# Multifractal Analysis of MODIS Aqua and Terra Satellite Time Series of Normalized Difference Vegetation Index and Enhanced Vegetation Index of Sites Affected by Wildfires

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Preprocessing

_{NIR}denotes the atmospherically corrected reflectance in the spectral bands of NIR (near infrared) and ρ

_{red}is that of the visible red band. The reflectance values are derived from the satellite data. The vegetation area affected by fires produces an evident decrease of the NDVI, since the reflectance of the NIR band decreases and the reflectance of the red band increases [31].

_{blue}represents the atmospherically corrected reflectance of the visible blue spectral domain, and the coefficients G, C

_{1}, C

_{2}, and L are set to 2.5, 6, 7.5, and 1, respectively, according to the research in [30,32], among which L is the canopy background adjustment to solve the transmission of nonlinear, differential NIR, and red radiant through a canopy, C

_{1}and C

_{2}are the coefficients of the aerosol resistance term, the blue band is used to correct the aerosol influences of the red band, and G is the gain factor. Since the atmospheric aerosol scattering in the visible blue band is higher than that in the visible red band, the reflectance difference between these two bands can be used to stabilize the index value at different aerosol concentrations [30]. Additionally, the EVI can remove the influences of vegetation canopy and background soil variations using the equation formulation with the adjustment coefficients.

_{r}and EVI

_{r}series corresponding to the NDVI and EVI series shown in Figure 2, after removing the periodicities significant at 95% (as explained above). It can be found that compared to the NDVI and EVI time series, the periodic seasonal variations of the residual series were rather reduced, especially for fire-unaffected sites and those affected by one fire. The immediate changes provoked by fires were, however, retained. The further statistical analyses were performed on the NDVI

_{r}and EVI

_{r}series.

#### 2.2. The Multifractal Detrended Fluctuation Analysis (MFDFA)

_{ave}, by integrating it, we obtain the profile y(i):

_{m}= int(N/m) contiguous boxes of identical size m. In case N is not a multiple of m, a short part of the series could remain at the end; thus, the same procedure is applied from the end of the profile y(i). Then, in each of the 2N

_{m}obtained segments, a least square method is performed to fit the profile with a polynomial, and the following variance is calculated:

_{ν}(i) is a p-degree polynomial that fits the profile in the box ν; and removes the trends of order until p in the profile, and until p − 1 in the original time series.

_{q}(m) is computed as:

_{q}(m) increases with m. If the series is characterized by long-range power-law correlations, the F

_{q}(m) increases with m as a power-law:

_{q}is called the generalized Hurst exponent. For q = 0, F

_{0}(m) is calculated as follows:

_{0}is obtained. For q = 2, the MFDFA becomes the DFA and the exponent h

_{2}is the scaling exponent that is estimated through the DFA. The exponent h

_{2}can be used to quantitatively characterize the persistence of a time series: if it is lower than 0.5, the series is anti-persistent; if it is higher than 0.5, the series is persistent; if it is equal to 0.5, the series is random.

_{q}is nearly constant with q, the series is called monofractal, indicating that the scaling behavior of the small and large fluctuations is approximately identical. If the small and large fluctuations have different scaling behaviors, the h

_{q}decreases with q, which indicates that more exponents are necessary to describe the fractality of the series, that in this case is multifractal with a more complex structure.

_{0}is the maximum. The width, W, of the multifractal spectrum is defined as:

_{max}and α

_{min}are the two zeros of the fitted second-degree polynomial. W is often employed to quantify the multifractality in a series. The larger the value of W, the higher the multifractal degree of the series. Another parameter describing the multifractal spectrum is the asymmetry, R, defined as:

#### 2.3. The Binomial Multifractal Model

^{k}, and ν = 1,…,N, the BMM is defined by:

_{q}has the following theoretical expression [27]:

_{−10}and F

_{10}for the BMM series generated with a = 0.75 and length N = 16,384 (k = 14) (Figure 7a) and N = 512 (k = 9) (Figure 7b). In particular, the length of the last binomial series is comparable with that of the data analyzed in this study.

_{q}spectrum and those obtained for the simulated binomial series, although such matching is generally better for the longer series and for q < 0 (Figure 8).

_{q}spectrum of a series simulated by the BMM, with a = 0.9 and length N = 512 = 2

^{9}(blue circles). From this series, 100 replicas were derived, but with 25 gaps (i.e., 5% of the length of the series) randomly placed along the series (thus, each replica is different from the other for the position of the gaps). Then, the gaps of each replica were filled by using each one of the four methods indicated above. We calculated the h

_{q}spectrum for each replica and averaged among them all. The mean h

_{q}spectrum of the replicas for each of the four filling methods is very similar to that of the original binomial series. The average root-mean-squared error (RMS) between the h

_{q}spectrum of each replica and the h

_{q}spectrum of the binomial series is shown in Figure 10, along with its standard deviation, varying the percentage of missing data and the parameter a.

