**2. Downscaling of NDVI based on fractal IFS**

#### **2.1 Review of downscaling of RS land surface parameters**

Liang [1] has reviewed several current downscaling methods, including linear decomposition methods and nonlinear statistical decomposition methods, methods for generating continuous regions, normalized difference vegetation index (NDVI) time series decomposition, multi-resolution data fusion, the statistical downscaling method of global climate model products (GCM), etc. Further, Gao et al. [5], Zhu et al. [6], and Huang et al. [7, 8] have done systematic and effective work in the spatiotemporal fusion downscaling of land surface reflectance, which has become a research hot topic. The spectral-spatial feature fusion by Wang et al. [9–12] and Shi and Wang [13] also achieved good results for subpixel mapping. These studies, however, scarcely considered the scale conversion process from the perspective of dynamics, which studies of surface parameter downscaling based on the fractal iterated function system (IFS) have paid attention to.

As a fractal branch of mathematics, because of its complete and rigorous theoretical system, it can systematically study the performance, nature, and causes of multi-scale characteristics of natural phenomena. In the fractal geometry theory system, in addition to the familiar fractal phenomenon description and fractal measurement, the internal causes or dynamic processes of mathematical fractals (interaction, feedback, and iteration, represented by IFS-iteration function system) and the physical causes of statistical fractals (such as critical or abrupt changes) are also important research contents of fractal geometry, and fractal geometry has become a part of nonlinear dynamics research [14]. Although the current research on fractal dynamics has just started, there are still many problems waiting to be solved, but its potential value and significance in dynamics research cannot be denied.

In quantitative remote sensing research, fractal methods are mostly used in the mapping of surface morphology (spatial structure) such as active radar imagery and snow and ocean imagery [15], but it also has important applications in scale conversion research and is further deepened and expanded. The use of fractals for surface parameter scale conversion modeling usually contains two important research components:


The mathematical basis of fractal generation is IFS. Kim and Barros [21] first constructed the *r* function from the dynamic factors (soil sediment content,

*Establishing the Downscaling and Spatiotemporal Scale Conversion Models of NDVI Based on… DOI: http://dx.doi.org/10.5772/intechopen.91359*

vegetation water content) of soil moisture scale conversion and then established the IFS to describe the soil moisture downscaling, and the conversion effect was good. The model can describe the dynamic process of soil moisture scale conversion, which has physical significance and demonstrates the advantages of downscaling surface parameters based on fractal IFS. In general, there is currently little research into the causes of fractal dynamics. In mathematics, the fractal IFS is a continuously iterative calculation based on the whole research object [14], and the RS land surface parameter image is created in units of local pixels. This ensures that the mathematical IFS vertical conversion factor (*r* function) is usually constant [21], while the vertical conversion factor of RS land surface parameters (such as soil moisture) is based on the physical elements of each pixel (such as sandy soil). The amount of space and the vegetation water content varies dynamically and temporally [21]. This is why the IFS function can describe the scale switching dynamics of surface parameters and why the model has certain physical meanings. The vertical conversion factor is used to describe the interscale conversion of surface parameter values and is the key to determining the IFS function. Different surface parameters have different values due to the spatial distribution and scale conversion factors (or dynamic factors), and the vertical conversion factor (*r* function) contains different types of variables and function forms. How to determine the *r* function is the difficulty in determining the IFS function, which is also an important reason why the latter is less frequently applied in descriptions of quantitative RS land surface parameter scale conversion. Therefore, the NDVI downscaling model based on the fractal IFS function can be considered to describe the dynamic process of scale conversion. This research covers a wide area and is of great significance. The following is a description of a preliminary implementation [22].

### **2.2 Methodology**

physical relationship of RS land surface parameter images at different resolutions (scales) and can quantitatively describe scale effects. This paper will also focus on the research progress of the spatial down-scaling and the spatiotemporal scaling.

