | 26.7 | 157 | 49.3 | 30.2 | 46.7 | 94.8 | 23.1 | 11.2 | 69.6 | 53.6 | 69.6 | 48.5 | 129.3 | 37.6 | 27.2 | 31 | 23.8 | 75.5 | 55.4 |

Max | 247.8 | 898.9 | 894.1 | 343.5 | 241.6 | 530.9 | 608.1 | 315.0 | 444.0 | 444.3 | 519.2 | 290.3 | 425.2 | 625.3 | 402.1 | 309.9 | 255.0 | 673.1 | 792.9 | 279.4 |

Skewness | 1.302 | 1.523 | 1.998 | 0.847 | 1.947 | 2.045 | 0.900 | 1.900 | 2.312 | 1.166 | 1.034 | 1.071 | 1.535 | 1.163 | 1.042 | 1.477 | 1.165 | 2.081 | 1.360 | 1.523 |

Kurtosis | 3.179 | 3.777 | 5.415 | 1.309 | 7.355 | 6.048 | 2.125 | 3.851 | 7.525 | 1.983 | 1.287 | 1.309 | 3.305 | 2.500 | 0.915 | 2.626 | 4.930 | 7.626 | 3.315 | 2.636 |

Transformed rainfall data with a square root | ||||||||||||||||||||

Mean | 10.1 | 16.2 | 18.3 | 12.0 | 8.8 | 12.5 | 15.9 | 9.3 | 10.3 | 13.4 | 14.5 | 11.9 | 11.6 | 16.1 | 13.6 | 10.1 | 9.6 | 14.1 | 16.2 | 10.5 |

SD | 1.71 | 4.00 | 2.95 | 2.04 | 1.53 | 2.64 | 2.69 | 2.44 | 2.95 | 2.19 | 2.82 | 1.75 | 2.47 | 2.43 | 2.58 | 2.25 | 1.61 | 3.21 | 3.46 | 1.85 |

Skewness | 0.524 | 0.526 | 1.327 | 0.282 | 0.985 | 1.178 | 0.239 | 1.239 | 1.008 | 0.679 | 0.455 | 0.653 | 0.888 | 0.633 | 0.465 | 0.853 | 0.218 | 0.621 | 0.575 | 1.020 |

Kurtosis | 1.578 | 1.236 | 2.904 | 0.368 | 2.820 | 2.464 | 0.660 | 1.544 | 2.563 | 0.518 | 0.362 | 0.433 | 0.932 | 0.768 | 0.561 | 0.771 | 1.619 | 3.179 | 0.850 | 1.146 |

**Table 2.**Linear correlation coefficients between the quarterly and the monthly average rainfall and satellite soil moisture for Poland and neighbouring countries.

Correlation coefficients between the quarterly average rainfall and satellite soil moisture (P_SM) in the years 2010–2014. Bold, the correlation coefficients are significant with p < 0.05, n = 76. | |||||||||||||||

P_SM_I 2010 | P_SM_II 2010 | P_SM_III 2010 | P_SM_IV 2010 | P_SM_I 2011 | P_SM_II 2011 | P_SM_III 2011 | P_SM_IV 2011 | ||||||||

−0.022 | 0.099 | −0.062 | 0.237 | 0.234 | 0.238 | 0.289 | 0.335 | ||||||||

P_SM_I 2012 | P_SM_II 2012 | P_SM_III 2012 | P_SM_IV 2012 | P_SM_I 2013 | P_SM_II 2013 | P_SM_III 2013 | P_SM_IV 2013 | ||||||||

0.191 | 0.184 | 0.187 | 0.195 | 0.038 | 0.307 | 0.2 | 0.196 | ||||||||

P_SM_I 2014 | P_SM_II 2014 | P_SM_III 2014 | P_SM_IV 2014 | ||||||||||||

0.021 | 0.189 | 0.158 | 0.336 | ||||||||||||

Correlation coefficients between the monthly average rainfall and satellite soil moisture (P_SM) in the years 2014–2017. Bold, the correlation coefficients are significant with p < 0.05, n = 391. | |||||||||||||||

