**4. Conclusions**

spatial-resolution feature of medium spatial-resolution images, and carried out a series of studies on spatiotemporal fusion of remotely sensed surface parameters such as surface reflectance [34], land surface temperature [35, 36], vegetation indexes [37], leaf area index [38], and so on. And then Huang et al. [39] reviewed this and presented the systematic achievements in theory and application. From the theoretical basis of spatiotemporal scale conversion fusion (the spatial scale consistency of the time-phase variation model and the time consistency of the spatial downscaling model) to the type division of the spatiotemporal scale conversion fusion algorithms (the algorithms based on features' components, the algorithms based on surface spatial information, the algorithms based on features' temporal change, and the combination algorithms of the ones above), and then to the key problems and challenges encountered in existing research (the imaging geometry and radiation characteristics, differences between multi-source RS images, the complexity of subpixel unmixing models, the complexity of features' temporal change models, etc.), and the possible development trend in future (improvement in the versatility and robustness of the algorithms), he made a detailed and in-depth explanation, so that we have a more comprehensive understanding of the development of spatiotemporal scale conversion fusion research. In fact, in addition to this method, the multi-fractal method has important potential to solve the above problems [21, 40]. The following is an example of NDVI analysis and how to establish a spatiotemporal scale conversion model (or spatiotemporal scaling fusion model)

**3.2 Spatiotemporal scale conversion models of NDVI based on multi-fractal**

As the best indicator of vegetation growth status and vegetation coverage, NDVI has typical phenological characteristics. This means that in the same area where the surface cover type is unchanged, the physiological characteristics and external forms of the plant can change significantly in different growth stages, and this change will be directly reflected in the changes in image spectrum and NDVI of the surface. Furthermore, the NDVI spatial scale conversion model based on RS images of different growth periods (i.e., different phases) will also change. How to effectively reflect the influence of the phase characteristics of RS images on the construction of this model and then construct a more universal NDVI spatial scale conversion model that can be integrated with surface phenological features, namely, NDVI spatiotemporal scale conversion model? This issue has important research value. Kim and Barros [21] proposed the idea of multi-fractal method for multi-temporal remote sensing soil moisture spatial down-scaling model to describe the phase characteristics of soil moisture spatial down-scaling, but did not do

Referring to the existing knowledge, the specific method of establishing the NDVI spatiotemporal scale conversion model is given here: first, analyze the surface condition of the study area, determine the type of the main cover of the study area, and based on its phenological knowledge, select enough low and medium-high spatial resolution images finely corresponding to important "nodes" of vegetation throughout the growing season; secondly, the NDVI spatial downscaling models for different growing stages "nodes" are constructed based on the down-scaling methods such as fractal IFS; third, according to multi-fractal theory and method, using the time phase as a factor in the fractal dimension calculation method, the models corresponding to each growth stages are "fused" to obtain a unified and full growth period NDVI scale conversion model (i.e., NDVI spatiotemporal scale conversion model). At this time, the time phase (i.e., different growth stages) has been

based on multi-fractal theory and method.

**theory and method**

*Fractal Analysis - Selected Examples*

specific research.

**48**

Taking normalized difference vegetation index (NDVI) as an example, the establishment of scaling models based on fractal theory was described and analyzed in the paper. It was concluded that fractal iterated function system was an effective methodology to establish downscaling models for remote sensing land surface parameters such as NDVI and multi-fractal modeling may be a novel methodology to establish spatiotemporal scale conversion models for land surface parameters such as NDVI in the future.

### **Acknowledgements**

This work was supported by the National Natural Science Foundation of China "Coupling of NDVI's up-scaling and down-scaling fusing with ground objects classification" (No. 41601350), the Natural Science Foundation of Fujian Province, China "Research on NDVI's scaling fusing with ground objects classification" (No. 2017J05069), and the "Scientific Research Climbing Plan" Project from Xiamen University of Technology "Spatial Distribution Estimation and Dynamic Monitoring of Soil Organic Matter Based on Multi-source and Heterogeneous Data" (No. XPDKT19010).

*Fractal Analysis - Selected Examples*
