# Analysis of Multifractal and Organization/Order Structure in Suomi-NPP VIIRS Normalized Difference Vegetation Index Series of Wildfire Affected and Unaffected Sites by Using the Multifractal Detrended Fluctuation Analysis and the Fisher–Shannon Analysis

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## Abstract

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## 1. Introduction

_{X}), being defined through an exponential transformation of SE, the Fisher–Shannon (FS) information plane can be derived, where the discrimination of different dynamics governing the same time series could be carried out. The FS information plane was applied, for instance, to investigate the physical quantities of Langevin equation of friction in earthquake rupture processes [29] and the seismic waves of strong earthquakes [30]. In this paper, the satellite NDVI time series of Suomi-NPP VIIRS that were extracted from four different study sites were analyzed using the MFDFA and FS to investigate the changes of vegetation dynamics that are induced by the Camp Fire (California, USA, 2018). The rest of the paper is structured, as follows: the satellite data, the sites and the statistical methods are described in Section 2. Section 3 and Section 4 present the results of the data analysis and their discussion, respectively, while Section 5 summarizes the conclusions.

## 2. Materials and Methods

#### 2.1. Data

_{NIR}refers to the atmospherically corrected reflectance of the near-infrared band, while ρ

_{red}represents that of the visible red band.

_{d}) series instead of the original NDVI series for the purpose of eliminating the effects of seasonal variations in the NDVI time series. The departure data NDVI

_{d}are widely used in the satellite data processing and filtering [19,35,36,37]. The NDVI

_{d}is calculated, as follows:

_{NDVI}is the average of all samples in all available years that were measured in the same calendar day.

#### 2.2. Study Sites

#### 2.3. Methods

#### 2.3.1. Multifractal Detrended Fluctuation Analysis

_{ave}its mean, the profile y(i) is calculated by integration:

_{m}segments, in order not to ignore this portion. For each of the 2N

_{m}segments, the polynomial local trend is calculated by a least square fit, and the following variance is obtained:

_{ν}(i) is the fitting polynomial of p-th degree in the segment ν. The fitting polynomial of degree p eliminates all of the trends in the profile of order up to p, thus up to p − 1 in the original series [22].

_{q}(m) is computed averaging over all the segments, as follows:

_{q}(m), the moment order q can assume any real value other than zero, while for q = 0, the fluctuation function is calculated by a logarithmic averaging procedure:

_{q}(m) increases with m as a power-law if the series is characterized by long-range correlations:

_{q}(m) on log-log scales and fitting it by a least square method versus m, the slope of the fitting line is the so-called generalized Hurst exponent h

_{q}, which represents the scaling properties of the series at a certain moment order q. In particular, for stationary series, h

_{2}is the Hurst exponent H. If the series is monofractal, the exponent h

_{q}are nearly constant with q, which suggests that the scaling behavior of the small and large variations is approximately identical; however, if the series is multifractal, the exponent h

_{q}decreases with q, which suggests that the small and large fluctuations scale differently and the series has a more complex structure.

#### 2.3.2. Fisher–Shannon Analysis

_{X}that is commonly utilized in statistical analysis:

_{X}are interrelated, which suggests that a better description of the dynamics of a time series would be given by using jointly both the measures. The equality stands in case of Gaussian processes. The use of both measures provides a more complete description of the time series, since the FIM focuses on the local properties of the probability density function, while the Shannon entropy on the global properties. It was also shown in Martin et al. (1999) [43] that FIM allowed for the detection of some non-stationary behavior in situations where the Shannon entropy showed a limited dynamics.

_{X}depends on the probability density function, attention has to be paid to its good estimation. In this study, we used the kernel-based approach to estimate f(x), which has been shown to have a better performance than the discrete-based approach in calculating the value of FIM and SE for the Gaussian distributed series [44]. The kernel-based approach for estimating the probability density function is based on the kernel density estimator technique [45,46]:

_{X}and FIM. For scalar signals, the line FIM·N

_{X}= 1 divides the FS information plane into two parts, and each signal is represented by a point that lies exclusively in the half-space of FIM·N

_{X}> 1.

