# Geographically Weighted Area-to-Point Regression Kriging for Spatial Downscaling in Remote Sensing

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Generic Formulation

^{2}observations for the latter in the study area G, where F is the ratio between the spatial resolutions of the coarse and fine pixels. Geospatial data can be considered to consist of trends and residuals. Hence, the generic formulation for the prediction of the target variable at pixel ${s}_{j}$ can be expressed as follows:

#### 2.2. ATPRK

^{2}fine pixels ${h}_{k}({s}_{j})$ within coarse pixel ${S}_{i}$. The relationship between the target variable and the covariate at coarse spatial resolution is modeled by linear regression:

#### 2.3. GWATPRK

_{j}) is the weight matrix at pixel S

_{j}. Further details can be found in Ref. [34]. The error variance of GWR in modeling the trend is

## 3. Experimental Design

#### 3.1. Study Area

#### 3.2. Data Description

#### 3.2.1. Ground Measurements of SSM

#### 3.2.2. Brightness Temperature

#### 3.2.3. Coarse SSM Products

#### 3.2.4. MODIS Products

#### 3.3. Process of Experiment Implementation

## 4. Results and Discussion

^{3}·m

^{−3}), mean error (ME) (m

^{3}·m

^{−3}), correlation coefficient (R), and slope (SLOP) of linear regression between ground observations and downscaled predictions. The lower the ME, the lower the RMSE and higher the value of R that would be expected. The closer SLOP is to 1, the better the results would be. This direct validation was applied only in the Naqu area because of the lack of in situ measurements from the upper HBR area during the period of interest. Hence, indirect validation was used in the upper HBR area by adopting the H-Tb as a reference dataset. The R and the histogram-matching method [61], used for the comparisons between the 1 km H-Tb data and the 1 km downscaled predictions, involved calculating the intersection distance (ID) and Kullback–Leibler divergence (K–L). For the cross validation of the downscaled results in both experimental regions, TC analysis was repeated for each of the three datasets for the ascending cases (two cases for ESA CCI) and descending cases (two cases for ESA CCI).

#### 4.1. Downscaled Results

#### 4.2. Direct Validation

^{3}·m

^{−3}, smallest ME absolute value of 0.002 m

^{3}·m

^{−3}, highest R value of 0.772 and better SLOP value of 1.036, the GWATPRK method using the ESACCI_C product resulted in the greatest accuracy. On average, when using the GWATPRK method, the RMSE values decreased by 26.4%, 13.2%, and 13.0% compared with the QRM, ATPRK, and SVR downscaling methods, respectively.

#### 4.3. Indirect Validation

#### 4.4. Cross Validation

^{3}·m

^{−3}) < ESACCI_P (0.033 m

^{3}·m

^{−3}) < FY3B_A (0.048 m

^{3}·m

^{−3}) < AMSR2_A (0.054 m

^{3}·m

^{−3}) < FY3B_D (0.064 m

^{3}·m

^{−3}) < AMSR2_D (0.072 m

^{3}·m

^{−}

^{3}) for the upper HBR area, and are ESACCI_C (0.031 m

^{3}·m

^{−3}) < ESACCI_P (0.037 m

^{3}·m

^{−3}) < FY3B_A (0.045 m

^{3}·m

^{−3}) < SMOS_D (0.047 m

^{3}·m

^{−3}) < FY3B_D (0.048 m

^{3}·m

^{−3}) < AMSR2_A (0.056 m

^{3}·m

^{−3}) < SMOS_A (0.069 m

^{3}·m

^{−}

^{3}) < AMSR2_D (0.070 m

^{3}·m

^{−3}) for the Naqu area. The EASCCI_C produced downscaled predictions with the best accuracy of the various coarse SSM products. This might be attributable to the better quality of the EASCCI_C product, which combines different active and passive SM products.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Locations and elevations of the two experimental regions: the upper HBR area and the Naqu area. The spatial distribution of the ground stations in the Naqu area and the arrangements of the coarse grid pixels for both areas are also shown.

**Figure 2.**Daily SSM values for each coarse surface soil moisture (SSM) product during August 2012 in Naqu area. Dots denote the average value and the bars denote ±1 standard deviation.