## 3. Results

_{2}. Figure 13 shows the box plot of h

_{2}for the selected pixel time series of the three types of sites. It can be observed that the series tended to be more persistent when the site was affected by two fires, while they tended to be characterized by a random dynamic if the site was fire-unaffected.

_{2}> of the three types of sites for each satellite were significantly different. Considering the difference as significant if the p-value < 0.05, we found that most of the comparisons were significantly different (Table 4). Moreover, as shown by the box plots (Figure 13), the increasing trend of h

_{2}from fire-unaffected sites to two-fire-affected sites was consistent with the spectral behavior of the pixel time series analyzed by means of the periodogram. The larger value of h

_{2}for two-fire-affected sites indicates a larger strength of the long-range correlation structure consistent with the relatively larger value of the periodogram at long periods observed for pixels affected by two fires.

_{q}range, which is the difference between the largest and the smallest h

_{q}, could be used to discriminate between monofractal and multifractal signals, since a very small value of the h

_{q}range indicates that small and large fluctuations of the series are almost similarly scaled. Consistent with [37], for an h

_{q}range > 0.15, the series can be considered multifractal. Therefore, the calculation of the multifractal parameters was performed only for those pixel residual series with an h

_{q}range > 0.15. Figure 14 shows, as an example, the multifractal spectrum of a MODIS Terra NDVI

_{r}series of a site not affected by any fire. The spectrum is characterized by the well-known parabolic shape, whose width, W, asymmetry, R, and maximum, α

_{0}, are the main parameters. Figure 15, Figure 16, Figure 17 and Figure 18 show the box plots of the h

_{q}range, W, R, and α

_{0}, respectively, for each type of satellite and site. On average, the fire-unaffected sites were characterized by a slightly smaller multifractality degree than the fire-affected sites, as indicated by the h

_{q}range and W box plots. The asymmetry was, on average, positive for most of the series of any type and site, while the box plot of the maximum, α

_{0}, indicates a clear increase from fire-unaffected sites to sites affected by two fires.

_{q}range>, and the p-values are listed in Table 5. For most of the site and satellite types, the <h

_{q}range> for the three types of sites was significantly different.

## 4. Discussion

_{2}exponent, might suggest a larger capability of recovery after a fire. A larger persistence could indicate that “the investigated ecosystems are governed by positive feedback mechanisms, which tend to destabilize the system under external forces, driving unstable growth-generating phenomena” [25]. The average largest persistence of the VI of the two-fire-affected sites could be due to a cumulative status of the effects of the first and second fire, each one contributing to drive growth-generating phenomena, leading to an overall larger recovery capability.

_{q}range, the multifractal width, W, the asymmetry, R, and the maximum, α

_{0}, are generally used to quantify the multifractality of a time series. The larger the values of the h

_{q}range and multifractal width, W, the higher the multifractality degree of the series. A higher multifractality degree indicates a larger heterogeneity of the time series, which means that the series is characterized by more complex dynamics. The multifractal asymmetry, R, measures the skewness of the multifractal spectrum, which quantifies the relative dominance of the small/large fluctuations in the series. The time dynamics of a series featured by the right-skewed multifractal spectrum is mainly dominated by the small fluctuations, while that of series characterized by a left-skewed multifractal spectrum is mainly governed by the large fluctuations. The maximum, α

_{0}, conveys information about the structure of the series: small values of α

_{0}mean that the underlying process loses the fine structure and appears more regular, while large values indicate that the underlying system is finer in structure and the series is more complex [38].

_{q}range and W increased from the sites not affected by any fire to the sites affected by two fires, taking intermediate values for the sites affected by only one fire. Such increasing trend of the h

_{q}range and width, W, indicated that the VI series of fire-affected sites were characterized by a more heterogeneous behavior than the fire-unaffected sites. The larger heterogeneity of the time dynamics of NDVI and EVI of fire-affected sites could be in relation to the more effective positive feedback mechanisms that would drive the recovery processes. The growth-generation processes lead the vegetation recovery of fire-affected sites, contrarily to the vegetation of fire-unaffected sites, whose relatively more random (or less persistent) fluctuations could explain the larger homogeneous character of their time dynamics. The occurrence of the first fire changes the dynamics of vegetation, which gradually recovers after fire. The occurrence of the second fire further strengthens such recovery process, and vegetation dynamics would be characterized by a “more focused” behavior, exerting “more focused actions” aimed at restabilizing those conditions existing before the fire. This situation could be reflected in the more heterogeneous behavior of the time dynamics of VI series.