## 3. Results

_{d}that we analyzed by the MFDFA and the FS method. Figure 2 shows, as an example, the NDVI and the NDVI

_{d}of some pixels of each site. Although the NDVI of needleleaf forests (Figure 2a,c) appears to be higher than that of woody savannas (Figure 2e,g), however the departure NDVI

_{d}of both vegetation covers do not show significant difference, thus confirming the effectiveness of the seasonal removal by the procedure of Equation (2). Furthermore, even by visual inspection, it is clearly seen that both NDVI and NDVI

_{d}drop sharply down at the time of Camp Fire occurrence, towards increase, then, gradually through time to the pre-fire conditions.

_{q}. Figure 4 shows as an example the fluctuation functions of the NDVI

_{d}of pixel P1 of site L3 for q ranging from −5 to 5.

_{q}that were calculated for all NDVI

_{d}series. Most of the h

_{q}decrease with the increase of the moment order q for all of the pixels; however, for a few pixels (P1, P2, P3, and P8 of site L4), h

_{q}fluctuates around the average value. We applied the MFDFA to one hundred shuffles for each NDVI

_{d}series and calculated their h

_{q}in order to recognize the source of multifractality (long-range correlations or type of distribution). Since the shuffling just removes the correlations but preserves the distribution of the series, we can discriminate between the multifractality due to the existence of long-range correlations (the h

_{q}of the shuffles fluctuates around the value of 0.5) or to the probability density function of the series (the h

_{q}of the shuffles behave in nearly the same manner as those of the original series). Figure 6 shows, as an example, the generalized Hurst exponents h

_{q}(black dots) of NDVI

_{d}series and the 95% confidence band (red dotted lines) of the h

_{q}of the shuffles for the pixel P7 of site L1; we can see that the h

_{q}of the original series are well beyond the 95% confidence band of the shuffles, whose generalized Hurst exponents, however, slightly decrease with the moment order q; this indicates that the multifractality of the NDVI

_{d}series could be due to both the presence of long-range correlations in the series and the distribution of the values.

_{q}-range) that is the difference between the maximum and the minimum generalized Hurst exponent to quantify the multifractality. Figure 7 shows the h

_{q}-range of all the analyzed NDVI

_{d}series. From a visual inspection, for evergreen needleleaf forests, the hq-range of burned and unburned sites is approximately the same (Figure 7a), while a certain difference seems to appear in the case of savannas (Figure 7b), indicating that the wildfire could have changed the multifractal characteristics of savannah, but not for evergreen needleleaf vegetation. In order to check whether the averages of h

_{q}, <h

_{q}>, are significantly different, we applied the two sample Student’s t-test to the group of h

_{q}calculated for L1 and L2 and for L3 and L4. We found that at 95% confidence, L1 and L2 are not significantly distinguishable (p-value = 0.381106), while L3 and L4 are significantly distinguishable (p-value = 0.010356), which indicates that the fire-affected and fire-unaffected sites are distinguishable for savanna vegetation covers, and not for evergreen needleleaf forests.

_{q}-range is 0.15 for FGN03, 0.18 for FGN05, and 0.16 for FGN08; the small value of the h

_{q}-range confirms the monofractal character of the three time series. Furthermore, the generalized Hurst exponents of FGN03 and FGN08 are well beyond the 95% confidence band that is based on the shuffles, and this strengthens the robustness of the MFDFA in detecting multifractality, even in short time series.

_{d}; thus, we also analyzed the mean NDVI

_{d}, <NDVI

_{d}>, which is the mean of NDVI

_{d}over the nine pixels for each site, and that might represent the behavior of vegetation in a site in a more global manner.

