D. Schertzer, ENPC, Paris
S. Lovejoy, Physics, McGill
Part 2: Introduction to scaling in precipitation and hydrology
Dam Safety Interest Group,Expert’s meeting 9 November, 2004
Overview
• Fractal Sets, dimensions
• Co-dimensions
• Spectra
• Examples from hydrology
• Nonclassical Probability distributions
Some Classical Fractal sets and their dimension
– Cantor set• Cantor square
• Devil’s staircase
– Koch snowflake– Sierpinski Triangle
• Sierpinski Pyramid
– Peano curve
Cantor set
• Let us start with:
and let us iterate:
Koch snowflake
Let us start with:
and let us iterate:
Fractal Dimension• THE SIMPLEST METHOD
– take advantage of self-similarity. – a 1-dimensional line segment with the
magnification of 2, yields 2 identical line segments,
– a 2-dimensional (e.g.square or a triangle), with the magnification of 2, yields 4 identical shapes,
– a 3-dimensional cube, magnify it 2 times. you will get 8 identical cubes,
Fractal Dimension (2)
• Let’s use a variable D for dimension, for magnification, and N for the number of identical shapes:
» DN=
• Or:
• D = log N / log
First Examples
– Cantor Set:
D = log N / log log 2 / log ≈ 0.63
-Koch Snowflake:
D = log N / log log 4 / log ≈1.26.
– Daily Rain Fall Events in Dedougou (SW Africa):1922-1966
– Each line is a different year,
– eachblack point a rainy day.
– Cantor-like set– D≈log(7)/log(12):
• i.e. divide into 12 parts keep only 7..
– (Hubert et Carbonnel, 1990)
Rain Fall Events
Cantor Square
D= log 4/ log 3≈1.26
CASCADES
• Isotropic Cascade. The left hand side shows an non-intermittent (“homogeneous”) cascade, the right hand side shows how intermittency which can be modeled by assuming that sub-eddies are either “alive” or “dead” (model”).
L
LL/2
L/2
L/2
L/2
N(L) ã L- D N(L) ã L- Ds
Ln 4
Ln 2Ln 3™™
D = ™™Ln 2
= 2
D =s= 1.58
ISOTROPIC
= self similarity
-model
•The -model for 2, C = 0.2. •the set of the surviving “active” regions has a dimension equal to D= 2-C = 1.8.•the cascade process is iterated an infinite number of times, here it is followed for only four generations on a 256 256 point grid,
•Novikov and Steward (1963), Mandelbrot (1974), Frisch et al (1978)
Sierpinski Triangle
• or :
Sierpinski Triangle (2)
• D= log 3 /log 2 ≈ 1.58.
Sierpinski Pyramid
• First iteration:
D = log 4/ log 2 But DT = 1
10 th iteration:
Topological Dimension
– Definition:• the empty set Ø has topological dimension -1. • the topological dimension of a set is DT if and only if you can
disconnect it (by cutting it) by taking out a subset of topological dimension DT-1.
– Classical examples:» isolated point(s) DT = 0 can be cut only by Ø» lines DT =1 can be cut by an isolated point» Surfaces DT = 2 can be cut by a line.
– indeed a topological invariant, i.e. invariant under 1:1 and bicontinuous transformations.
Peano Curve
• motif:
D=log 9 / log 3 = 2i.e. it is a plane filling line
iterations:
A model of hydraulic network, From Steinhaus 1962
Geometric Method•The similarity method is
great for a fractal composed of a certain number of identical versions of itself…
•A way out: graph log(size) against log(magnification),
•details add additional irregularities, which add to the measurement.
• Fractal dimension = slope
Cloud perimeters over 5 decades yield D≈1.35 (Lovejoy, 1982)
Isotropic Scale Invariance and fractal sets
Fractal Dimension:
L
d=dimension of spaceD= fractal dimension of set
C=d-D= fractal codimension
Scale invariance:
Same form after zoom by factor .
n L( ) ∝ L
D
ρ L( ) =
n L( )
L
d
= ∝ L
D − d
= L
− C
n L( ) =
D
n L( ) D=scale invariant
Box Counting Dimension
• Sketch of the lava flow field from Etna (1900-1974) using box-counting technique:
• resolution is decreased by factors of 2 at each step. The finest resolution was 43 m.
• From Gaonac'h et al (1992).
N( λB ∩ A)~ −Dg(A)λ
“Defined” as the scaling exponent of the number of (nearly) disjoint boxes necessary to cover A:
Better understood as a crude “approximation” of the Hausdorff dimension
Are classical geostatistics Applicable to rain?