_{2}exponent, and of the heterogeneity of the series, depicted by the width, W; in fact, the occurrence of a fire forces the vegetation system to “react” with a higher resilience, reflected in the higher complexity of the fire-affected sites.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Land covers of the study area and the burned area of the Camp Fire (pink boundary), Btu Fire (dark red boundary), and Humboldt Fire (blue boundary).

**Figure 2.**Normalized Difference Vegetation Index (NDVI) and Enhanced Vegetation Index (EVI) time series for a pixel affected by two fires (

**a**), one fire (

**b**), and no fire (

**c**). Upper panels: Terra MODIS collection; Lower panels: Aqua MODIS collection.

**Figure 3.**Periodogram of an Aqua EVI pixel time series of a site not affected by any fire (

**a**), affected by one fire (

**b**), and affected by two fires (

**c**). The dotted lines represent the 95% confidence level (see text for details). Two main periodicities at about 6 and 12 months are significant at 95%. They represent the meteo-climatic fluctuations.

**Figure 4.**Mean periodogram of Aqua EVI pixel time series in site not affected by any fire, affected by one, and affected by two fires. The two periodicities at about 6 and 12 months are well-identified.

**Figure 5.**Residuals of the pixels, whose periodograms are shown in Figure 3, after removing the two periodicities at 6 and 12 months by Fourier filtering.

**Figure 6.**Residual Normalized Difference Vegetation Index (NDVI

_{r}) and Residual Enhanced Vegetation Index (EVI

_{r}) time series corresponding to the example pixel time series shown in Figure 2, affected by two fires (

**a**), one fire (

**b**), and no fire (

**c**). Upper panels: Terra MODIS collection; Lower panels: Aqua MODIS collection.

**Figure 7.**Fluctuation functions for q = −10 and q = 10 of the binomial series generated with a = 0.75 and (

**a**) N = 2

^{14}and (

**b**) N = 2

^{9}.

**Figure 8.**The h

_{q}spectrum of the binomial series. The comparison is between the theoretical case for a = 0.75 and the simulated cases with size N = 2

^{14}and N = 2

^{9}.

**Figure 9.**Comparison among the mean h

_{q}spectra of the binomial series with different gap filling for a = 0.9, size N = 2

^{9}, and 5% of gap percentage.

**Figure 10.**Comparison among the mean RMS with different gap filling and 3% to 10% of gap percentage for a BMM with size N = 2

^{9}and (

**a**) a = 0.75 and (

**b**) a = 0–9.

**Figure 11.**Fluctuation functions for q = 10 (

**a**) and q = −10 (

**b**) of the residual NDVI of a pixel affected by two fires (polynomial degrees, p, from 1 to 5). (

**c**) Fluctuation functions for q = −10 of the residual EVI of a pixel affected by two fires. (

**d**) Fluctuation functions from q = −10 to q = 10 for the residual NDVI of an Aqua pixel not affected by any fire. (

**e**) Fluctuation functions from q = −10 to q = 10 for the residual EVI of a Terra pixel not affected by any fire.

**Figure 12.**Distribution of the coefficient of determination, R, for each series and each q value for the data of Aqua NDVI with no fire occurrence for (

**a**) T = 0.9 and (

**b**) T = 0.95. The white boxes correspond to R values smaller than the threshold.

**Figure 13.**Box plots of h

_{2}for (

**a**) residual NDVI and EVI of MODIS/Aqua and (

**b**) residual NDVI and EVI of MODIS/Terra.

**Figure 15.**Box plots of the h

_{q}range for (

**a**) NDVI

_{r}and EVI

_{r}of MODIS/Aqua and (

**b**) NDVI

_{r}and EVI

_{r}of MODIS/Terra.

**Figure 16.**Box plots of the width, W, of the multifractal spectrum of (

**a**) NDVI

_{r}and EVI

_{r}of MODIS/Aqua and (

**b**) NDVI

_{r}and EVI

_{r}of MODIS/Terra.

**Figure 17.**Box plots of the asymmetry, R, of (

**a**) NDVI

_{r}and EVI

_{r}of MODIS/Aqua and (

**b**) NDVI

_{r}and EVI

_{r}of MODIS/Terra.

**Figure 18.**Box plots of the maximum, α

_{0}, of (

**a**) NDVI

_{r}and EVI

_{r}of MODIS/Aqua and (

**b**) NDVI

_{r}and EVI

_{r}of MODIS/Terra.

