Transcript Document 7169195

Are Forest Fires
HOT?
Jean Carlson, UCSB
Background
•
•
•
Much attention has been given to “complex
adaptive systems” in the last decade.
Popularization of information, entropy, phase
transitions, criticality, fractals, self-similarity,
power laws, chaos, emergence, selforganization, etc.
Physicists emphasize emergent complexity via
self-organization of a homogeneous substrate
near a critical or bifurcation point (SOC/EOC)
18 Sep 1998
Forest Fires: An Example of Self-Organized
Critical Behavior
Bruce D. Malamud, Gleb Morein, Donald L. Turcotte
4 data sets
Criticality and power laws
• Tuning 1-2 parameters  critical point
• In certain model systems (percolation, Ising, …) power
laws and universality iff at criticality.
• Physics: power laws are suggestive of criticality
• Engineers/mathematicians have opposite interpretation:
–
–
–
–
Power laws arise from tuning and optimization.
Criticality is a very rare and extreme special case.
What if many parameters are optimized?
Are evolution and engineering design different? How?
• Which perspective has greater explanatory power for
power laws in natural and man-made systems?
Highly
Optimized
Tolerance (HOT)
•
•
•
•
•
Complex systems in biology, ecology, technology,
sociology, economics, …
are driven by design or evolution to highperformance states which are also tolerant to
uncertainty in the environment and components.
This leads to specialized, modular, hierarchical
structures, often with enormous “hidden” complexity,
with new sensitivities to unknown or neglected
perturbations and design flaws.
“Robust, yet fragile!”
“Robust, yet fragile”
• Robust to uncertainties
– that are common,
– the system was designed for, or
– has evolved to handle,
• …yet fragile otherwise
• This is the most important feature of
complex systems (the essence of HOT).
Robustness of
HOT systems
Fragile
Robust
(to known and
designed-for
uncertainties)
Fragile
(to unknown
or rare
perturbations)
Robust
Uncertainties
Complexity
Robustness
Interconnection
Aim: simplest
possible story
The simplest possible spatial model of HOT.
Square site
percolation
or
simplified
“forest fire”
model.
Carlson and Doyle,
PRE, Aug. 1999
empty square lattice
occupied sites
not connected
connected clusters
20x20 lattice
Assume one
“spark” hits the
lattice at a single
site.
A “spark” that hits
an empty site does
nothing.
A “spark” that hits
a cluster causes
loss of that cluster.
Yield = the density after one spark
yield
density
loss
Average over configurations.
yield =
density=.5
4 * .25  2 * .375
0.2917 
6
avg avg
trees spark
1
no sparks
0.9
“critical point”
0.8
Y=
(avg.)
yield
0.7
0.6
0.5
0.4
N=100
0.3
sparks
0.2
0.1
0
0
0.2
0.4
0.6
0.8
 = density
1
1
0.9
Y=
0.8
(avg.)
yield
0.7
“critical point”
limit
N
0.6
0.5
0.4
0.3
0.2
c = .5927
0.1
0
0
0.2
0.4
0.6
0.8
 = density
1
Y
Fires
don’t
matter.
Cold