Years | P_SM _1 | P_SM _2 | P_SM _3 | P_SM _4 | P_SM _5 | P_SM _6 | P_SM _7 | P_SM _8 | P_SM _9 | P_SM _10 | P_SM _11 | P_SM _12 | |||

2014 | 0.208 | −0.028 | −0.199 | −0.186 | −0.333 | −0.090 | −0.150 | −0.063 | −0.003 | 0.072 | −0.206 | 0.131 | |||

2015 | −0.087 | −0.342 | 0.095 | 0.049 | 0.008 | 0.171 | 0.141 | −0.118 | 0.309 | −0.029 | 0.053 | 0.362 | |||

2016 | −0.147 | −0.260 | −0.007 | 0.088 | 0.037 | 0.309 | 0.296 | 0.459 | 0.160 | −0.155 | 0.454 | 0.316 | |||

2017 | 0.360 | −0.012 | 0.200 | 0.054 | 0.025 | 0.210 | 0.255 | 0.376 | 0.203 | 0.429 | 0.271 | 0.441 |

^{3}m

^{−3}), 1–12 months.

**Table 3.**Fitted semivariogram models for quarterly rainfall (P) and soil moisture (SM) data used in ordinary kriging interpolation method and cross-semivariogram models between rainfall and soil moisture (P_SM)—1° in the ordinary cokriging method corresponds to about 100 km.

Semivariogram_P | C_{0}/(C_{0} + C) | Semivariogram_SM | C_{0}/(C_{0} + C) | Cross-semivariogram_P_SM | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Quarter | Model | C_{0} | C_{0} + C | A (°) | A_{z} (°) | Model | C_{0} | C_{0} + C | A (°) | A_{z} (°) | Model | C_{0} | C_{0} + C | C_{0}/(C_{0} + C) | A (°) | A_{z} (°) | ||

2010 | I | Exp. | 0.641 | 3.11 | 0.206 | 1.50 | 116 | Exp. | 0.00001 | 0.00335 | 0.003 | 1.00 | 105 | Exp. | 0.000 | 1.500 | 0.000 | 4.01 | 33 |

II | Exp. | 1.263 | 18.24 | 0.069 | 6.47 | 50 | Exp. | 0.00000 | 0.00176 | 0.001 | 1.62 | 58 | Exp. | −0.002 | −0.016 | 0.125 | 3.46 | 12 | |

III | Exp. | 0.900 | 8.73 | 0.103 | 1.26 | 113 | Exp. | 0.00003 | 0.00168 | 0.017 | 1.54 | 66 | Exp. | 0.000 | −0.016 | 0.000 | 8.11 | 12 | |

IV | Exp. | 0.540 | 4.16 | 0.130 | 2.00 | 70 | Exp. | 0.00013 | 0.00205 | 0.063 | 1.88 | 116 | Exp. | 0.000 | 0.011 | 0.000 | 5.58 | 63 | |

2011 | I | Exp. | 0.327 | 2.39 | 0.137 | 1.29 | 117 | Exp. | 0.00098 | 0.00327 | 0.298 | 2.73 | 97 | Exp. | 0.000 | 0.017 | 0.000 | 4.35 | 107 |

II | Exp. | 2.680 | 7.29 | 0.367 | 2.82 | 69 | Exp. | 0.00007 | 0.00114 | 0.060 | 1.73 | 124 | Exp. | 0.000 | 0.019 | 0.000 | 3.16 | 115 | |

III | Exp. | 1.440 | 7.16 | 0.201 | 1.53 | 113 | Exp. | 0.00000 | 0.00131 | 0.002 | 1.70 | 41 | Exp. | 0.004 | 0.022 | 0.179 | 2.08 | 105 | |

IV | Exp. | 0.100 | 5.48 | 0.018 | 1.49 | 120 | Exp. | 0.00013 | 0.00119 | 0.113 | 2.39 | 58 | Gaus. | 0.005 | 0.040 | 0.122 | 6.20 | 105 | |