A
B C D
E F G
NT (L) ≈ −D(T)L
A) the field 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-Monofractal: D(T)=const <2 , -Multifractal: D(T)<2,
decreasing function
Test using functional box
counting
Area =LimL→ 0 L2NT (L)( )
≈LimL→ 02−D(T )L → 0;
D T( ) < 2
Functional Box counting on 3D radar rain scans
102101100
L
101
102
103
104
N(L
)
101
L
101
102
103
104
105
N(L
)
reflectivity thresholds increasing (top to bottom) by factors of 2.5 (dat from Montreal).
NT (L) ≈ −D(T)L
Log10 N(L)
Log10 L
horizontal
Vertical and horizontal
Science: Lovejoy, Schertzer and Tsonis 1987
100km1km 1km 10km
Classical geostatistics
Functional box counting on French topography: 1 -1000km
103102101100100
101
102
103
104
105
106
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
1800m100m
km
N(L)L-D
Multifractal: slopes vary with threshold
Lovejoy and Schertzer 1990
Slope =2 (required for classical geostatistics -regularity of Lebesgue measures)
Systematic resolution dependence
Fractal Codimension– Geometric definition:
• natural extension of vector subspace codimension
• If the set is A included in E (embedding space): D(A)=dim(A)< dim(E)=d
• Geometrical codimension: Cg (A)=d-D(A)
– As a consequence Cg is bounded:
• O≤Cg (A)≤d
Ex. In 3D space (dim(E)=3), the codimension of a line (dim(A)=1) is: Cg=3-1=2
Fractal Codimension (2)– stochastic processes:
• Probability of events, not the number of occurences
– Statistical definition:
• Codimension = scaling exponent of the
probability that a -dimensional ball of
resolution coversintersects A
Pr(Bλ ∩ A)=λ−c(A)
Pr(B3λ
∩ A)Pr(Bλ ∩ A)
=23
=3−C(A) ⇒ C(A)=log(3/2)log(3)
=1−log(2)log(3)
Example of the Cantor Set
Fractal Codimension (3)
Pr( λB ∩ A)~N( λB ∩ A)N( λB ∩ E)
~ −Dg(A)λ
−dλ
C(A)>d⇒ C(A) >Cg(A) (≡d)C(A)>d
D(A)≡d−C(A)⎫ ⎬ ⎭
⇒ D(A)<0
gC (A)<d=dim(E) <∞⇒ Cg(A)≡C(A)
• Relating the two definitions:
‘ latent dimension ’ paradox, in fact a statistical exponent !
Unbounded codimension
bounded codimension
Meteorological measuring network
L
Fractal set: each point is a station
9962 stations (WMO)
Number
n L( ) ∝ LD
Density ρ L( ) = n L( )L−2 ∝ L−C ; C = d − D; d = 2
The fractal dimension of the network=
1.75
Slope=D=1.75
L et al 86
C=2-1.75=0.25
Intersection Theorem
• if independent:
E1 ⊂E E2 ⊂E
C(E1 ∩ E2)=C(E1)+C(E2)
Pr(E1 ∩ E2) =Pr(E1)Pr(E2)
a trivial consequence of:
no trivial results for geometrical codimensions !
Ex. Sparse but violent regions of storms - no matter how “large” - with D<0.25 cannot be detected by the global network
Consequence: if C (E1 ∩E2 ) > D then the intersection is almost surely empty
Energy Spectra
Correlation functions, structure functions
For stationary processes X (statisticallytranslationally invariant along the time axis), wedefine
R τ( ) = X t( ) X t − τ( )
S τ( ) = X t( ) − X t − τ( )( )
2
R( )t is the correlation function, S( )t is thestructurefunctionandtherandomprocessX( )t isnonzerooverthe intervalfrom- /T 2to+ /T 2.The y arerelate d b:y
S τ( ) = 2 R 0( ) − R τ( )( )
wh en R(τ),R(0)diverge, (S τ)ca nconverge.
tT/2-T/2
X(t)
Spectral densities
%X ω( ) = X t( )−∞
∞
∫ e−iωtdt
P ω( ) =
1T
%X ω( )2
in 1-D "spectral density" (e.g. time) is defined as:
in D dimensions (e.g. space):
%X k( ) = X r( )∫ e−i k⋅rdr
P k( ) =
1N
%X k( )2
The(isotropic)
“spectrum”E k( ) = P k( )dD k
k=k∫
In D dimensions:
E ω( ) =P ω( ) + P −ω( ) =2P ω( )
In 1-D:
k = kwhere
Wiener-Khintchin Theorem
P k( ) = R r( )−∞
∞
∫ e−ik⋅rdr
This is the "Wiener Khintchin theorem" which relates the spectral density to autocorrelation function of a stationary process.