Liang [1] has reviewed several current downscaling methods, including linear decomposition methods and nonlinear statistical decomposition methods, methods for generating continuous regions, normalized difference vegetation index (NDVI) time series decomposition, multi-resolution data fusion, the statistical downscaling method of global climate model products (GCM), etc. Further, Gao et al. [5], Zhu et al. [6], and Huang et al. [7, 8] have done systematic and effective work in the spatiotemporal fusion downscaling of land surface reflectance, which has become a research hot topic. The spectral-spatial feature fusion by Wang et al. [9–12] and Shi and Wang [13] also achieved good results for subpixel mapping. These studies, however, scarcely considered the scale conversion process from the perspective of dynamics, which studies of surface parameter downscaling based on the fractal

As a fractal branch of mathematics, because of its complete and rigorous theoretical system, it can systematically study the performance, nature, and causes of multi-scale characteristics of natural phenomena. In the fractal geometry theory system, in addition to the familiar fractal phenomenon description and fractal measurement, the internal causes or dynamic processes of mathematical fractals (interaction, feedback, and iteration, represented by IFS-iteration function system) and the physical causes of statistical fractals (such as critical or abrupt changes) are also important research contents of fractal geometry, and fractal geometry has become a part of nonlinear dynamics research [14]. Although the current research on fractal dynamics has just started, there are still many problems waiting to be solved, but its potential value and significance in dynamics research cannot be

In quantitative remote sensing research, fractal methods are mostly used in the mapping of surface morphology (spatial structure) such as active radar imagery and snow and ocean imagery [15], but it also has important applications in scale conversion research and is further deepened and expanded. The use of fractals for surface parameter scale conversion modeling usually contains two important

1.The performance of fractal features, that is, fractal metrics, and also the fractal dimension of the research object. For example, Zhang et al. [16, 17] used the information dimension method to describe the fractal dimension of leaf area index (LAI) scale conversion. Luan et al. [18, 19] and Wu et al. [20] used the similar dimension method to measure the fractal dimension of NDVI and LAI

2.The intrinsic nature of the fractal phenomenon, that is, the dynamics produced, which is the combined effect of multifactor surface effects.

The mathematical basis of fractal generation is IFS. Kim and Barros [21] first constructed the *r* function from the dynamic factors (soil sediment content,

**2. Downscaling of NDVI based on fractal IFS**

*Fractal Analysis - Selected Examples*

iterated function system (IFS) have paid attention to.

denied.

**40**

research components:

scale-up conversion, respectively.

**2.1 Review of downscaling of RS land surface parameters**

How does one build an NDVI downscaling model based on the fractal IFS function? The following points need to be considered: first, how to identify the sensitive factors affecting the spatial distribution and scale effect of NDVI for NDVI; second, how to use this sensitive factor to establish the vertical scale conversion factor *r* function in the IFS and then determine the IFS function to achieve NDVI downscaling; and finally, how to evaluate the downscaled conversion results. The solution incorporating these considerations is described below.

#### *2.2.1 Identify sensitive factors*

According to the above description, water body is an important parameter affecting the spatial distribution and scale effect of NDVI; thus it can be determined that the pixel water parameter is one of the important dynamic factors of NDVI scale conversion. In addition, Wen et al. [23] gave a method for albedo conversion from small-scale to large-scale images and used the pixel topographical influencing factors to correct the converted results, which demonstrated that the method was effective for albedo scale conversion of rugged terrain. Considering the close relationship between the surface reflectivity and the surface albedo, and that the surface reflectance is the basic parameter for calculating NDVI, the topographic factor parameter can be determined as one of the important kinetic factors for NDVI scale conversion. Therefore, the important dynamic factors in NDVI spatial distribution and scale conversion are determined to be the pixel water parameters and topographic factors.

#### *2.2.2 Determine the vertical conversion factor r function and establish the IFS function*

Referring to Kim [21], IFS formula (1), horizontal transformation formula (2), and vertical transformation formula (3) for large-scale surface parameter pixel downscaling are obtained as follows. The IFS formula is calculated by pixel-by-pixel sliding. Get the full image downscaling results:

$$\left.I\text{FS}^{i,j}\right|\_{n,m}(\mathfrak{x}^i,\mathfrak{y}^j,\mathfrak{s}^{\vec{y}}) = \left(p\_n(\mathfrak{x}^i), q\_m(\mathfrak{y}^j), I\_{n,m}(\mathfrak{x}^i,\mathfrak{y}^j,\mathfrak{s}^{\vec{y}})\right),\tag{1}$$

$$\begin{cases} p\_n(\mathbf{x}^i) = \mathbf{x}\_{n-1}^i + a(\mathbf{x}^i - \mathbf{x}\_0^i) \\\\ q\_m(\mathbf{y}^i) = \mathbf{y}\_{m-1}^j + a(\mathbf{y}^j - \mathbf{y}\_0^j) \end{cases} \tag{2}$$

Furthermore,

<sup>1</sup> <sup>¼</sup> *<sup>r</sup>*<sup>1</sup> *xi*

be constructed:

*<sup>n</sup>*�1, *<sup>y</sup> <sup>j</sup> m*�1 *<sup>s</sup>*

*ij*

*Re*

*DOI: http://dx.doi.org/10.5772/intechopen.91359*

*R f*

Therefore, the calculation of the *<sup>r</sup>*1ð*x<sup>i</sup>*

magnitude of *r* are calculated as follows:

*2.2.3 Evaluation of downscaling results*

largely guarantee the accuracy of the result.