_{q}and the h

_{q}-range of the <NDVI

_{d}> of the four sites. A general pattern seems to be evidenced from the analysis of <NDVI

_{d}>; in fact, for both the evergreen needleleaf forests and woody savannas, the <NDVI

_{d}> is characterized by a value of h

_{q}-range for burned sites (L1 and L3) larger than that of unburned sites (L2 and L4), although for savanna covers the difference between the burned and unburned sites is much larger.

_{d}series and the results are shown in the Fisher–Shannon information plane (Figure 10). It can be seen that a clear separation exists between the burned and unburned sites for both vegetation covers. The two sample Student’s t-test for the N

_{X}and FIM indicates a significant difference between the two groups of the obtained parameters of burned and unburned sites (p-value < 0.05) (Table 1).

## 4. Discussion

_{X}), by which the discrimination between burned and unburned sites was investigated.

_{q}-range is used to quantify the degree of multifractality of a series. A larger multifractality indicates that the series is characterized by a larger heterogeneity that means the large fluctuations of the series scale differently from the small fluctuations since the MFDFA highlights the scaling behavior of the time dynamics of a series. The NDVI of unburned sites of savannas type are featured by a relatively small value of h

_{q}-range, which indicated that the NDVI is rather monofractal and the scaling behavior of the large and small variations is nearly the same; the vegetation is probably only subjected to fluctuations induced by the seasonal and climatic effects. The NDVI of fire-affected sites of woody savannas covers appear, instead, more heterogeneous, suggesting that small and large fluctuations scale differently; the wildfire impacted on this vegetation that responded to such stress by intermittently fluctuating and thus increasing its multifractality. Such a difference between burned and unburned sites covered by woody savannas becomes much more evident when analyzing the <NDVI

_{d}>, which might more globally represent the status of vegetation of a site, filtering out the local variability that could affect single pixels. The <NDVI

_{d}> of woody savannas, in fact, allows for very clearly discriminating between burned and unburned sites, with the last being signaled by a lower multifractality degree. Additionally, the <NDVI

_{d}> of unburned sites of evergreen needleleaf forests is characterized by a relatively lower multifractalty degree than burned sites, although no significant difference has been found between the single pixels of fire-affected and fire-unaffected sites. Such not so clear multifractal difference between burned and unburned sites that are covered by evergreen needleleaf forests could be due to the fact that fire mainly affects the grass, whose vegetation fine texture makes it more sensitive to the burning than is true of leaves and stems of forests, as was observed in [50].

_{X}and larger FIM values, whereas those of the unburned sites are characterized by larger N

_{X}and smaller FIM in agreement with previous researches [21]. A larger FIM and a smaller N

_{X}quantifies a larger organization and less disorder of the series, where, for larger organization and less disorder, we mean that the distribution of the values of the series is rather peaked, and some values are more frequent than others. Since the vegetation undergoes the disturbance of wildfire, the sudden decrease of NDVI soon after the wildfire and its gradual increase during the post-fire vegetation recovery makes the distribution of the values less uniform than that characterizing the NDVI of the unburned sites; thus, the vegetation does not uniformly fluctuate and the FIM and N

_{X}are larger and smaller, respectively.

## 5. Conclusions

_{q}-range), Shannon entropy power (N

_{X}), and Fisher Information Measure (FIM), were utilized to measure the degree of heterogeneity and organization of the vegetation series in conjunction with their separating ability to discriminate fire-affected and fire-unaffected pixels.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Nomenclatures