Tauberian theorem
E ω( ) ≈ω−
POWER LAWS F.T. POWER LAWS
S τ( ) ≈τ 2H ; H = −1( ) / 2
Note this is valid for 1<<3 (0≤H≤1) for S(τ), 1< (H<0) for R(τ).
Fourier scaling:
Structure function scaling:
Spectra in hydrology
1m
QuickTime™ and aAnimation decompressor
are needed to see this picture.
f295, 11293 drops
The angle averaged drop spectra5 storms, 18 triplets
Top: f142, 2nd= f145, 3rd=f295, 4th=f229, 5th=f207 thick line has theoretical slope: -2-5/3
1m-1
White noise(standard theory)
Corrsin-Obukov passive scalar theory
E k( ) ≈k−5/
Temporal Scaling of radar rain reflectivities
Temporal spectrum of the radar reflection from a single 30X27X37m pulse volume at 1km altitude. ω is in Hz.
(Duncan 1993)
E k( ) ≈k−5/
Hourly Rainfall
11 Years of hourly rainfall in Assink.De Lima 1998.
Synoptic maximum
Log10 frequency (hr-1)
Log
10
En e
r gy
0.00001 0.0001 0.001 0.01 0.1 1 r-
.
0.01
0.10.2
7
6
5
4
3
log
10
E(
ω
)
-.5 -.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
log10
ω (days
-1
)
16days
1Year
Temporal scaling and the Synoptic maximum
6.0
5.5
5.0
4.5
4.0
3.5
3.0
log
10
E(
ω
)
-.5 -.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0
log10
ω (days
-1
)
16days
1Year
16 days = « synoptic maximum » = the lifetime of planetary size structures.
Tessier, Lovejoy, Schertzer, 1996
2 days11 years
RAIN
RIVER FLOWAverage daily river flow and rain for 30 French stations (catchments< 200km2).
a= Mississippib=Susquahanac=Arkansas d= Osagee=Colorado f=McCloudg=North Nashuah=Milli= Pendeltonj=Rocky Brook
Synoptic maximum
Pandey, Lovejoy, Schertzer, 1998
Spectra of
rivers from 1 day to
70 years
Ensemble average
Low frequency rain spectra
Fraedrich and Larnder 1993
=0.5
The average spectrum from 13 stations in Germany (daily precip)
Annual peak
Synoptic maximum
Climate: Northern Hemisphere average temperatures
From Lovejoy and Schertzer 1985
=1.8
Scaling of paleotemperatures: GRIP Greenland Ice Core
2001501005000
-44
-42
-40
-38
-36
-34
-32
-30
Figure 1
Time (kyr BP)
δΟ18
(%)
0,50,0-0,5-1,0-1,5-2,0-2,5
1
2
3
4
5
6
Figure 2
log (f) (kyr)
log (E(f))
f
-1.4
10
10
-1
High resolution (200 yr average) record the GRIP Greenland ice
core (Johnsen et al., 1992; GRIP members, 1993; Dansgaard et al.,
1993): •3,000 m long, 1,200 data points • sharp fluctuations at small time scales.
The power spectrum of the data (log-log plot); •global straight line is an indication of scaling. •no obvious frequency at (20 kyr)-1 or (40 kyr)-1
Schmitt, Schertzer Lovejoy 1995
Spatial Spectrum of radar reflectivity
k-1.45
Horizontal spectrum of 256X256 (McGill) radar scan with 75m resolution (from Tessier et al 1993).
Montreal Clouds: spectra
10
15
20
25
30
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5
log E(k)
log k (1/m)
10
10
5
10
15
20
25
30
35
-3.5 -3 -2.5 -2 -1.5 -1 -0.5
log E(k)
log k (1/m)
10
10
Spectra of the smaller scenes, seperated in the vertical for clarity, with power law regressions shown
Sachs, Lovejoy, Schetzer 2002
Larger scenes
What is the outer scale of atmospheric turbulence?
Spectra of hundreds of satellite images spanning the scale range 1-5000 km, and 38 clouds spectra (1m-1km) from ground camera.
A multifractal analysis is more informative (see below).
Leff > 5 000 km !!
7
8
9
10
11
12
13
14
15
-4 -3 -2 -1 0 1 2 3
log E(k)10
log k (1/km)
AVHRR 12
GMS 5
SPOT
Photography
10
Lovejoy and Schertzer 2002
Topo-graphy
Spectrum
(20000km)-1 (1m)-1
Energy spectra over a scale range of 108 Global (ETOPO5, 10km), continental US (GTOPO30: 1km and 90m), Lower Saxony, 20cm).