**43**

determination of the *r* function (containing *r*<sup>1</sup> *xi*

0,0 � *<sup>r</sup>*<sup>1</sup> *xi*

<sup>1</sup> <sup>¼</sup> *<sup>r</sup>*<sup>1</sup> *<sup>x</sup><sup>i</sup>*

<sup>1</sup> <sup>¼</sup> *<sup>r</sup>*<sup>1</sup> *xi*

*<sup>n</sup>*, *<sup>y</sup> <sup>j</sup> m*�1 *<sup>s</sup>*

*<sup>n</sup>*�1, *<sup>y</sup> <sup>j</sup> m*�1 *<sup>s</sup>*

*<sup>n</sup>*�1, *<sup>y</sup> <sup>j</sup> m*�1 *<sup>s</sup>*

> *Rk* <sup>1</sup> <sup>¼</sup> *<sup>r</sup>*<sup>1</sup> *<sup>x</sup><sup>i</sup>*

*ij*

*Establishing the Downscaling and Spatiotemporal Scale Conversion Models of NDVI Based on…*

*ij*

*ij* 0,0 � *<sup>r</sup>*<sup>1</sup> *xi*

*<sup>n</sup>*, *y <sup>j</sup> m s*

Based on the above sensitivity factors, a vertical conversion factor *r* function can

where *S*water represents the pixel water parameter; *s* represents the topographic information, taking into account the magnitude of the *r* function; the normalized difference water index (NDWI) and slope (calculated from the digital elevation model (DEM) image) represent, respectively, the water body effect and the topographic influence in the pixel; *γ* and *β* are the coefficients of the two parameters, respectively; and *δ* represents the adjustment constant. Two different orders of

For NDVI, the *γ*, *β*, and *δ* coefficients can be calculated by linear regression

Following construction of the *r* function, formulas (1)–(3) can be solved in combination with other known conditions, and NDVI downscaling can be achieved.

In order to obtain more accurate downscaling results, if the resolution of the low resolution image is too different from the resolution of the target resolution image (such as downscaling from 250 m MODIS NDVI to 30 m NDVI), a hierarchical downscaling method will be adopted. First, the low-resolution surface parameter image is downscaled to an intermediate resolution image, and then the intermediate resolution image is further downscaled to the target resolution image, which can

Referring to the study by Kim and Barros [21], the accuracy of the downscaled results can be evaluated using statistical indicators such as the maximum, minimum, variance, and standard deviation (compared to high-resolution NDVI images). Moreover, the histograms of the downscaled NDVI and true NDVI images were drawn and compared, and their correlation coefficient was calculated. With those indexes, the accuracy of the downscaled images and methodology could be validated.

between the high-resolution NDVI image and its NDWI/slope images.

, *y<sup>j</sup>*

(0 ≤*r*<sup>1</sup> ≤1) is used to adjust the NDVI surface roughness. The following treatment focuses on establishing the vertical transformation formula for NDVI, that is, the

*<sup>N</sup>*,0 � *<sup>r</sup>*<sup>1</sup> *<sup>x</sup><sup>i</sup>*

0,0 � *<sup>r</sup>*<sup>1</sup> *xi*

*ij*

*<sup>n</sup>*�1, *<sup>y</sup> <sup>j</sup> m s*

*<sup>n</sup>*, *<sup>y</sup> <sup>j</sup> m*�1 *<sup>s</sup>*

*<sup>n</sup>*�1, *<sup>y</sup> <sup>j</sup> m s*

, *y<sup>j</sup>* and *r*<sup>2</sup> *xi*

*r* ¼ *γ* � *S*water þ *β* � *s* þ *δ*, (12)

*r*<sup>1</sup> ¼ *γ*<sup>1</sup> � *S*water þ *β*<sup>1</sup> � *s* þ *δ*1, (13) *r*<sup>2</sup> ¼ *γ*<sup>2</sup> � *S*water þ *β*<sup>2</sup> � *s* þ *δ*2*:* (14)

*ij*

*ij*

*ij*

0,*<sup>M</sup>* <sup>þ</sup> *<sup>r</sup>*<sup>1</sup> *xi*

*<sup>N</sup>*,*M:* (11)

Þ function is significant, and *r*1ð Þ *n*, *m*

, *y<sup>j</sup>* ).