2012 | I | Exp. | 1.230 | 9.34 | 0.132 | 1.30 | 120 | Exp. | 0.00078 | 0.00371 | 0.209 | 2.03 | 58 | Exp. | 0.000 | 0.039 | 0.003 | 7.05 | 105 |

II | Exp. | 0.192 | 4.73 | 0.041 | 1.85 | 112 | Exp. | 0.00011 | 0.00108 | 0.106 | 1.63 | 154 | Exp. | 0.001 | 0.024 | 0.045 | 8.98 | 113 | |

III | Exp. | 1.374 | 8.19 | 0.168 | 5.20 | 26 | Exp. | 0.00091 | 0.00142 | 0.642 | 2.73 | 0 | Exp. | 0.004 | 0.019 | 0.196 | 3.65 | 112 | |

IV | Exp. | 0.312 | 3.17 | 0.098 | 1.42 | 106 | Exp. | 0.00092 | 0.00135 | 0.684 | 3.85 | 170 | Exp. | 0.000 | −0.002 | 0.000 | 3.69 | 115 | |

2013 | I | Exp. | 1.220 | 6.39 | 0.191 | 2.53 | 69 | Exp. | 0.00047 | 0.00290 | 0.163 | 1.81 | 131 | Exp. | 0.000 | −0.025 | 0.000 | 2.30 | 45 |

II | Exp. | 1.690 | 6.75 | 0.250 | 6.12 | 76 | Exp. | 0.00000 | 0.00222 | 0.000 | 2.10 | 146 | Exp. | 0.000 | 0.008 | 0.013 | 2.67 | 122 | |

III | Exp. | 2.534 | 7.32 | 0.346 | 3.28 | 105 | Exp. | 0.00057 | 0.00091 | 0.621 | 1.87 | 1 | Exp. | 0.008 | 0.019 | 0.403 | 5.04 | 112 | |

IV | Exp. | 0.420 | 5.13 | 0.082 | 1.64 | 105 | Exp. | 0.00088 | 0.00154 | 0.570 | 4.08 | 172 | Exp. | 0.002 | 0.011 | 0.152 | 1.58 | 116 | |

2014 | I | Exp. | 0.493 | 2.76 | 0.179 | 2.53 | 150 | Exp. | 0.00000 | 0.00127 | 0.001 | 1.49 | 86 | Exp. | 0.000 | −0.006 | 0.000 | 5.88 | 112 |

II | Exp. | 1.420 | 10.70 | 0.133 | 1.64 | 108 | Exp. | 0.00000 | 0.00095 | 0.001 | 1.59 | 145 | Exp. | 0.000 | 0.012 | 0.000 | 2.91 | 114 | |

III | Exp. | 2.119 | 12.39 | 0.171 | 5.73 | 108 | Exp. | 0.00078 | 0.00120 | 0.650 | 2.54 | 65 | Exp. | 0.000 | 0.016 | 0.000 | 2.34 | 117 | |

IV | Exp. | 0.374 | 3.42 | 0.109 | 1.62 | 108 | Exp. | 0.00092 | 0.00184 | 0.498 | 1.73 | 145 | Exp. | 0.000 | 0.017 | 0.000 | 4.79 | 112 | |

Max | 2.680 | 18.24 | 0.367 | 6.47 | 150 | 0.0010 | 0.0037 | 0.684 | 4.08 | 172.0 | 0.008 | 1.500 | 0.403 | 8.98 | 122 | ||||

Min | 0.100 | 2.39 | 0.018 | 1.26 | 26 | 0.0000 | 0.0009 | 0.000 | 1.00 | 0.0 | −0.002 | −0.025 | 0.000 | 1.58 | 12 | ||||

1 | Exp. | 12.1 | 135.6 | 0.089 | 4.29 | 130 | Exp. | 0.00143 | 0.00340 | 0.421 | 3.46 | 101 | Gaus. | 0.000 | 0.115 | 0.001 | 2.40 | 103 | |