Y
Everything burns.
Burned

Critical point
Y

This picture is very generic and “universal.”
Y
critical
phase
transition

Statistical physics:
Phase transitions,
criticality, and
power laws
2
10
Power laws
cumulative
frequency
Criticality
1
10
0
10
-1
10
0
10
1
10
2
10
3
10
cluster size
4
10
Average
cumulative
distributions
fires
clusters
size
high density
cumulative
frequency
low density
Power laws:
only at the
critical point
cluster size
Self-organized criticality
(SOC)
Create a dynamical
system around the
critical point
yield
density
Self-organized criticality (SOC)
N  N lattice
Iterate on:
1. Pick n sites at random, and
grow new trees on any which
are empty.
2. Spark 1 site at random. If
occupied, burn connected
cluster.
2
1  n  N 2
Use n  .1N in examples .
lattice
distribution
fire
density
yield
fires
0.8
0.6
0.4
0.2
0
0
200
400
600
800
3
10
-.15
2
10
1
10
0
10 0
10
1
10
2
10
3
10
4
10
1000
18 Sep 1998
Forest Fires: An Example of Self-Organized
Critical Behavior
Bruce D. Malamud, Gleb Morein, Donald L. Turcotte
4 data sets
3
10
2
10
-1/2
1
10
SOC FF
0
10
-2
10
-1
10
0
10
1
10
2
10
Exponents are way off
3
10
4
10
Edge-of-chaos, criticality,
self-organized criticality
(EOC/SOC)
Essential claims:
• Nature is adequately described yield
by generic configurations (with
generic sensitivity).
• Interesting phenomena is at
criticality (or near a bifurcation).
yield
density
• Qualitatively appealing.
• Power laws.
• Yield/density curve.
• “order for free”
• “self-organization”
• “emergence”
• Lack of alternatives?
• (Bak, Kauffman, SFI, …)
• But...
• This is a testable hypothesis
(in biology and engineering).
• In fact, SOC/EOC is very rare.
Self-similarity?
?
Forget random,
generic
configurations.
Would you design a
system this way?
What about high
yield configurations?
Barriers
What about high
yield configurations?
Barriers
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
• Rare, nongeneric, measure zero.
• Structured, stylized configurations.
• Essentially ignored in stat. physics.
• Ubiquitous in
• engineering
• biology
• geophysical phenomena?
What about high
yield configurations?
Highly Optimized Tolerance (HOT)
critical
Cold
Burned
Why power laws?
Almost any
distribution
of sparks

Optimize
Yield
Power law
distribution
of events
both analytic and numerical results.
Special cases
Singleton
(a priori
known spark)
Uniform
spark
Optimize
Yield
Optimize
Yield
No fires
Uniform
grid
Special cases
No fires
In both cases, yields 1 as N .
Uniform
grid
Generally….
1.
2.
3.
4.
Gaussian
Exponential
Power law
….
Optimize
Yield
Power law
distribution
of events
Probability distribution (tail of normal)
2.9529e-016
0.1902
High probability region
5
10
15
20
25
30
5
2.8655e-011
10
15
20
25
30
4.4486e-026
Grid design:
optimize the
position of “cuts.”
cuts = empty sites in
an otherwise fully
occupied lattice.
Compute the global optimum for this constraint.
Optimized grid
Small events likely
large events
are unlikely
density = 0.8496
yield = 0.7752
Optimized grid
density = 0.8496
yield = 0.7752
1
0.9
0.8
High yields.
0.7
0.6
grid
0.5
0.4
random
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
Local incremental
algorithm
“grow” one site at a
time to maximize
incremental (local)
yield
density= 0.8
yield = 0.8
“grow” one site at a
time to maximize
incremental (local)
yield
density= 0.9
yield = 0.9
“grow” one site at a
time to maximize
incremental (local)
yield
Optimal
density= 0.97
yield = 0.96
“grow” one site at a
time to maximize
incremental (local)
yield
Very sharp “phase transition.”
1
0.9
optimized
0.8
0.7
0.6
0.5
0.4
random
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
density
1
0
10
“critical”
-1
10
Cum.
Prob.
“grown”
-2
10
-3
10
All produce
Power laws
grid
-4
10
0
10
1
10
2
3
10
10
size
SOC and HOT have very
different power laws.
d
d=1
HOT
d
d=1
SOC
d 1

