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

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d ependence i n Remo t e S en s i ng : The ex amp l e

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Nonlinear variability in Geophysics Groupe d'Analyse de la variabilité Nonlinéaire en Géophysique

• S. Lovejoy,

Physics, McGill

• D. Schertzer,

CNRS, Paris

• H. Gaonac’h,

GEOTOP, UQAM April 19, 2002

Wha t i s t h e t a ngent

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t h e c

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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

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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

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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

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B r i t t a ny . . . depend i ng

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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

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t i

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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

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t r a r y , a n e s s en t i a l f e a t u r e

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s t i s t h a t . . . w i t hou t mak i ng t h em

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t , a t e a ch s c a l e we gue s st h e de t a i l s wh i ch p r

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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

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s t

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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

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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

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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

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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

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e a l i t y wou l d b e t

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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

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r e c t i f i a b l e . Th i s s t a t emen t i s c

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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

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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

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c

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s t

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B r i t a i n a nd

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s e v e r a l f r

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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

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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

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a f r a c t a l d imen s i

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1980 ' s : Thef r a c t a l i t yo f c

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But… the coastline is a level set of the topography. So what are the statistics of the topography field h(x,y)?

Vening-Meinesz 1951: E(k)=k−β; β=2 Balmino et al 1973, Bell 1975 : E(k)=k−β • • • • β • • • • 2 Some early scaling results:

Mono ….

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Mu l t i f r a c t a l ?

Monofractal:

∆h λ ∆x

( )

q = λ ξ q

( ) ∆h ∆x

( )

q ∆h ∆x

( )= h x + ∆x ( )− h x ( )

large scale small scale Scaling exponent

H = C = d − D = β −1

( )/ 2

Unique fractal dimension of surface Spectral exponent of topography

ξ q

( ) = qH − K q ( )

Multifractal:

Nonlinear, convex function

• r

ξ q

( ) = qH

(e.g. fractional Brownian or Levy motion)

(e.g. Fractionally Integrated Flux model)

Multifractality and Functional Box Counting

A B C D E F G

NT (L) ≈

−D(T)

L

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.

• Classical geostatistics :D(T)=2
• Monofractality: D(T) <2 ,

constant

• Multifractality, D(T)<2,

decrasing functions

Functional box counting on French topography: 1 -1000km

10 3 10 2 10 1 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6

L N(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).

3600m

1800m 100m km

N(L)≈L-D

Multifractal: slopes

vary with threshold

Lovejoy and Schertzer 1990

Slope =2 (required for classical geostatistics - regularity of Lebesgue measures)

complexity variability heterogeneity

200 m 200 m 400 m 30 m

Etnean lava flow geometry

Fractality

ETOPO5 data set

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

4320X2160 points at 5 minutes arc (roughly 10km), including bathymetry

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

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.

Lower Saxony (with trees, top), Lower Saxony (without trees, bottom),

ETOPO5 ±22o

US 90m

US GTOPO30

Reference slope -2.1 (10,000km)-1

(1m)-1

Four of the 8 Mies channels

• St. Lawrence estuary, 7 m resolution, narrow visible

channels (airborne data)

Functional box counting of ocean colour data

N λ

( ) ≈ λ

DT ;

λ = L0 / L

Mies Spectra

E k

( ) ≈ k

−β

β = 1.18

Channels 1-8 offset for clarity

(14m)-1 (210km)-1

Multifractal properties

Multiscaling of moments

φλ

q = λ K q

( )

φΛ = ∆hΛ Conservative cascade quantity; Λ=

finest resolution

Probability distributions

Pr φλ >l > λ

γ

( )≈ λ

−c γ

( )

Universality classes

K q

( ) =

C1 α −1 q

α − q

( )

Divergence of statistical moments

Pr φλ >l > s

( )≈ s −qD ;

s >>1

φλ

q → ∞;

q > qD

K q

( )↔ c γ ( )

L .T.

Trace moments- intercomp arison

US DEM 90m Lower Saxony (50cm) GTOPO30 1km ETOPO5 (strip; 10km) Log10<φλq> Log10<φλq> Log10λ Log10λ

10km 20,000km 1km

4000km 5900km 90m 3km 50cm

φλ

q = λ K q

( )

The dotted line corresponds to the "mean" K(q) with parameters α=1.8 and C1=0.12.

The convexity shows the topography is multifractal, fractional brownian motion implies K(q)=0; the monofractal beta model, that K(q) is linear.

K(q) individual data sets

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).

The simplest hypothesis: isotropic statistics with α=1.8, C1=0.12

Lower Saxony

subsections without trees q=1.77, 2.18

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GTOPO30 (US) ETOPO5

20000 km

50cm

Multifractal topography simulation on a sphere

With J. Tan, 1996

Self-similar (isotropic) multifractal topographies: continents, oceans

H=0.45 (Oceans) H=0.7 (Continents) α=1.8, C1=0.12

Linear GSI topography: “texture”

100 200 100 200 0.2 0.4 0.2 0.4 100 200 100 200 0.2 0.4 0.2 0.4

G = 1.3 0.6 −0.4 0.7      

Different random seed

Anisotropy position- independent

α=1.8, C1=0.12, H=0.7 Rotation dominant e=0.3 Stratification dominant f=0.3

Position dependent anisotropy: Nonlinear GSI

Atmos- pheric Cascades from cloud imagery

909 Satellite images, moments q=0.2, 0.4, …1.8 Lovejoy, Schertzer, Stanway 2001

Infra Red Visible

NOAA12 NOAA14 GMS

ελ

q = λ K q

( )

Radiative transfer on multifractal clouds

With B. Watson

Radiative transfer on multifractals

The left hand column shows the results of the simulation of 200 million photons through a cloud with the mean optical thickness vertical optical thickness = 0.844, and isotropic phase functions. For the simulation, the sun was incident at θ (azimuthal angle), = φ (polar angle)= 0.1 radians, periodic boundary conditions were used in the horizontal. The cloud was generated with a continuous multifractal cascade process with the universal multifractal parameters of observed satellite radiances: α=1.35, C1=0.15, H=0.3. The simulation was performed on a 1283 grid for θ = 15 degrees, φ = 0. The right hand column shows the optical thickness as a function of space for the angles of the corresponding pictures in the left column. Note that in all cases, the images were normalized by the maximum so as to enhance contrast.