Mul t i f r a c t a l sa n d r e s o l u t i on - PowerPoint PPT Presentation

Mul t i f r a c t a l sa n d r e s o l u t i on d ependence i n Remo t e S en s i ng : The ex amp l e o f ocean co l ou r S. Lovejoy, Physics, McGill Nonlinear variability in Geophysics Nonlinéaire en Géophysique D. Schertzer, CNRS, Paris H. Gaonac’h, GEOTOP, UQAM Groupe d'Analyse de la variabilité April 19, 2002

Wha t i s t h e t a ngent o f t h e c o a s t o f B r i t t a ny? Perrin 1913: "A l t hough d i f f e r en t i a b l e f u n c t i on s may b e t h e s imp l e s t , t h e y a r e none t h e l e s s t h e exc ep t i o n s . . . i n g eome t r i c l a nguage , c u r v e s w i t hou t t a ngen t a r e t h e r u l e wh i l e r e gu l a r cu r v e s . . . a r e v e r y s p e c i a l . . . Cons i d e r t h e d i f f i c u l t y i n f i n d i ng t h e t a ngen t t o a p o i n t o f t h e c o a s t o f B r i t t a ny . . . depend i ng on t h e r e s o l u t i o n o f t h e map t h e t a ng en t wou l d c h ange . Th e po i n t i s t h a t a map i s s imp l y a c onven t i on a l d r aw ing i n wh i ch e a ch l i n e h a s a t a ng en t . On t he c on t r a r y , a n e s s en t i a l f e a t u r e o f t h e c o a s t i s t h a t . . . w i t hou t mak i ng t h em ou t , a t e a ch s c a l e we gue s s t h e de t a i l s wh i ch p r oh i b i t u s f r om d r aw in g a t a ngen t . . . "

How Long i s t h eVi s t u l a… t h e c o a s t o f Br i t a i n ? S t e inhaus 1954 : " . . . Th e l e f t b ank o f t h e V i s t u l a when mea su r ed w i t h i n c r e a s ed p r e c i s i on wou ld f u r n i s h l e ng t h s t e n , h und r ed , a nd e v en a t h ou s and t ime s a s g r e a t a s t h e l e ng t h r e ad o f f a s choo l map . A s t a t emen t n e a r l y a d e qua t e t o r e a l i t y wou l d b e t o c a l l mo s t a r c s e n coun t e r ed i n na t u r e a s n o t r e c t i f i a b l e . Th i s s t a t emen t i s c on t r a r y t o t h e b e l i e f t h a t n o t r e c t i f i a b l e a r c s a r e a n i n v en t i on o f ma t h ema t i c i a n s a nd t h a t n a t u r a l a r c s a r e r e c t i f i a b l e : i t i s t h e oppo s i t e wh i ch i s t r u e . . . " Richard son 1961 : Emp i r i c a l s c a l i n g o f c o a s t o f B r i t a i n a nd o f s e v e r a l f r on t i e r s u s i ng "R i ch a r d son d i v i d e r s " me t hod . Mande lbro t1967 : p ape r "How l ong i s t h e coa s t o f B r i t a i n ? " i n t e r p r e t s R i ch a r d son ' s s c a l i ng e xponen t i n t e rms o f a f r a c t a l d imen s i on .

But… the coastline is a level set of the topography. So what are the statistics of the topography field h(x,y)? Some early scaling results: Vening-Meinesz 1951: E(k)=k −β ; β=2 Balmino et al 1973, Bell 1975 : E(k)=k −β � • • • • � β • • • • � 2 1980 ' s : Thef r a c t a l i t yo f c o a s t l i n e s b e comes "obv i ou s " !

Mono …. o r Mu l t i f r a c t a l ? Scaling exponent q = λ ( ) ∆ h ∆ x ( ) ( ) ξ q q ∆ h λ ∆ x ( ) = h x + ∆ x ( ) − h x ( ) ∆ h ∆ x small scale large scale Spectral exponent of topography ( ) = qH ξ q ( ) / 2 H = C = d − D = β − 1 Monofractal: or (e.g. fractional Brownian or Levy motion) Unique fractal dimension of surface ( ) = qH − K q ( ) ξ q Multifractal: Nonlinear, convex function (e.g. Fractionally Integrated Flux model)

Multifractality and Functional Box Counting − D ( T ) N T ( L ) ≈ L A -Classical geostatistics :D(T)=2 B C D -Monofractality: D(T) <2 , constant -Multifractality, D(T)<2, E F G decrasing functions A) the � eld is shown with two isolines that have thresholds values; the box size is unity. In B), C) and D), we cover areas whose value exceeds by boxes that decrease in size by factors of two. In E), F) and G) the same degradation in resolution is applied to the set exceeding the threshold.