10
1

d
• HOT  decreases with dimension.
• SOC increases with dimension.
• HOT yields compact events of nontrivial size.
• SOC has infinitesimal, fractal events.
HOT
SOC
infinitesimal
size
large
A HOT forest fire abstraction…
Fire suppression
mechanisms must
stop a 1-d front.
Burnt regions are 2-d
Optimal strategies
must tradeoff
resources with risk.
Generalized
“coding” problems
Optimizing d-1 dimensional
cuts in d dimensional spaces.
Data compression
Web
Fires
6
5
Cumulative
(Crovella)
4
Frequency
Data
compression
WWW files
Mbytes
(Huffman)
d=1
d=0
3
2
Forest fires
1000 km2
1
(Malamud)
0
d=2
Los Alamos fire
-1
-6
-5
-4
-3
-2
-1
0
1
Decimated data
Size of events
Log (base 10)
(codewords, files, fires)
2
6
Web files
5
Codewords
4
Cumulative
Frequency
-1
3
Fires
2
-1/2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
Size of events
Log (base 10)
2
Data + Model/Theory
6
DC
5
WWW
4
3
2
1
SOC  = .15
Forest fire
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
Data + PLR HOT Model
6
DC
5
WWW
4
3
FF
2
1
0
-1
-6
-5
-4
-3
-2
-1
0
1
2
SOC and HOT are extremely different.
SOC
HOT
Data
Max event size
Infinitesimal
Large
Large
Large event shape
Fractal
Compact
Compact
Slope 
Small
Large
Large
Dimension d
d-1
1/d
1/d
HOT
SOC
SOC and HOT are extremely different.
SOC
HOT & Data
Max event size
Infinitesimal
Large
Large event shape
Fractal
Compact
Slope 
Small
Large
Dimension d
d-1
1/d
HOT
SOC
HOT: many mechanisms