*<sup>n</sup>*, *y <sup>j</sup> m s*

*<sup>N</sup>*,0, (9)

0,*M*, (10)

*ij <sup>N</sup>*,*M*, (8)

*Rg*

$$I\_{n,m}(\mathbf{x}^i, \mathbf{y}^j, \mathbf{s}^j) = \left(e\_{n,m}\mathbf{x}^i + f\_{n,m}\mathbf{y}^j + \mathbf{g}\_{n,m}\mathbf{x}^i\mathbf{y}^j + r\_1(\mathbf{x}^i, \mathbf{y}^j)\mathbf{s}^{ji} + k\_{n,m}\right) \times r\_2(\mathbf{x}^i, \mathbf{y}^j), \tag{3}$$

where *IFS<sup>i</sup>*,*<sup>j</sup>* � � *<sup>n</sup>*,*<sup>m</sup> xi* , *y<sup>j</sup>* , *s ij* � � represents the surface parameter of the pixel at the ð Þ *<sup>i</sup>*, *<sup>j</sup>* location when the large-scale pixel of the surface parameter is downscaled to the small-scale image of the *<sup>n</sup>* � *<sup>m</sup>* dimension; *xi* , *y<sup>j</sup>* , and *s ij* correspond, respectively, to the *<sup>x</sup>*-direction coordinate *pn <sup>x</sup><sup>i</sup>* � �, the *<sup>y</sup>*-direction coordinate *qm <sup>y</sup><sup>j</sup>* � �, and the surface parameter values *In*,*<sup>m</sup> xi* , *y<sup>j</sup>* , *s ij* � � of the three-dimensional data of the pixel; *xi <sup>n</sup>*�<sup>1</sup> and *xi* <sup>0</sup> represent, respectively, the *x*-direction starting coordinate of the ð Þ *i*, *j* pixel in the *n* � *m* dimensional small-scale image and the *x*-direction starting coordinate of the large-scale pixel; *α* represents the downscaling ratio (small-scale/large-scale, which is less than or equal to 1); *en*,*<sup>m</sup>*, *f <sup>n</sup>*,*<sup>m</sup>*, *gn*,*<sup>m</sup>*, and *kn*,*<sup>m</sup>* are, respectively, functions of the *x* and *y* coordinates of the lower left corner and upper right corner of the large-scale pixel, the downscaled surface parameter data, and the vertical scale conversion surface function; and *r*<sup>1</sup> *xi* , *y<sup>j</sup>* � � and *r*<sup>2</sup> *xi* , *y<sup>j</sup>* � � represent, respectively, the two different vertical conversion factors in the vertical scale conversion surface function. Reference [21] should be consulted for the parameters or factors not represented in the formula, which will not be explained here. Generally, the *pn <sup>x</sup><sup>i</sup>* � � and *qm <sup>y</sup><sup>j</sup>* � � coordinates of the ð Þ *<sup>i</sup>*, *<sup>j</sup>* pixel are obtained by dividing the large-scale pixel equally into 1/*α* parts, and the *In*,*<sup>m</sup> xi* , *y<sup>j</sup>* , *s ij* � � calculation is the key. In formula (3), *r*<sup>2</sup> *xi* , *y<sup>j</sup>* � � is the same as the *r*<sup>1</sup> *xi* , *y<sup>j</sup>* � � function, but their argument coefficients are different.