4 | Exp. | 101.9 | 308.3 | 0.331 | 4.96 | 93 | Sph. | 0.00097 | 0.00320 | 0.303 | 5.84 | 103 | Gaus. | −0.001 | −0.290 | 0.003 | 4.00 | 68 | |

7 | Sph. | 290.0 | 3422.0 | 0.085 | 4.66 | 93 | Sph. | 0.00119 | 0.00311 | 0.382 | 5.23 | 103 | Gaus. | −0.001 | −0.723 | 0.001 | 4.11 | 73 | |

10 | Sph. | 1.0 | 626.0 | 0.002 | 5.77 | 110 | Exp. | 0.00087 | 0.00278 | 0.313 | 4.47 | 81 | Gaus. | −0.001 | −0.067 | 0.015 | 3.87 | 73 | |

2015 | 1 | Sph. | 29.0 | 434.0 | 0.067 | 2.28 | 86 | Exp. | 0.00160 | 0.00429 | 0.373 | 4.25 | 64 | Gaus. | −0.001 | −0.166 | 0.007 | 4.49 | 100 |

4 | Exp. | 86.7 | 173.6 | 0.499 | 3.86 | 86 | Exp. | 0.00049 | 0.00252 | 0.194 | 2.49 | 62 | Gaus. | −0.001 | −0.111 | 0.009 | 4.48 | 87 | |

7 | Exp. | 231.5 | 551.9 | 0.419 | 2.67 | 71 | Exp. | 0.00103 | 0.00419 | 0.246 | 4.57 | 69 | Gaus. | 0.001 | 0.284 | 0.004 | 3.94 | 87 | |

10 | Exp. | 4.6 | 173.0 | 0.027 | 3.50 | 72 | Exp. | 0.00070 | 0.00273 | 0.256 | 4.77 | 58 | Gaus. | 0.000 | −0.129 | 0.001 | 3.02 | 105 | |

2016 | 1 | Exp. | 28.9 | 112.5 | 0.257 | 3.99 | 115 | Sph. | 0.00246 | 0.00863 | 0.285 | 5.46 | 95 | Gaus. | 0.000 | −0.190 | 0.001 | 3.46 | 90 |

4 | Exp. | 1.0 | 478.7 | 0.002 | 4.87 | 90 | Exp. | 0.00080 | 0.00224 | 0.357 | 5.16 | 66 | Gaus. | −0.001 | −0.181 | 0.006 | 3.38 | 110 | |

7 | Exp. | 145.0 | 2386.0 | 0.061 | 2.97 | 65 | Exp. | 0.00117 | 0.00383 | 0.305 | 4.66 | 74 | Gaus. | 0.000 | 0.359 | 0.000 | 3.61 | 120 | |

10 | Exp. | 320.0 | 1095.0 | 0.292 | 4.53 | 73 | Exp. | 0.00090 | 0.00345 | 0.261 | 4.98 | 78 | Gaus. | −0.084 | −1.240 | 0.068 | 4.28 | 120 | |

2017 | 1 | Exp. | 0.0 | 210.0 | 0.000 | 3.73 | 77 | Sph. | 0.00127 | 0.00813 | 0.156 | 5.05 | 84 | Gaus. | 0.026 | 0.550 | 0.047 | 4.18 | 120 |

4 | Sph. | 1.0 | 1603.0 | 0.001 | 4.35 | 83 | Exp. | 0.00104 | 0.00299 | 0.348 | 4.00 | 106 | Gaus. | −0.001 | −0.512 | 0.002 | 3.32 | 115 | |

7 | Exp. | 392.0 | 1592.0 | 0.246 | 4.17 | 60 | Exp. | 0.00090 | 0.00252 | 0.357 | 4.65 | 90 | Gaus. | 0.001 | 0.572 | 0.002 | 3.88 | 115 | |

10 | Exp. | 125.0 | 1239.0 | 0.101 | 3.75 | 90 | Exp. | 0.00075 | 0.00389 | 0.193 | 4.62 | 90 | Gaus. | 0.001 | 0.648 | 0.002 | 4.79 | 120 | |