grid
grown or evolved
DDOF
All produce:
• High densities
• Modular structures reflecting external disturbance patterns
• Efficient barriers, limiting losses in cascading failure
• Power laws
Robust,
yet fragile?
Extreme robustness and extreme hypersensitivity.
Small
flaws
Robust,
yet fragile?
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
If probability of sparks changes.
disaster
Tradeoffs?
Sensitivity to:
sparks
assumed
p(i,j)
flaws
Optimized
Critical percolation
and SOC forest fire
models
HOT forest fire models
• SOC & HOT have completely different characteristics.
• SOC vs HOT story is consistent across different models.
Characteristic Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile
Events/structure
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
External behavior Complex
Internally
Simple
Statistics
Nominally simple
Complex
Power laws
Power laws
only at criticality at all densities
Characteristic
Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile.
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
Events/structure
Characteristics
External behavior Complex
Internally
Simple
Nominally simple
Complex
Statistics
Power laws
at all densities
Toy models
Power laws
only at criticality
?
• Power systems
• Computers
Examples/
• Internet
• Software
Applications
• Ecosystems
• Extinction
• Turbulence
Characteristic
Critical
HOT
Densities
Yields
Robustness
Low
Low
Generic
High
High
Robust, yet fragile.
Events/structure
Generic, fractal
self-similar
Structured, stylized
self-dissimilar
External behavior Complex
Internally
Simple
Nominally simple
Complex
Statistics
Power laws
at all densities
Power laws
only at criticality
But when we look in detail
at any of these examples...
…they have all the HOT
features...
• Power systems
• Computers
• Internet
• Software
• Ecosystems
• Extinction
• Turbulence
HOT features of ecosystems
• Organisms are constantly challenged by
environmental uncertainties,
• And have evolved a diversity of mechanisms to
minimize the consequences by exploiting the
regularities in the uncertainty.
• The resulting specialization, modularity, structure,
and redundancy leads to high densities and high
throughputs,
• But increased sensitivity to
novel perturbations not included
in evolutionary history.
• Robust, yet fragile!
HOT and evolution : mutation and natural selection in a community
• Begin with 1000 random lattices, equally divided
between tortoise and hare families
• Each parent gives rise to two offspring
• Small probability of mutation per site
• Sparks are drawn from P(i,j)
• Fitness= Yield (1 spark for hares, full P(i,j) for tortoises)
• Death if Fitness<0.4
• Natural selection acts on remaining lattices
• Competition for space in a community of bounded size
Barriers to cascading failure: an abstraction of biological mechanisms
for robustness
Tong Zhou
Fast mutators
(hares)
Slow mutators
(tortoises)
Hares:
-noisy patterns
-lack protection
for rare events
Genotype (heritable traits): lattice layout
Phenotype (characteristics which can be observed in the
environment): cell sizes and probabilities
Fitness (based on performance in the organisms lifetime): Yield
(Primitive) Punctuated Equilibrium:
Hares win in the
short run.
But face episodic
extinction due to
rare events (niche
protects 50).
Tortoises take
over, and
diversity increases.
Until hares win
again.
hares
tortoises
Tortoise population exhibits power laws
Hares have excess large events
Convergent Evolution: Species which evolve in spatially
separate, but otherwise similar habitats develop similar
phenotypic traits. They are not genetically close, but have
developed similar adaptations to their environmental niches.
Our analogy: different runs with the same P(i,j) evolve towards
phenotypically similar, genotypically dissimilar lattice populations
The five great extinctions are associated with a rate of
disappearance of species well in excess of the background, as
deduced from the fossil record.
Paleontologists attribute these to rare disturbances, such as
meteor impacts. Robust, yet Fragile!
Punctuated Equilibrium (left) vs. Gradualism (right): PE: rapid, bursts
of change
(horizontal lines),
followed by
extended periods
of relative
stability (vertical
lines), followed by
extinction.
Our analogy: after
a transient period
of rapid evolution
lattices have
barrier patterns,
which are
relatively
stable until
extinction
Large extinction
events are typically
followed by
increased diversity.
The recovery period
is the time lapse
between the peak
extinction rate, and
the maximum rate
of origination of
new species.
Our analogy:
extinction of the
hares is are
followed by
diversification of
both families
The current mass extinction is frequently attributed to overpopulation
and causes which can be attributed to humans, such as deforestation
Our analogy: large events
can be due to rare
disturbances, especially
if they are not not part of
the evolutionary
history of the (vulnerable)
species.
Robust, yet Fragile!
Evolution by natural selection in
coupled communities with different environments:
Uniform
Sparks
Skewed
Sparks
Fitness based on a single spark.
Eliminate protective niches.
Fixed maximum capacity for each habitat.
Fast and slow mutation rates (rate subject to mutation).
Coupled Habitats: Fast and slow mutators compete with each other
in each habitat, with a small chance of migration from one habitat
to the other.
Efficient barrier patterns develop in the uniform habitat. After
an extinction in the skewed habitat, uniform lattices invade, and
subsequently lose their lower right barriers: a successful strategy
in the short term, but leads to vulnerability on longer time scales