For NDVI, *gn*,m, *en*,m, *f <sup>n</sup>*,m, and *kn*,m represent the functions of the ð Þ *n*, *m* pixel of the downscaled NDVI image. Based on the special downscaled NDVI 3D values of the four corner pixels, *gn*,m, *en*,m, *f <sup>n</sup>*,m, and *kn*,m can be calculated as formulas (4)–(11):

$$\mathcal{g}\_{n,m} = \frac{s\_{n-1,m-1} - s\_{n-1,m} - s\_{n,m-1} + s\_{n,m} - R\_1^g}{\varkappa\_0 \mathcal{y}\_0 - \varkappa\_N \mathcal{y}\_0 - \varkappa\_0 \mathcal{y}\_M + \varkappa\_N \mathcal{y}\_M},\tag{4}$$

$$e\_{n,\mathbf{m}} = \frac{s\_{n-1,m-1} - s\_{n,m-1} - g\_{n,\mathbf{m}}(\mathbf{x}\_0 y\_0 - \mathbf{x}\_N y\_0) - R\_1^\epsilon}{\mathbf{x}\_0 - \mathbf{x}\_N},\tag{5}$$

$$f\_{n,\mathbf{m}} = \frac{s\_{n-1,m-1} - s\_{n-1,m} - g\_{n,\mathbf{m}}(\boldsymbol{\kappa}\_0 \boldsymbol{y}\_0 - \boldsymbol{\kappa}\_0 \boldsymbol{y}\_M) - R\_1^f}{\boldsymbol{y}\_0 - \boldsymbol{y}\_M},\tag{6}$$

$$k\_{n, \mathbf{m}} = \varepsilon\_{n, \mathbf{m}} - \varepsilon\_{n, \mathbf{m}} \mathbf{x}\_N - f\_{n, \mathbf{m}} \mathbf{y}\_M - \mathbf{g}\_{n, \mathbf{m}} \mathbf{x}\_N \mathbf{y}\_M - R\_1^k. \tag{7}$$

*Establishing the Downscaling and Spatiotemporal Scale Conversion Models of NDVI Based on… DOI: http://dx.doi.org/10.5772/intechopen.91359*

Furthermore,

*2.2.2 Determine the vertical conversion factor r function and establish the IFS function*

sliding. Get the full image downscaling results:

*ij* � � <sup>¼</sup> *en*,*mx<sup>i</sup>* <sup>þ</sup> *<sup>f</sup> <sup>n</sup>*,*<sup>m</sup>y<sup>j</sup>* <sup>þ</sup> *gn*,*<sup>m</sup>x<sup>i</sup>*

small-scale image of the *<sup>n</sup>* � *<sup>m</sup>* dimension; *xi*

conversion surface function; and *r*<sup>1</sup> *xi*

pixel equally into 1/*α* parts, and the *In*,*<sup>m</sup> xi*

, *y<sup>j</sup>* � � is the same as the *r*<sup>1</sup> *xi*

, *y<sup>j</sup>* , *s*

*IFSi*,*<sup>j</sup>* � � *<sup>n</sup>*,*<sup>m</sup> xi* , *yj* , *s*

*Fractal Analysis - Selected Examples*

*In*,*<sup>m</sup> xi*

*xi*

(3), *r*<sup>2</sup> *xi*

**42**

are different.

, *yj* , *s*

where *IFS<sup>i</sup>*,*<sup>j</sup>*

parameter values *In*,*<sup>m</sup> xi*

� � *<sup>n</sup>*,*<sup>m</sup> xi* , *y<sup>j</sup>* , *s*

Referring to Kim [21], IFS formula (1), horizontal transformation formula (2), and vertical transformation formula (3) for large-scale surface parameter pixel downscaling are obtained as follows. The IFS formula is calculated by pixel-by-pixel

*ij* � � <sup>¼</sup> *pn <sup>x</sup><sup>i</sup>* � �, *qm <sup>y</sup><sup>j</sup>* � �,*In*,*<sup>m</sup> <sup>x</sup><sup>i</sup>*

� �

location when the large-scale pixel of the surface parameter is downscaled to the

the *<sup>x</sup>*-direction coordinate *pn <sup>x</sup><sup>i</sup>* � �, the *<sup>y</sup>*-direction coordinate *qm <sup>y</sup><sup>j</sup>* � �, and the surface

<sup>0</sup> represent, respectively, the *x*-direction starting coordinate of the ð Þ *i*, *j* pixel in the *n* � *m* dimensional small-scale image and the *x*-direction starting coordinate of the large-scale pixel; *α* represents the downscaling ratio (small-scale/large-scale, which is less than or equal to 1); *en*,*<sup>m</sup>*, *f <sup>n</sup>*,*<sup>m</sup>*, *gn*,*<sup>m</sup>*, and *kn*,*<sup>m</sup>* are, respectively, functions of the *x* and *y* coordinates of the lower left corner and upper right corner of the large-scale pixel, the downscaled surface parameter data, and the vertical scale

