Part 2: Introduction to scaling in precipitation and hydrology

1
Part 2 Introduction to scaling in precipitation
and hydrology
S. Lovejoy, Physics, McGill
D. Schertzer, ENPC, Paris
Dam Safety Interest Group, Experts meeting 9
November, 2004
2
Overview

• Fractal Sets, dimensions
• Co-dimensions
• Spectra
• Examples from hydrology
• Nonclassical Probability distributions

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Some Classical Fractal sets and their dimension

• Cantor set
• Cantor square
• Devils staircase
• Koch snowflake
• Sierpinski Triangle
• Sierpinski Pyramid
• Peano curve

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Cantor set

• Let us start with

and let us iterate
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Koch snowflake

• Let us start with

and let us iterate
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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,

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Fractal Dimension (2)

• Lets use a variable D for dimension, l for
  magnification, and N(l) for the number of
  identical shapes
• D
• N l
• Or
• D log N / log l

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First Examples

• Cantor Set

D log N / log l log 2 / log 3 0.63
-Koch Snowflake
D log N / log l log 4 / log 3 1.26.
9
Rain Fall Events

• Daily Rain Fall Events in Dedougou (SW
  Africa)1922-1966
• Each line is a different year,
• eachblack point a rainy day.
• Cantor-like set
• Dlog(7)/log(12)
• i.e. divide into 12 parts keep only 7..
• (Hubert et Carbonnel, 1990)

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Cantor Square
D log 4/ log 31.26
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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 (b-model).

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b-model

• The b-model for l  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 in?nite 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)

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Sierpinski Triangle

• or

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Sierpinski Triangle (2)

• D log 3 /log 2 1.58.

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Sierpinski Pyramid

• First iteration

10 th iteration

D log 4/ log 2 But DT 1
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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 11 and bicontinuous transformations.

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Peano Curve

• motif

iterations
Dlog 9 / log 3 2 i.e. it is a plane filling
line
A model of hydraulic network, From Steinhaus 1962
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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 D1.35
(Lovejoy, 1982)
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Isotropic Scale Invariance and fractal sets
Dscale invariant
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Box Counting Dimension
Defined as the scaling exponent of the number
of (nearly) disjoint boxes necessary to cover A

• 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).

Better understood as a crude approximation of
the Hausdorff dimension
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Are classical geostatistics Applicable to rain?
Test using functional box counting
-Classical geostatistics D(T)2 -Monofractal
D(T)const lt2 , -Multifractal D(T)lt2,
decreasing function
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.
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Functional Box counting on 3D radar rain scans
Classical geostatistics
Log10 N(L)
Vertical and horizontal
horizontal
100km
1km
1km
10km
Log10 L
L
L
reflectivity thresholds increasing (top to
bottom) by factors of 2.5 (dat from Montreal).
Science Lovejoy, Schertzer and Tsonis 1987
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Functional box counting on French topography 1
-1000km
Slope 2 (required for classical geostatistics
-regularity of Lebesgue measures)
Multifractal slopes vary with threshold
N(L)?L-D
100m
1800m
Systematic resolution dependence
3600m
km
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
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Fractal Codimension

• Geometric definition
• natural extension of vector subspace codimension
• If the set is A included in E (embedding space)
  D(A)dim(A)lt dim(E)d
• Geometrical codimension Cg (A)d-D(A)
• As a consequence Cg is bounded
• OCg (A)d

Ex. In 3D space (dim(E)3), the codimension of a
line (dim(A)1) is Cg3-12
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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 l
  covers/intersects A

Example of the Cantor Set
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Fractal Codimension (3)

• Relating the two definitions

bounded codimension
Unbounded codimension
 latent dimension  paradox, in fact a
statistical exponent !
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Meteorological measuring network
Fractal set each point is a station
9962 stations (WMO)
L
Number
Density
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The fractal dimension of the network 1.75
SlopeD1.75
C2-1.750.25
L et al 86
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Intersection Theorem