NDVI | Normalized Difference Vegetation Index |

MFDFA | multifractal detrended fluctuation analysis |

FS | Fisher-Shannon |

VIIRS | Visible Infrared Imaging Radiometer Suite |

Suomi-NPP | Suomi National Polar-Orbiting Partnership |

MODIS | Moderate Resolution Imaging Spectroradiometer |

EOS | Earth Observing System |

VI | vegetation indices |

ROI | region of interest |

DFA | Detrended Fluctuation Analysis |

FIM | Fisher Information Measure |

SE | Shannon entropy |

N_{X} | Shannon entropy power |

BRDF | bidirectional reflectance distribution function |

SDS | science data sets |

NDVI_{d} | departure NDVI |

PG&E | Pacific Gas and Electric Co. |

IGBP | International Geosphere-Biosphere Programme |

h_{q} | Generalized Hurst exponents |

h_{q}-range | range of the generalized Hurst exponent |

FGN | Fractional Gaussian Noise |

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**Figure 2.**Time series of NDVI and NDVI

_{d}of an example pixel for each study site. The NDVI of L1-P6, L2-P4, L3-P3 and L4-P7 are shown in (

**a**), (

**c**), (

**e**), and (

**g**), while the NDVI

_{d}of which are shown in (

**b**), (

**d**), (

**f**), and (

**h**).

**Figure 3.**q-th order fluctuation function with the polynomial degree p ranging from 1 to 5 of NDVI

_{d}for the pixel P8 of site L1. (

**a**) q = −5 and (

**b**) q = 5.

**Figure 4.**Fluctuation functions with q ranging from −5 to 5 of NDVI

_{d}for the pixel P1 of site L3.

**Figure 5.**Generalized Hurst exponents h

_{q}of NDVI

_{d}for all the pixels of the four study sites L1 (

**a**), L2 (

**b**), L3 (

**c**), and L4 (

**d**).

**Figure 6.**Generalized Hurst exponents h

_{q}of NDVI

_{d}for the pixel P7 of site L1 (black dots) and the 95% confidence band of the h

_{q}of the shuffles (red dotted lines).

**Figure 7.**Range of the generalized Hurst exponent (h

_{q}-range) of NDVI

_{d}for (

**a**) sites L1, L2 and (

**b**) sites L3, L4.

**Figure 8.**Generalized Hurst exponent of the Fractional Gaussian Noise simulations with three different Hurst exponent values of (

**a**) 0.3, (

**b**) 0.5, and (

**c**) 0.8.

**Figure 10.**Fisher–Shannon information plane for the (

**a**) NDVI

_{d}and (

**b**) <NDVI

_{d}> comparisons of L1 and L2, (

**c**) NDVI

_{d}, and (

**d**) <NDVI

_{d}> comparisons of L3 and L4.

L1–L2 | L3–L4 | |
---|---|---|

N_{X} | 1.4743E−14 | 0.000431 |

FIM | 0.000007 | 0.001475 |

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**MDPI and ACS Style**

Ba, R.; Song, W.; Lovallo, M.; Lo, S.; Telesca, L. Analysis of Multifractal and Organization/Order Structure in Suomi-NPP VIIRS Normalized Difference Vegetation Index Series of Wildfire Affected and Unaffected Sites by Using the Multifractal Detrended Fluctuation Analysis and the Fisher–Shannon Analysis. *Entropy* **2020**, *22*, 415.
https://doi.org/10.3390/e22040415

**AMA Style**

Ba R, Song W, Lovallo M, Lo S, Telesca L. Analysis of Multifractal and Organization/Order Structure in Suomi-NPP VIIRS Normalized Difference Vegetation Index Series of Wildfire Affected and Unaffected Sites by Using the Multifractal Detrended Fluctuation Analysis and the Fisher–Shannon Analysis. *Entropy*. 2020; 22(4):415.
https://doi.org/10.3390/e22040415

**Chicago/Turabian Style**

Ba, Rui, Weiguo Song, Michele Lovallo, Siuming Lo, and Luciano Telesca. 2020. "Analysis of Multifractal and Organization/Order Structure in Suomi-NPP VIIRS Normalized Difference Vegetation Index Series of Wildfire Affected and Unaffected Sites by Using the Multifractal Detrended Fluctuation Analysis and the Fisher–Shannon Analysis" *Entropy* 22, no. 4: 415.
https://doi.org/10.3390/e22040415