Inadequate dynamic range
Slope -1.8
Fractional Integration and Differentiation
Brownian motion
Brownian motion X(t) is given by:dX
dt
= p ( t ) X t( ) = p ′t( ) d ′t
− ∞
t
∫
p(t) is a white noise.
In Fourier space:
i ω
˜
X ( ω ) =˜
p ( ω )
The structure function:
S τ( ) = τ σ
2
Fractional Integration, Fractional Brownian Motion
Consider the generalization:
d
s
X
s
dt
s
= p ( t )
withsoluti :on
X
s
t( ) = I
s
p t( )( )
whereIsisthefractionalintegral(s>0)derivative( <s 0) of the
function .pInfourierspace this isparticularlysimple:
i ω( )
s
˜
X
s
t( ) =˜p ω( )
˜
X
s
t( ) =˜p ω( ) i ω( )
− s
henc e for thespectr :um
E
Xs
ω( ) = E
p
ω( ) ω
− 2 s
= σ
2
ω
− 2 s
Real space properties
The structure function of Xs is obtained by the Tauberian theoremas above:
S τ( ) = τ
2 H
withH=-1s /2.For thesolutionXs(t):
X
s
t( ) = p t( ) ∗
Θ t( )
t
1 − s
⎡
⎣
⎢
⎤
⎦
⎥
whereΘ t( ) istheHeavisidefunction(=1,t<0,=0, ≥0t )andwehaveusedthetheconvol ution theorem ontheequation
˜
Xs
t( ) = ˜p ω( ) i ω( )
− s
and:
Thi s yields:
i ω( )
− s
→
F . T .
1
Γ s( )
Θ t( )
t
1 − s
⎡
⎣
⎢
⎤
⎦
⎥
X
s
t( ) =
1
Γ s( )
p ′t( )
t − ′t( )
s − 1
d ′t
− ∞
t
∫
Dimensions and spectraFor monofractal functions:
C=H=(-1)/2
So that for surfaces defined by d dimensional processes:
Dsurf=d+1-C=d+(3-)/2
Although this relation has been frequently used to estimate Dsurf
from , it is only valid for monofractals
Probability Distributions
Pr x > s( ) = p x( )dxs
∞
∫
Pr x > s( ) < e−as; s>>1
Pr x > s( ) ≈s−qD ; s>>1
(Tail) Cumulative Distribution Function
“Thin tailed” distributions
“Fat tailed” distributions (e.g. “Pareto” / power law)
Moments xq = xqdPr
−∞
∞
∫ xq → ∞; q≥qD
Hydrometeorological long time series
Return period 100 years (algebraic law)
Return period 1000 years (exponential law)
Padova series (Italy): empirical probability distribution (dots), normal fit (continouous line) and aymptotic power-law (dashed line).
Bendjoudi et Hubert Rev. Sci. Eau,1999
qD ≈4
Radar reflectivity of rain:
Probability distributions
Probability distribution of radar reflectivities from 10 constant altitude maps (resolution varying from 0.25 to 2.5 km, range 20 to 200km).
qD=1.06
From Schertzer and Lovejoy 1987.
French rivers and precipitation (<200km2)
-6
-5
-4
-3
-2
-1
0
210-1
log Pr (P' > P)
10
log P
10
210-1
-6
-5
-4
-3
-2
-1
0
log Q
10
log Pr (Q' > Q)
10
Prob of a daily river flow Q' exceeding Q from 30 time series from the corresponding river.
Prob of a daily rainfall accumulation P' exceeding P from 30 time series,
France.qD = 2.7
qD = 3.6
Pr ′P > P( )≈P−qD
PqD → ∞; q>qD
French river, small basins Tessier, L+S 1996
Probability distributions of Normalized US Rivers: Divergence of moments
Pr ′Q >Q( ) ≈Q−qD
QqD → ∞; q> qDExtreme events
20 US rivers with basins in the range 4-106 km2; 10-75 years in length
Pandey, Lovejoy, Schertzer 1998
Temperature distributions, northern hemisphere
4 years
16 years
64 years
qD=5
Lovejoy and Schertzer 1985 (data from Jones et al 1982)
Temperature probability distributions for paleotemperatures
qD=5
350 years
1400 years
5600 years22400 years
Lovejoy and Schertzer 1985 (data from Greenland Camp Century core)
Conclusions1. Scaling/scale invariant sets are fractalsFractal dimensions and codimensions are scaling exponents
2. Scaling fields multifractals, spectral analysis.
3. Rain, temperature, topography, river flow show wide range scaling.
4. Probabilities can have “fat tails”: slow, algebraic fall-off.
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