• if independent

a trivial consequence of
no trivial results for geometrical codimensions !
then the intersection is almost surely empty
Consequence if
Ex. Sparse but violent regions of storms - no
matter how large - with Dlt0.25 cannot be
detected by the global network
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Energy Spectra
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Correlation functions, structure functions
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Spectral densities
in 1-D "spectral density" (e.g. time) is defined
as
in D dimensions (e.g. space)
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The(isotropic) spectrum
In 1-D
In D dimensions
where
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Wiener-Khintchin Theorem
This is the "Wiener Khintchin theorem" which
relates the spectral density to autocorrelation
function of a stationary process.
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Tauberian theorem
POWER LAWS ?F.T. POWER LAWS
Fourier scaling
Structure function scaling
Note this is valid for 1lt?lt3 (0H1) for S(?),
1lt? (Hlt0) for R(?).
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Spectra in hydrology
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f295, 11293 drops
1m
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The angle averaged drop spectra5 storms, 18
triplets
Corrsin-Obukov passive scalar theory
White noise (standard theory)
1m-1
Top f142, 2nd f145, 3rdf295, 4thf229,
5thf207 thick line has theoretical slope -2-5/3
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Temporal Scaling of radar rain reflectivities
Temporal spectrum of the radar reflection from a
single 30X27X37m pulse volume at 1km altitude. w
is in Hz.
(Duncan 1993)
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Hourly Rainfall
Synoptic maximum
Log10 Energy
Log10 frequency (hr-1)
11 Years of hourly rainfall in Assink. De Lima
1998.
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Temporal scaling and the Synoptic maximum
Average daily river flow and rain for 30 French
stations (catchmentslt 200km2).
RIVER FLOW
16 days  synoptic maximum  the lifetime of
planetary size structures.
RAIN
11 years
2 days
Tessier, Lovejoy, Schertzer, 1996
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Spectra of rivers from 1 day to 70 years
a Mississippi bSusquahana cArkansas d
Osage eColorado fMcCloud gNorth
Nashua hMill i Pendelton jRocky Brook
Synoptic maximum
Ensemble average
Pandey, Lovejoy, Schertzer, 1998
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Low frequency rain spectra
b0.5
Annual peak
Synoptic maximum
The average spectrum from 13 stations in Germany
(daily precip)
Fraedrich and Larnder 1993
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Climate Northern Hemisphere average temperatures
b1.8
From Lovejoy and Schertzer 1985
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Scaling of paleotemperatures GRIP Greenland Ice
Core

• 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
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Spatial Spectrum of radar reflectivity
k-1.45
Horizontal spectrum of 256X256 (McGill) radar
scan with 75m resolution (from Tessier et al
1993).
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Montreal Clouds spectra
Spectra of the smaller scenes, seperated in the
vertical for clarity, with power law regressions
shown
Larger scenes
Sachs, Lovejoy, Schetzer 2002
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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 gt 5 000 km !!
Lovejoy and Schertzer 2002
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Topo-graphy Spectrum
Inadequate dynamic range
Slope -1.8
(20000km)-1
(1m)-1
Energy spectra over a scale range of 108 Global
(ETOPO5, 10km), continental US (GTOPO30 1km and
90m), Lower Saxony, 20cm).
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Fractional Integration and Differentiation
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Brownian motion
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Fractional Integration, Fractional Brownian Motion
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Real space properties
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Dimensions and spectra
For monofractal functions CH(b-1)/2 So that
for surfaces defined by d dimensional
processes Dsurfd1-Cd(3-b)/2 Although this
relation has been frequently used to estimate
Dsurf from b, it is only valid for monofractals
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Probability Distributions
(Tail) Cumulative Distribution Function
Thin tailed distributions
Fat tailed distributions (e.g. Pareto / power
law)
Moments
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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
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Radar reflectivity of rain Probability
distributions
qD1.06
Probability distribution of radar reflectivities
from 10 constant altitude maps (resolution
varying from 0.25 to 2.5 km, range 20 to 200km).
From Schertzer and Lovejoy 1987.
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French rivers and precipitation (lt200km2)
qD  3.6
Prob of a daily rainfall accumulation P'
exceeding P from 30 time series, France.
qD  2.7
Prob of a daily river flow Q' exceeding Q from
30 time series from the corresponding river.
Tessier, LS 1996
French river, small basins
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Extreme events
Probability distributions of Normalized US
Rivers Divergence of moments
Pandey, Lovejoy, Schertzer 1998
20 US rivers with basins in the range 4-106 km2
10-75 years in length
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Temperature distributions, northern hemisphere
qD5
64 years
4 years
16 years
Lovejoy and Schertzer 1985 (data from Jones et al
1982)
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Temperature probability distributions for
paleotemperatures
qD5
22400 years
350 years
5600 years
1400 years
Lovejoy and Schertzer 1985 (data from Greenland
Camp Century core)
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Conclusions
1. Scaling/scale invariant sets are
fractals Fractal dimensions and codimensions are
scaling exponents
2. Scaling fields multifractals, spectral
analysis.

1. Rain, temperature, topography, river flow show
   wide range scaling.

4. Probabilities can have fat tails slow,
algebraic fall-off.