Patterns of extinction, invasion, evolution
Over an extended time window, spanning the two previous extinctions,
we see the long term fitness <Y> initially increases as the invading
lattices adapt to their new environment. This is followed by a sudden
decline when the lattices lose a barrier. This adaptation is beneficial for
common events, but fatal for rare events.
Evolution and extinction
HOT
fitness
Specialization
Disturbance
density
HOT and Evolution
• Robustness in an uncertain environment
provides a mechanism which leads to a
variety of phenomena consistent with
observations in the fossil record (large
extinctions associated with rare disturbance,
punctuated equilibrium, genotypic divergence,
phenotypic convergence).
• In a model which retains abstract notions of genotype, phenotype,
and fitness, highly evolved lattices develop efficient barriers to
cascading failure, similar to those obtained by deliberate design.
• Robustness barriers are central in natural and man made complex
systems. They may be physical (skin) or in the state space
(immune system) of a complex, interconnected system.
• Forest Fires: a case where a common disturbance type (fires)
• Acts over a broad range of scales (terrestrial ecosystems)
• Power law statistics describing the distribution of fire sizes.
• Exponents are consistent with the simplest HOT model involving
optimal allocation of resources (suppress fires).
• Evolutionary dynamics are much more complex.
18 Sep 1998
Forest Fires: An Example of Self-Organized
Critical Behavior
Bruce D. Malamud, Gleb Morein, Donald L. Turcotte
4 data sets
All four data sets are fit with the PLR model with α=1/2.
4
10
3
10
Rank
order
2
10
1
10
0
10
-4
10
-3
10
-2
10
-1
10
Size (1000 km2)
0
10
1
10
Forest fires
dynamics
Weather
Spark sources
Intensity
Frequency
Extent
Flora and fauna
Topography
Soil type
Climate/season
Los Padres National Forest
Max Moritz
Red: human ignitions
(near roads)
Yellow: lightning (at
high altitudes in
ponderosa pines)
Brown: chaperal
Pink: Pinon
Juniper
Ignition and vegetation patterns in Los Padres National Forest
Santa Monica Mountains
Max Moritz and Marco Morais
SAMO Fire History
Fires are compact regions
of nontrivial area.
Fires 1930-1990
Fires 1991-1995
4
10
4 Science data sets
+LPNF
+ HFIREs (SA=2)
3
10
Cumulative
P(size)
PLR
2
10
SM
1
10
0
10 -4
10
-3
10
-2
10
-1
10
Rescaling data for frequency and large size cutoff
gives excellent agreement, except for the SM data set
0
1
10
10
size
We are developing a realistic fire spread model
HFIREs:
GIS data for
Landscape images
Modelsbyfor
Fuel Succession
Regrowth modeled
vegetation
succession models
1996 Calabasas Fire
Historical fire spread
Simulated fire spread
Suppression?
HFIREs Simulation Environment
• Topography and vegetation initialized with recent observations
(100 m GIS resolution) for Santa Monica Mountains
• Weather based on historical data (SA rate treated as a separate
parameter)
• Fire spread modeled using Rothermel equations
• Fuel regrowth based on succession models
• 8 ignitions per year
• Weather sampled stochastically from distribution (4 day SA at
prescribed rate)
• Fire terminates in a cell when rate of spread (RoS) falls below
a specified value
• Generate 600 year catalogs, omit data for first 100 years in our
statistics
Preliminary results from the HFIRES simulations
(no extreme weather conditions included)
(we have generated many 600 year catalogs varying both
extreme weather and suppression)
Fire scar shapes are compact
Data: typical five year period
HFIREs simulations: typical five
year period
4
10
PLR
3
10
HFIRE
SA=2,
10
RoS=
.033 m/s,
FC=46 yr
2
1
10
LPNF
0
10 -4
10
-3
10
-2
10
-1
10
0
10
1
10
Excellent agreement between data, HFIREs and the PLR HOT model
4
10
4 Science
+LPNF
+ Hfire (SA=2)
3
10
PLR
2
10
SM
1
10
0
10 -4
10
-3
10
-2
10
-1
10
0
10
• small: incomplete
SM
discrepancy? • large: short catalog, or aggressive human intervention
(inhomogeneous)
1
10
Deviations from typical regional values for suppression (RoS) and
the number of extreme weather events (SA), lead to deviations from
the α=1/2 fit, and unrealistic values of the fire cycle (FC)
3
10
SA= 1, 2, 4, 6
2
10
1
10
SA=0, vary stopping rate
0
10 0
10
1
10
2
10
3
10
4
10
5
10
SA=0, =.65
SA=2, =.5
SA=4, =.5
SA=6, =.3
Increased rate of SA leads to
flatter curves, shorter fire cycles
Type conversion!
3
10
2
10
1
10
0
10 2
10
3
10
4
10
5
10
What is the optimization problem?
(we have not answered this question for fires today)
Plausibility Argument:
• Fire is a dominant disturbance which shapes terrestrial ecosystems
• Vegetation adapts to the local fire regime
• Natural and human suppression plays an important role
• Over time, ecosystems evolve resilience to common variations
• But may be vulnerable to rare events
• Regardless of whether the initial trigger for the event is large or small
HFIREs Simulations:
• We assume forests have evolved this resiliency (GIS topography
and fuel models)
• For the disturbance patterns in California (ignitions, weather models)
• And study the more recent effect of human suppression
• Find consistency with HOT theory
• But it remains to be seen whether a model which is optimized or
evolves on geological times scales will produce similar results
The shape of trees
by Karl Niklas
Simulations of selective
pressure shaping early
plants
• L: Light from the sun (no overlapping branches)
• R: Reproductive success (tall to spread seeds)
• M: Mechanical stability (few horizontal branches)
• L,R,M: All three look like real trees
Our hypothesis is that robustness in an uncertain environment is the
dominant force shaping complexity in most biological, ecological, and
technological systems