Max | 392.0 | 3422.0 | 0.499 | 5.77 | 130 | 0.0025 | 0.0086 | 0.421 | 5.84 | 106.0 | 0.026 | 0.648 | 0.068 | 4.79 | 120 | ||||

Min | 0.0 | 112.5 | 0.000 | 2.28 | 60 | 0.0005 | 0.0022 | 0.156 | 2.49 | 58.0 | −0.084 | −1.240 | 0.000 | 2.40 | 68 |

_{0}—nugget variance, C

_{0}+ C—sill, A—effective range, A

_{z}—anisotropy.

**Table 4.**Performance of inverse distance weighting (IDW), ordinary kriging (OK), and ordinary cokriging (OCK) for estimation of quarterly rainfall in Poland and neighbouring countries.

IDW | Kriging (OK) | Cokriging (OCK) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Quarter | a | SE | R^{2} | b | SE Pre. | a | SE | R^{2} | b | SE Pre. | a | SE | R^{2} | b | SE Pre. |

2010 | I | 0.552 | 0.230 | 0.048 | 47.9 | 35.4 | 0.610 | 0.248 | 0.051 | 42.4 | 36.4 | 1.441 | 0.029 | 0.957 | −43.4 | 7.5 |

II | 1.062 | 0.082 | 0.599 | −19.7 | 88.7 | 1.011 | 0.072 | 0.634 | −1.3 | 87.8 | 1.066 | 0.009 | 0.992 | −16.7 | 12.7 | |

III | 0.983 | 0.194 | 0.183 | 2.3 | 108.9 | 1.030 | 0.216 | 0.166 | −14.4 | 110.1 | 1.318 | 0.019 | 0.977 | −108.8 | 18.3 | |

IV | 0.996 | 0.159 | 0.256 | 6.6 | 43.6 | 1.000 | 0.165 | 0.244 | 7.2 | 44.0 | 1.229 | 0.019 | 0.974 | −31.0 | 8.2 | |

2011 | I | 0.651 | 0.302 | 0.039 | 29.6 | 29.3 | 0.734 | 0.339 | 0.040 | 23.7 | 29.3 | 1.428 | 0.019 | 0.980 | −32.6 | 4.2 |

II | 1.206 | 0.151 | 0.358 | −33.8 | 60.0 | 1.252 | 0.150 | 0.379 | −39.4 | 59.0 | 1.386 | 0.032 | 0.944 | −60.3 | 17.6 | |

III | 1.393 | 0.209 | 0.285 | −103.9 | 74.6 | 1.517 | 0.258 | 0.237 | −136.8 | 77.1 | 1.371 | 0.020 | 0.978 | −95.7 | 13.1 | |

IV | 1.129 | 0.195 | 0.227 | −6.6 | 46.7 | 1.140 | 0.217 | 0.195 | −6.3 | 47.7 | 1.216 | 0.012 | 0.989 | −17.5 | 5.6 | |

2012 | I | 1.051 | 0.296 | 0.104 | −0.5 | 67.9 | 1.100 | 0.325 | 0.095 | −4.4 | 68.3 | 1.428 | 0.020 | 0.980 | −43.4 | 10.2 |

II | 1.257 | 0.196 | 0.267 | −44.2 | 53.8 | 1.206 | 0.210 | 0.226 | −33.3 | 55.2 | 1.188 | 0.011 | 0.991 | −33.2 | 6.0 | |

III | 1.114 | 0.132 | 0.386 | −22.4 | 67.7 | 0.983 | 0.115 | 0.391 | 7.7 | 67.4 | 1.162 | 0.019 | 0.970 | −33.2 | 15.0 | |

IV | 1.102 | 0.181 | 0.245 | −11.1 | 38.3 | 1.119 | 0.200 | 0.200 | −12.0 | 39.0 | 1.266 | 0.016 | 0.981 | −36.6 | 6.1 | |