Wildfire Event | Duration | Cause | Burned Area (Acres) | Tree Coverage (%) |
---|---|---|---|---|

Humboldt Fire | 11–21 June 2008 | Arson | 23,344 | Grasslands 54.59%, Savannas 32.02%, Woody Savannas 12.86% |

Btu Fire | 21 June–29 July 2008 | Lightning strikes | 57,815 | Woody Savannas 59.94%, Evergreen Needleleaf Forests 23.78%, Savannas 7.29%, Grasslands 5.71% |

Camp Fire | 8–25 November 2018 | Electrical transmission lines | 153,336 | Evergreen Needleleaf Forests 37.20%, Woody Savannas 31.20%, Grasslands 16.06%, Savannas 9.38% |

VI Product | VI | Spatial Resolution (m) | Frequency (Days) | Temporal Extent (Year) | Sensor/Satellite |
---|---|---|---|---|---|

MOD13Q1 | NDVI, EVI | 250 | 16 | 2000–2020 | MODIS/Terra |

MYD13Q1 | NDVI, EVI | 250 | 16 | 2002–2020 | MODIS/Aqua |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire | 54 | 39 | 37 | 48 |

One Fire | 136 | 62 | 174 | 113 |

Two Fires | 149 | 80 | 184 | 138 |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire–One Fire | 0.073337 | 0.001878 | 0.666646 | 0.778663 |

One Fire–Two Fires | 1.83 × 10^{−23} | 0.325468 | 6.25 × 10^{−10} | 6.94 × 10^{−14} |

No Fire–Two Fires | 3.20 × 10^{−16} | 2.06 × 10^{−6} | 0.001233 | 3.14 × 10^{−5} |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire–One Fire | 0.897719 | 0.042757 | 0.321087 | 0.960532 |

One Fire–Two Fires | 6.95 × 10^{−11} | 0.595574 | 3.14 × 10^{−7} | 0.688014 |

No Fire–Two Fires | 8.71 × 10^{−9} | 0.007061 | 1.77 × 10^{−7} | 0.679548 |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire–One Fire | 0.733290 | 0.045961 | 0.288707 | 0.348462 |

One Fire–Two Fires | 7.66 × 10^{−10} | 0.669236 | 1.26 × 10^{−6} | 0.941305 |

No Fire–Two Fires | 3.04 × 10^{−8} | 0.010015 | 7.83 × 10^{−7} | 0.230172 |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire–One Fire | 0.909562 | 0.562129 | 0.194883 | 0.904868 |

One Fire–Two Fires | 0.001069 | 0.745943 | 0.009704 | 0.711969 |

No Fire–Two Fires | 0.003398 | 0.778580 | 0.790529 | 0.880401 |

Aqua-EVI | Aqua-NDVI | Terra-EVI | Terra-NDVI | |
---|---|---|---|---|

No Fire–One Fire | 0.225453 | 3.77 × 10^{−4} | 0.125898 | 0.002197 |

One Fire–Two Fires | 3.39 × 10^{−25} | 0.757963 | 8.55 × 10^{−13} | 1.70 × 10^{−8} |

No Fire–Two Fires | 7.87 × 10^{−17} | 5.84 × 10^{−6} | 0.002117 | 2.45 × 10^{−13} |

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## Share and Cite

**MDPI and ACS Style**

Ba, R.; Lovallo, M.; Song, W.; Zhang, H.; Telesca, L.
Multifractal Analysis of MODIS Aqua and Terra Satellite Time Series of Normalized Difference Vegetation Index and Enhanced Vegetation Index of Sites Affected by Wildfires. *Entropy* **2022**, *24*, 1748.
https://doi.org/10.3390/e24121748

**AMA Style**

Ba R, Lovallo M, Song W, Zhang H, Telesca L.
Multifractal Analysis of MODIS Aqua and Terra Satellite Time Series of Normalized Difference Vegetation Index and Enhanced Vegetation Index of Sites Affected by Wildfires. *Entropy*. 2022; 24(12):1748.
https://doi.org/10.3390/e24121748

**Chicago/Turabian Style**

Ba, Rui, Michele Lovallo, Weiguo Song, Hui Zhang, and Luciano Telesca.
2022. "Multifractal Analysis of MODIS Aqua and Terra Satellite Time Series of Normalized Difference Vegetation Index and Enhanced Vegetation Index of Sites Affected by Wildfires" *Entropy* 24, no. 12: 1748.
https://doi.org/10.3390/e24121748