, *y<sup>j</sup>* � � and *r*<sup>2</sup> *xi*

, *y<sup>j</sup>* , *s*

For NDVI, *gn*,m, *en*,m, *f <sup>n</sup>*,m, and *kn*,m represent the functions of the ð Þ *n*, *m* pixel of the

*x*0*y*<sup>0</sup> � *xNy*<sup>0</sup> � *x*0*yM* þ *xNyM*

*x*<sup>0</sup> � *xN*

*y*<sup>0</sup> � *yM*

*kn*,m <sup>¼</sup> *sn*,*<sup>m</sup>* � *en*,m*xN* � *<sup>f</sup> <sup>n</sup>*,m *yM* � *gn*,m*xNyM* � *<sup>R</sup><sup>k</sup>*

downscaled NDVI image. Based on the special downscaled NDVI 3D values of the four corner pixels, *gn*,m, *en*,m, *f <sup>n</sup>*,m, and *kn*,m can be calculated as formulas (4)–(11):

*gn*,m <sup>¼</sup> *sn*�1,*m*�<sup>1</sup> � *sn*�1,*<sup>m</sup>* � *sn*,*m*�<sup>1</sup> <sup>þ</sup> *sn*,*<sup>m</sup>* � *<sup>R</sup><sup>g</sup>*

*en*,m <sup>¼</sup> *sn*�1,*m*�<sup>1</sup> � *sn*,*m*�<sup>1</sup> � *gn*,m *<sup>x</sup>*0*y*<sup>0</sup> � *xNy*<sup>0</sup>

*<sup>f</sup> <sup>n</sup>*,m <sup>¼</sup> *sn*�1,*m*�<sup>1</sup> � *sn*�1,*<sup>m</sup>* � *gn*,m *<sup>x</sup>*0*y*<sup>0</sup> � *<sup>x</sup>*0*yM*

the two different vertical conversion factors in the vertical scale conversion surface function. Reference [21] should be consulted for the parameters or factors not represented in the formula, which will not be explained here. Generally, the *pn <sup>x</sup><sup>i</sup>* � � and *qm <sup>y</sup><sup>j</sup>* � � coordinates of the ð Þ *<sup>i</sup>*, *<sup>j</sup>* pixel are obtained by dividing the large-scale

*<sup>n</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>α</sup> xi* � *<sup>x</sup><sup>i</sup>*

*<sup>m</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>α</sup> <sup>y</sup><sup>j</sup>* � *<sup>y</sup> <sup>j</sup>*

*<sup>y</sup><sup>j</sup>* <sup>þ</sup> *<sup>r</sup>*<sup>1</sup> *xi*

, *y<sup>j</sup>*

0 � �

0

, *y<sup>j</sup>* � �*s*

*ij* � � represents the surface parameter of the pixel at the ð Þ *<sup>i</sup>*, *<sup>j</sup>*

, and *s*

*ij* � � of the three-dimensional data of the pixel; *xi*

( *pn xi* � � <sup>¼</sup> *<sup>x</sup><sup>i</sup>*

*qm <sup>y</sup><sup>i</sup>* � � <sup>¼</sup> *<sup>y</sup> <sup>j</sup>*

, *yj* , *s ij* � � � � , (1)

*ij* <sup>þ</sup> *kn*,*<sup>m</sup>*

� � , (2)

� *<sup>r</sup>*<sup>2</sup> *xi*

*ij* correspond, respectively, to

, *y<sup>j</sup>* � � represent, respectively,

*ij* � � calculation is the key. In formula

1

1

1

, (4)

, (5)

, (6)

<sup>1</sup> *:* (7)

, *y<sup>j</sup>* � � function, but their argument coefficients

� � � *<sup>R</sup><sup>e</sup>*

� � � *<sup>R</sup><sup>f</sup>*

, *y<sup>j</sup>* � �, (3)