2013 | I | 1.029 | 0.162 | 0.268 | 1.4 | 54.5 | 1.039 | 0.160 | 0.278 | 1.2 | 54.1 | 1.259 | 0.024 | 0.960 | −33.3 | 12.7 |

II | 1.226 | 0.155 | 0.357 | −61.4 | 66.2 | 1.119 | 0.133 | 0.387 | −29.6 | 64.7 | 1.241 | 0.024 | 0.960 | −62.3 | 16.6 | |

III | 1.428 | 0.203 | 0.306 | −73.7 | 61.4 | 1.388 | 0.192 | 0.317 | −64.3 | 60.8 | 1.435 | 0.030 | 0.953 | −76.8 | 16.0 | |

IV | 1.102 | 0.180 | 0.251 | −6.4 | 43.4 | 1.142 | 0.190 | 0.243 | −9.0 | 43.7 | 1.240 | 0.015 | 0.985 | −23.1 | 6.2 | |

2014 | I | 0.755 | 0.165 | 0.157 | 23.5 | 29.1 | 0.796 | 0.161 | 0.179 | 19.9 | 28.7 | 1.260 | 0.024 | 0.960 | −23.3 | 6.3 |

II | 1.062 | 0.184 | 0.227 | −12.9 | 87.1 | 1.132 | 0.195 | 0.229 | −26.6 | 87.0 | 1.290 | 0.018 | 0.977 | −57.6 | 15.0 | |

III | 1.099 | 0.092 | 0.557 | −29.7 | 80.2 | 1.016 | 0.082 | 0.574 | −3.8 | 78.6 | 1.129 | 0.020 | 0.965 | −33.6 | 22.5 | |

IV | 0.927 | 0.927 | 0.127 | 12.2 | 39.9 | 0.928 | 0.246 | 0.117 | 13.0 | 40.1 | 1.306 | 0.018 | 0.981 | −32.4 | 5.9 | |

Max | 1.428 | 0.927 | 0.599 | 47.9 | 108.9 | 1.517 | 0.339 | 0.634 | 42.4 | 110.1 | 1.441 | 0.032 | 0.992 | −16.7 | 22.5 | |

Min | 0.552 | 0.082 | 0.039 | −103.9 | 29.1 | 0.610 | 0.072 | 0.040 | −136.8 | 28.7 | 1.066 | 0.009 | 0.944 | −108.8 | 4.2 | |

1 | 1.046 | 0.039 | 0.640 | −2.1 | 6.7 | 0.930 | 0.037 | 0.612 | 3.2 | 7.1 | 1.077 | 0.011 | 0.964 | −3.5 | 2.2 | |

4 | 1.021 | 0.048 | 0.538 | −0.9 | 11.4 | 1.004 | 0.047 | 0.571 | −0.2 | 10.9 | 1.162 | 0.023 | 0.864 | −6.9 | 6.2 | |

7 | 1.013 | 0.036 | 0.665 | −1.45 | 29.5 | 0.970 | 0.034 | 0.675 | 2.3 | 29 | 1.087 | 0.018 | 0.901 | −7.4 | 16.0 | |

10 | 1.03 | 0.026 | 0.793 | −1.08 | 9.6 | 0.947 | 0.024 | 0.797 | 1.8 | 9.5 | 1.026 | 0.005 | 0.992 | −1.0 | 1.9 | |

2015 | 1 | 1.058 | 0.042 | 0.636 | −3 | 12.1 | 0.927 | 0.038 | 0.619 | 3.8 | 12.4 | 1.088 | 0.014 | 0.941 | −4.9 | 4.9 |

4 | 1.129 | 0.054 | 0.534 | −3.6 | 9.0 | 1.086 | 0.049 | 0.559 | −2.5 | 8.7 | 1.222 | 0.028 | 0.824 | −6.8 | 5.5 | |

7 | 0.989 | 0.067 | 0.352 | 0.6 | 18.3 | 0.929 | 0.064 | 0.348 | 6.1 | 18.3 | 1.283 | 0.031 | 0.810 | −20.5 | 9.9 | |

