Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems

Baveye, Philippe C.; Boast, C. W.; Gaspard, Sarra; Tarquis, Ana M.; Millán, H.

doi:10.1002/9780470511206.ch2

Cited by 8 publications

(11 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Equations ( 19 – 20 ) represent the Legendre transformation from the independent variables τ xy and q to the conjugate independent variables f xy and α xy (Arnold 1978 ; Baveye et al 2008 ).…”

Section: Methodsmentioning

confidence: 99%

Multifractality of the standardized precipitation index: influence of pan evaporation and virtual temperature-based potential evapotranspiration

Millán

Macías

Valderá³

2022

Meteorol Atmos Phys

View full text Add to dashboard Cite

Hydrological extremes experience an increase in some regions and a decrease in other zones. The objectives of the present work were (i) to introduce Class A pan evaporation data and virtual temperature-based potential evapotranspiration (PET v hereafter) into the Standardized Precipitation–Evapotranspiration Index (SPEIm hereafter) computation and (ii) to describe small and large fluctuations of SPI and SPEIm through multifractal detrended fluctuation (MF-DFA) and multifractal detrended cross-correlation (MF-DCCA) analyses. We used 40 years data (1974–2013) of monthly rainfall (P), mean, minimum and maximum air temperature, pan evaporation (E), relative humidity (RH) and relative sun brightness (RSB). Meteorological variables were collected from Puyo meteorological station, Pastaza Province, Ecuador. SPI time series for 1 and 6 months timescales were determined following two approaches. We computed SPI values using precipitation as the only input variable. Additionally, we incorporated pan evaporation and virtual temperature-based potential evapotranspiration into the standard SPEI computation (SPEIm). The SPEIm revealed some differences as compared with the classical SPI methodology. Five out of fifteen Asymmetry Index (AI) values were positive (0.095 ≤ AI ≤ 0.419).This indicates the relevance of high fluctuations at different time scales. Joint multifractal spectra between SPI (1,6)/SPEIm(1,6) versus RH and RSB rendered negative AI values which suggests the importance of low fluctuations at shorter time scales. The DCCA cross-correlation coefficient allows one to identify those time scales where SPI and SPEIm are influenced by other meteorological variables. Long-term correlation and sub-Gaussian behaviour of meteorological variables (apart from air temperature) are the main causes of multifractal structures.

show abstract

Section: Methodsmentioning

confidence: 99%

Multifractality of the standardized precipitation index: influence of pan evaporation and virtual temperature-based potential evapotranspiration

Millán

Macías

Valderá³

2022

Meteorol Atmos Phys

View full text Add to dashboard Cite

show abstract

“…Geometric (mono)fractals are self-similar sets of points. Multifractals (or multifractal measures) are self-similar measures defined on specific set of points [Baveye et al 2008;Evertsz et al 1992;Falconer 2003]. In general, the former can be generated by additive processes and the latter by multiplicative cascade of random processes.…”

Section: Multifractal Analysis (Mfa): Theory and Methodsmentioning

confidence: 99%

Multifractal analysis of three-dimensional grayscale images: Characterization of natural porous structures

Milazzo¹,

Pajor²

2016

Preprint

View full text Add to dashboard Cite

A multifractal analysis (MFA) is performed on three-dimensional grayscale images associated with natural porous structures (soil samples). First, computed tomography (CT) scans are carried out on the samples to generate 3D grayscale images. Then, a preliminary analysis is conducted to evaluate key quantities associated with the porosity, such as void fraction, pore volume, connectivity, and surface area. Finally, the samples are successfully identified and separated into two different structure families by using the MFA. A new software (Munari) to carry out the MFA of 3D grayscale images is also presented.

show abstract

“…The two most popular of these are GBL (Sect. 2.2) and sandbox lacunarity (described below) (Baveye et al 2008) and have been compared by Allain and Cloitre (1991) and Pendleton et al (2005). In some cases lacunarity indices have been named 'the lacunarity' in the literature, leading to confusion (Baveye et al 2008).…”

Section: Other Lacunarity Indicesmentioning

confidence: 99%

“…2.2) and sandbox lacunarity (described below) (Baveye et al 2008) and have been compared by Allain and Cloitre (1991) and Pendleton et al (2005). In some cases lacunarity indices have been named 'the lacunarity' in the literature, leading to confusion (Baveye et al 2008). Applications can depend crucially on the choice of the lacunarity index, and we recommend that the term 'lacunarity' is not used as if it were a single quantity.…”

Section: Other Lacunarity Indicesmentioning

confidence: 99%

Computation of Lacunarity from Covariance of Spatial Binary Maps

Hingee

Baddeley

Caccetta

et al. 2019

JABES

View full text Add to dashboard Cite

We consider a spatial binary coverage map (binary pixel image) which might represent the spatial pattern of the presence and absence of vegetation in a landscape. 'Lacunarity' is a generic term for the nature of gaps in the pattern: a popular choice of summary statistic is the 'gliding-box lacunarity' (GBL) curve. GBL is potentially useful for quantifying changes in vegetation patterns, but its application is hampered by a lack of interpretability and practical difficulties with missing data. In this paper we find a mathematical relationship between GBL and spatial covariance. This leads to new estimators of GBL that tolerate irregular spatial domains and missing data, thus overcoming major weaknesses of the traditional estimator. The relationship gives an explicit formula for GBL of models with known spatial covariance and enables us to predict the effect of changes in the pattern on GBL. Using variance reduction methods for spatial data, we obtain statistically efficient estimators of GBL. The techniques are demonstrated on simulated binary coverage maps and remotely sensed maps of local-scale disturbance and meso-scale fragmentation in Australian forests. Results show in some cases a fourfold reduction in mean integrated squared error and a twentyfold reduction in sensitivity to missing data.Supplementary materials accompanying the paper appear online and include a software implementation in the R language.

show abstract

Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures: Theory and Operational Aspects of their Application to Natural Systems

Cited by 8 publications

References 52 publications

Multifractality of the standardized precipitation index: influence of pan evaporation and virtual temperature-based potential evapotranspiration

Multifractality of the standardized precipitation index: influence of pan evaporation and virtual temperature-based potential evapotranspiration

Multifractal analysis of three-dimensional grayscale images: Characterization of natural porous structures

Computation of Lacunarity from Covariance of Spatial Binary Maps

Contact Info

Product

Resources

About