Functional box counting on French topography: 1 -1000km Slope =2 10 6 (required for classical Multifractal: slopes 10 5 geostatistics - vary with threshold regularity of 10 4 Lebesgue 100m N(L) ≈ L -D measures) 10 3 N(L) 1800m 10 2 10 1 3600m 10 0 10 0 10 1 10 2 10 3 km L N(L) = number of covering boxes for exceedance sets at various altitudes. The dimensions d increase from 0.84 (3600m) to 1.92 (at 100m). Lovejoy and Schertzer 1990

complexity 200 m heterogeneity variability 200 m 30 m 400 m

Etnean lava flow geometry

Fractality

ETOPO5 data set 4320X2160 points at 5 minutes arc (roughly 10km), including bathymetry The boxes show the regions used for comparing continental versus oceanic statistics The lines indicate the central strip used for analyses at (near) constant spatial resolution

GTOPO30 and 90m set GTOPO30 is the continental US at 30’’ (roughly 1km) resolution We also analysed a 90m resolution data set (rectangles)

Lower Saxony data set at 50cm resolution The lines indicate transects compared for the effects of trees

Spectral intercomparison ETOPO5 ±22 o US 90m Lower Saxony (with Reference trees, top), Lower slope -2.1 Saxony (without trees, bottom), US GTOPO30 (1m) -1 (10,000km) -1 Spectral energy versus the wavenumber for the four DEMs. The small arrows show the frequency beyond which the spectra are poorly estimated due to inadequate vertical resolution.

Four of the 8 Mies channels St. Lawrence estuary, 7 m resolution, narrow visible channels (airborne data)

Functional box counting of ocean colour data ( ) ≈ λ D T ; N λ λ = L 0 / L

Mies Spectra Channels 1-8 offset for clarity ( ) ≈ k − β E k β = 1.18 (210km) -1 (14m) -1

Multifractal properties φ Λ = ∆ h Λ Conservative cascade quantity; Λ = finest resolution q = λ ( ) φ λ K q Multiscaling of moments L . T . ( ) ↔ c γ ( ) K q ( ) ≈ λ ( ) γ − c γ Pr φ λ > l > λ Probability distributions α − q C 1 ( ) ( ) = K q α − 1 q Universality classes Divergence of statistical moments q → ∞ ; q > q D φ λ ( ) ≈ s − q D ; Pr φ λ > l > s s >> 1

ETOPO5 (strip; 10km) GTOPO30 1km Log 10 < φ λ q > Trace moments- 20,000km 10km 4000km 1km Log 10 λ Log 10 λ intercomp arison Log 10 < φ λ q > q = λ ( ) φ λ K q 5900km 90m 3km 50cm Lower Saxony (50cm) US DEM 90m

K(q) individual data sets The convexity shows the topography is multifractal, fractional brownian motion implies K(q)=0; the monofractal beta model, that K(q) is linear. The dotted line corresponds to the "mean" K(q) with parameters α =1.8 and C 1 =0.12.

The simplest hypothesis: isotropic statistics with α =1.8, C 1 =0.12 Lower Saxony subsections without trees q=1.77, 2.18 only) ETOPO5 GTOPO30 (US) 50cm 20000 km The solid lines are theory using α =1.8, C1=0.12 for q (from top to bottom, q=2.18, 1.77, 1.44, 1.17, 0.04, 0.12 and 0.51).

Multifractal topography simulation on a sphere With J. Tan, 1996

Self-similar (isotropic) multifractal topographies: continents, oceans α =1.8, C 1 =0.12 H=0.45 (Oceans) H=0.7 (Continents)

Linear GSI topography: “texture” 200 200 100 100 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 200 100 200 100   1.3 0.6   G = Different random seed − 0.4   0.7

Anisotropy position- independent α =1.8, C 1 =0.12, H=0.7 Rotation dominant e=0.3 Stratification dominant f=0.3

Position dependent anisotropy: Nonlinear GSI

Atmos- NOAA12 pheric Cascades from NOAA14 cloud imagery GMS 909 Satellite images, moments q=0.2, 0.4, …1.8 q = λ K q ( ) ε λ Infra Red Visible Lovejoy, Schertzer, Stanway 2001

Radiative transfer on multifractal clouds With B. Watson