10 | 1.07 | 0.029 | 0.777 | −2.5 | 6.2 | 1.013 | 0.024 | 0.815 | −0.5 | 5.6 | 1.053 | 0.009 | 0.971 | −1.9 | 2.3 | |

2016 | 1 | 1.087 | 0.057 | 0.497 | −2.9 | 7.3 | 1.004 | 0.050 | 0.525 | 0.0 | 7.1 | 1.213 | 0.019 | 0.915 | −7.8 | 2.9 |

4 | 1.018 | 0.025 | 0.807 | −0.6 | 9.1 | 0.983 | 0.022 | 0.828 | 0.8 | 8.6 | 1.029 | 0.004 | 0.993 | −1.2 | 1.7 | |

7 | 1.034 | 0.043 | 0.595 | −5.7 | 29.3 | 0.939 | 0.039 | 0.589 | 7.5 | 29.5 | 1.103 | 0.009 | 0.974 | −13.6 | 7.5 | |

10 | 1.05 | 0.036 | 0.678 | −5.3 | 18.6 | 1.051 | 0.032 | 0.734 | −5.4 | 16.9 | 1.146 | 0.016 | 0.924 | −15.6 | 9.0 | |

2017 | 1 | 1.052 | 0.035 | 0.759 | −1.1 | 6.9 | 0.973 | 0.033 | 0.756 | 0.8 | 7.0 | 1.039 | 0.005 | 0.983 | −1.1 | 1.1 |

4 | 1.027 | 0.022 | 0.844 | −2.5 | 13.8 | 0.982 | 0.020 | 0.865 | 1.1 | 12.8 | 1.018 | 0.004 | 0.995 | −1.2 | 2.5 | |

7 | 1.019 | 0.038 | 0.644 | −2.3 | 23.9 | 0.993 | 0.034 | 0.674 | 0.7 | 22.9 | 1.130 | 0.016 | 0.929 | −15.2 | 10.7 | |

10 | 1.09 | 0.041 | 0.645 | −9.1 | 19.5 | 0.996 | 0.035 | 0.672 | 0.7 | 18.7 | 1.101 | 0.010 | 0.967 | −10.8 | 5.9 | |

Max | 1.129 | 0.067 | 0.844 | 0.6 | 29.5 | 1.086 | 0.064 | 0.865 | 7.5 | 29.5 | 1.283 | 0.031 | 0.995 | −1.0 | 16.0 | |

Min | 0.989 | 0.022 | 0.352 | −9.1 | 6.2 | 0.927 | 0.020 | 0.348 | −5.4 | 5.6 | 1.018 | 0.004 | 0.810 | −20.5 | 1.1 |

^{2}—coefficient of determination, SE Pre.—standard error prediction.

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## Share and Cite

**MDPI and ACS Style**

Usowicz, B.; Lipiec, J.; Łukowski, M.; Słomiński, J.
Improvement of Spatial Interpolation of Precipitation Distribution Using Cokriging Incorporating Rain-Gauge and Satellite (SMOS) Soil Moisture Data. *Remote Sens.* **2021**, *13*, 1039.
https://doi.org/10.3390/rs13051039

**AMA Style**

Usowicz B, Lipiec J, Łukowski M, Słomiński J.
Improvement of Spatial Interpolation of Precipitation Distribution Using Cokriging Incorporating Rain-Gauge and Satellite (SMOS) Soil Moisture Data. *Remote Sensing*. 2021; 13(5):1039.
https://doi.org/10.3390/rs13051039

**Chicago/Turabian Style**

Usowicz, Bogusław, Jerzy Lipiec, Mateusz Łukowski, and Jan Słomiński.
2021. "Improvement of Spatial Interpolation of Precipitation Distribution Using Cokriging Incorporating Rain-Gauge and Satellite (SMOS) Soil Moisture Data" *Remote Sensing* 13, no. 5: 1039.
https://doi.org/10.3390/rs13051039