**Figure 3.**Daily coverage fractions of remote sensing products (i.e., eight groups of four coarse SSM products and MODIS LST) during August 2012 in two experimental regions: (

**a**) upper HBR area and (

**b**) Naqu area.

**Figure 5.**One kilometer trend predictions of GWATPRK from different coarse SSM products in two experimental regions. (

**a**) Upper HBR area on August 14 2012, and (

**b**) Naqu area on 13 August 2012.

**Figure 6.**Upper HBR area on 14 August 2012. (

**a**) Various 25 km SSM images and downscaled 1 km SSM images derived using the four methods: (

**b**) GWATPRK, (

**c**) QRM, (

**d**) ATPRK, and (

**e**) SVR.

**Figure 7.**Naqu area on 13 August 2012. (

**a**) Various 25 km SSM images and downscaled 1 km SSM images derived using the four methods: (

**b**) GWATPRK, (

**c**) QRM, (

**d**) ATPRK, and (

**e**) SVR.

**Figure 8.**CDF curves for various remotely sensed products on available days within one month in the upstream HBR area: (

**a**) coarse 25 km SSM products and downscaled 1 km SSM predictions derived using the four methods: (

**b**) GWATPRK, (

**c**) QRM, (

**d**) ATPRK, and (

**e**) SVR.

**Figure 9.**Simple Taylor diagram displaying statistical comparisons between 1 km SSM downscaled predictions combining two cases and ground-based measurements for AMSR-2, SMOS, FY-3B, and ESA CCI. The in situ point is for all the validation data.

**Figure 10.**Standard deviations of average errors of 1 km predictions from different coarse SSM products in two experimental regions: (

**a**) upper HBR area and (

**b**) Naqu area.

Data Source | Short Name | Spatial Resolution | Temporal Resolution | Coverage | |
---|---|---|---|---|---|

AMSR-2 | Ascending product | AMSR2_A | 25 km | Daily | Global 2012– |

Descending product | AMSR2_D | ||||

SMOS | Ascending product | SMOS_A | 25 km | Daily | Global 2010– |

Descending product | SMOS_D | ||||

FY-3B | Ascending product | FY3B_A | 25 km | Daily | Global 2011– |

Descending product | FY3B_D | ||||

ESA CCI | Combined product | ESACCI_C | 25 km | Daily | Global 1978–2016 |

Passive product | ESACCI_P |

Study Area | Coarse SSM | Other Variables | Downscaling Method | Validation | ||
---|---|---|---|---|---|---|

Name | Trend | Residual | ||||

Upper HRB area | AMSR-2 (25 km) ESA CCI (25 km) FY-3B (25 km) | LST (1 km/25 km) NDVI (1 km/25 km) BSA (1 km/25 km) Tb (1 km) | GWATPRK | GWR | ATPK | Cross validation using the TC method Indirect validation using Tb data |

QRM | Quadratic regression | Bilinear interpolation | ||||

ATPRK | Ordinary linear regression | ATPK | ||||

SVR | Support vector regression | Bilinear interpolation | ||||

Naqu area | AMSR-2 (25 km) ESA CCI (25 km) FY-3B (25 km) SMOS (25 km) | LST (1 km/25 km) NDVI (1 km/25 km) BSA (1 km/25 km) In situ (1 km) | GWATPRK | GWR | ATPK | Direct validation using ground observations Cross validation using the TC method |

QRM | Quadratic regression | Bilinear interpolation | ||||

ATPRK | Ordinary linear regression | ATPK | ||||

SVR | Support vector regression | Bilinear interpolation |

**Table 3.**Spatial statistics of comparisons between 1 km downscaled predictions and ground-based measurements for the different cases in the Naqu area. (The bold values represent the best performance of all downscaling predictions for each validation index. In each validation index, the values with * mean the best performance of different downscaling predictions from the same satellite-based SSM product.)