*<sup>n</sup>*�<sup>1</sup> and

$$R\_1^g = r\_1(\boldsymbol{\kappa}\_{n-1}^j, \boldsymbol{y}\_{m-1}^j)\boldsymbol{s}\_{0,0}^{\vec{\boldsymbol{y}}} - r\_1(\boldsymbol{\kappa}\_n^i, \boldsymbol{y}\_{m-1}^j)\boldsymbol{s}\_{N,0}^{\vec{\boldsymbol{y}}} - r\_1(\boldsymbol{\kappa}\_{n-1}^i, \boldsymbol{y}\_m^j)\boldsymbol{s}\_{0,M}^{\vec{\boldsymbol{y}}} + r\_1(\boldsymbol{\kappa}\_n^i, \boldsymbol{y}\_m^j)\boldsymbol{s}\_{N,M}^{\vec{\boldsymbol{y}}}.\tag{8}$$

$$R\_1^\epsilon = r\_1(\boldsymbol{\pi}\_{n-1}^i, \boldsymbol{y}\_{m-1}^j)\boldsymbol{s}\_{0,0}^{\vec{\boldsymbol{y}}} - r\_1(\boldsymbol{\pi}\_n^i, \boldsymbol{y}\_{m-1}^j)\boldsymbol{s}\_{N,0}^{\vec{\boldsymbol{y}}},\tag{9}$$

$$R\_1^f = r\_1(\boldsymbol{\pi}\_{n-1}^i, \boldsymbol{\mathcal{y}}\_{m-1}^j)\boldsymbol{s}\_{0,0}^{\boldsymbol{j}} - r\_1(\boldsymbol{\pi}\_{n-1}^i, \boldsymbol{\mathcal{y}}\_m^j)\boldsymbol{s}\_{0,M}^{\boldsymbol{j}}.\tag{10}$$

$$R\_1^k = r\_1(\boldsymbol{x}\_n^i, \boldsymbol{y}\_m^j) \boldsymbol{s}\_{N,M}^{ij}. \tag{11}$$

Therefore, the calculation of the *<sup>r</sup>*1ð*x<sup>i</sup>* , *y<sup>j</sup>* Þ function is significant, and *r*1ð Þ *n*, *m* (0 ≤*r*<sup>1</sup> ≤1) is used to adjust the NDVI surface roughness. The following treatment focuses on establishing the vertical transformation formula for NDVI, that is, the determination of the *r* function (containing *r*<sup>1</sup> *xi* , *y<sup>j</sup>* and *r*<sup>2</sup> *xi* , *y<sup>j</sup>* ).

Based on the above sensitivity factors, a vertical conversion factor *r* function can be constructed:

$$
\sigma = \chi \times \mathbb{S}\_{\text{water}} + \beta \times \mathfrak{s} + \delta,\tag{12}
$$

where *S*water represents the pixel water parameter; *s* represents the topographic information, taking into account the magnitude of the *r* function; the normalized difference water index (NDWI) and slope (calculated from the digital elevation model (DEM) image) represent, respectively, the water body effect and the topographic influence in the pixel; *γ* and *β* are the coefficients of the two parameters, respectively; and *δ* represents the adjustment constant. Two different orders of magnitude of *r* are calculated as follows:

$$r\_1 = \gamma\_1 \times \mathbb{S}\_{\text{water}} + \beta\_1 \times \mathfrak{s} + \delta \mathfrak{s},\tag{13}$$

$$r\_2 = \gamma\_2 \times \mathbb{S}\_{\text{water}} + \beta\_2 \times \mathfrak{s} + \delta\_2. \tag{14}$$

For NDVI, the *γ*, *β*, and *δ* coefficients can be calculated by linear regression between the high-resolution NDVI image and its NDWI/slope images.

Following construction of the *r* function, formulas (1)–(3) can be solved in combination with other known conditions, and NDVI downscaling can be achieved.

#### *2.2.3 Evaluation of downscaling results*

In order to obtain more accurate downscaling results, if the resolution of the low resolution image is too different from the resolution of the target resolution image (such as downscaling from 250 m MODIS NDVI to 30 m NDVI), a hierarchical downscaling method will be adopted. First, the low-resolution surface parameter image is downscaled to an intermediate resolution image, and then the intermediate resolution image is further downscaled to the target resolution image, which can largely guarantee the accuracy of the result.

Referring to the study by Kim and Barros [21], the accuracy of the downscaled results can be evaluated using statistical indicators such as the maximum, minimum, variance, and standard deviation (compared to high-resolution NDVI images). Moreover, the histograms of the downscaled NDVI and true NDVI images were drawn and compared, and their correlation coefficient was calculated. With those indexes, the accuracy of the downscaled images and methodology could be validated.