AMSR2_A | AMSR2_D | SMOS_A | SMOS_D | FY3B_A | FY3B_D | ESACCI_C | ESACCI_P | ||
---|---|---|---|---|---|---|---|---|---|

RMSE | GWATPRK | 0.114 * | 0.175 * | 0.114 * | 0.083 * | 0.079 * | 0.195 * | 0.056 * | 0.075 * |

QRM | 0.158 | 0.198 | 0.166 | 0.107 | 0.089 | 0.260 | 0.096 | 0.126 | |

ATPRK | 0.140 | 0.188 | 0.148 | 0.087 | 0.087 | 0.233 | 0.073 | 0.078 | |

SVR | 0.154 | 0.176 | 0.147 | 0.093 | 0.088 | 0.245 | 0.061 | 0.079 | |

ME | GWATPRK | 0.050 * | 0.093 * | −0.092 * | −0.023 * | 0.016 * | −0.189* | 0.002 * | 0.003 * |

QRM | 0.081 | 0.171 | −0.134 | −0.028 | 0.030 | −0.205 | −0.040 | 0.034 | |

ATPRK | 0.054 | 0.134 | −0.134 | −0.025 | 0.024 | −0.203 | −0.003 | −0.007 | |

SVR | 0.060 | 0.151 | −0.116 | −0.026 | 0.027 | −0.190 | 0.016 | 0.022 | |

R | GWATPRK | 0.469 * | 0.382 * | 0.449 * | 0.688* | 0.676* | 0.341 * | 0.772 * | 0.699 * |

QRM | 0.346 | 0.311 | 0.342 | 0.612 | 0.439 | 0.315 | 0.668 | 0.399 | |

ATPRK | 0.463 | 0.373 | 0.408 | 0.663 | 0.566 | 0.332 | 0.766 | 0.675 | |

SVR | 0.369 | 0.338 | 0.383 | 0.660 | 0.611 | 0.317 | 0.723 | 0.576 | |

SLOP | GWATPRK | 0.694 * | 0.586 * | 0.675 * | 0.630 * | 0.809 * | 0.534 * | 1.036 * | 0.811 * |

QRM | 0.621 | 0.536 | 0.527 | 0.556 | 0.597 | 0.508 | 0.816 | 0.663 | |

ATPRK | 0.679 | 0.576 | 0.675* | 0.602 | 0.744 | 0.519 | 1.045 | 0.766 | |

SVR | 0.680 | 0.542 | 0.568 | 0.535 | 0.624 | 0.509 | 1.076 | 0.701 |

**Table 4.**Spatial statistics of comparisons between 1 km downscaled predictions and 1 km Tb data for the different cases in the upstream HBR area.

AMSR2_A | AMSR2_D | ESACCI_C | ESACCI_P | ||
---|---|---|---|---|---|

R | GWATPRK | 0.514 | 0.478 | 0.703 | 0.647 |

QRM | 0.398 | 0.386 | 0.501 | 0.412 | |

ATPRK | 0.477 | 0.434 | 0.571 | 0.519 | |

SVR | 0.459 | 0.457 | 0.601 | 0.500 | |

ID | GWATPRK | 0.419 | 0.382 | 0.589 | 0.547 |

QRM | 0.328 | 0.313 | 0.422 | 0.361 | |

ATPRK | 0.347 | 0.332 | 0.469 | 0.416 | |

SVR | 0.431 | 0.351 | 0.511 | 0.482 | |

K-L | GWATPRK | 0.617 | 0.622 | 0.542 | 0.551 |

QRM | 0.686 | 0.700 | 0.631 | 0.647 | |

ATPRK | 0.665 | 0.681 | 0.586 | 0.602 | |

SVR | 0.583 | 0.652 | 0.570 | 0.573 |

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## Share and Cite

**MDPI and ACS Style**

Jin, Y.; Ge, Y.; Wang, J.; Heuvelink, G.B.M.; Wang, L. Geographically Weighted Area-to-Point Regression Kriging for Spatial Downscaling in Remote Sensing. *Remote Sens.* **2018**, *10*, 579.
https://doi.org/10.3390/rs10040579

**AMA Style**

Jin Y, Ge Y, Wang J, Heuvelink GBM, Wang L. Geographically Weighted Area-to-Point Regression Kriging for Spatial Downscaling in Remote Sensing. *Remote Sensing*. 2018; 10(4):579.
https://doi.org/10.3390/rs10040579

**Chicago/Turabian Style**

Jin, Yan, Yong Ge, Jianghao Wang, Gerard B. M. Heuvelink, and Le Wang. 2018. "Geographically Weighted Area-to-Point Regression Kriging for Spatial Downscaling in Remote Sensing" *Remote Sensing* 10, no. 4: 579.
https://doi.org/10.3390/rs10040579