Are Forest Fires HOT? Jean Carlson, UCSB. Background Much attention has been given to “complex adaptive systems” in the last decade. Popularization of

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, self-organization, etc.

• Physicists emphasize emergent complexity via self-organization of a homogeneous substrate near a critical or bifurcation point (SOC/EOC)

Forest Fires: An Example of Self-Organized Critical BehaviorBruce D. Malamud, Gleb Morein, Donald L. Turcotte

18 Sep 1998

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 high-performance 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

Robust

Fragile

Robust(to known anddesigned-foruncertainties)

Fragile(to unknown

or rareperturbations)

Uncertainties

ComplexityRobustness

Aim: simplest possible story

Interconnection

Square site percolation or simplified “forest fire” model.

The simplest possible spatial model of HOT.

Carlson and Doyle,PRE, Aug. 1999

empty square lattice occupied sites

connectednot connected clusters

20x20 lattice

A “spark” that hits an empty site does

nothing.

Assume one “spark” hits the

lattice at a single site.

A “spark” that hits a cluster causes

loss of that cluster.

Yield = the density after one spark

yield density loss

density=.5

yield =

6

375.*225.*40.2917

Average over configurations.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y =(avg.)yield

= density

“critical point”

N=100

no sparks

sparks

sparktreesavgavg

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

limit N

“critical point”Y =(avg.)yield

= density

c = .5927

Cold

Fires don’t matter.

Y

Y

Burned

Everything burns.

Critical point

Y

critical phase transition

This picture is very generic and “universal.”

Y

Statistical physics:Phase transitions,

criticality, and power laws

100

101

102

103

104

10-1

100

101

102

Power laws

Criticality

cluster size

cumulativefrequency

Averagecumulativedistributions

clusters

fires

size

Power laws: only at the critical pointlow density

high density

cluster size

cumulativefrequency

Self-organized criticality (SOC)

Create a dynamical system around the

critical point

yield

density

Self-organized criticality (SOC)

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.

.examplesin1.Use 2Nn

latticeNN

21 Nn

lattice

fire

distribution

density

yieldfires

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

100

101

102

103

104

100

101

102

103

-.15


18 Sep 1998

4 data sets

10-2

10-1

100

101

102

103

104

100

101

102

103

SOC FF

Exponents are way off

-1/2

Edge-of-chaos, criticality, self-organized criticality

(EOC/SOC)

yieldyield

density

Essential claims:

• Nature is adequately described by generic configurations (with generic sensitivity).

• Interesting phenomena is at criticality (or near a bifurcation).

• 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?

What about high yield

configurations?

?

Forget random, generic

configurations.

Would you design a system this way?

Barriers

Barriers


configurations?

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

• Rare, nongeneric, measure zero.• Structured, stylized configurations.• Essentially ignored in stat. physics.• Ubiquitous in

• engineering• biology• geophysical phenomena?


configurations?

critical

Cold

Highly Optimized Tolerance (HOT)

Burned

Why power laws?Why power laws?

Almost any distribution

of sparks

OptimizeYield

Power law distribution

of events

both analytic and numerical results.

Special casesSpecial cases

Singleton(a priori

known spark)

Uniformspark

OptimizeYield

Uniformgrid

OptimizeYield

No fires

Special casesSpecial cases

No fires

Uniformgrid

In both cases, yields 1 as N .

Generally….Generally….

1. Gaussian2. Exponential3. Power law4. ….

OptimizeYield

Power law distribution

of events

5 10 15 20 25 30

5

10

15

20

25

30

0.1902 2.9529e-016

2.8655e-011 4.4486e-026

Probability distribution (tail of normal)

High probability region

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

density = 0.8496yield = 0.7752

Small events likely

large events are unlikely

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

random

grid

High yields.

Optimized grid

density = 0.8496yield = 0.7752

“grow” one site at a time to maximize incremental (local) yield

Local incrementalalgorithm

density= 0.8yield = 0.8


density= 0.9yield = 0.9


Optimal density= 0.97yield = 0.96


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

random

density

Very sharp “phase transition.”

optimized

100

101

102

103

10-4

10-3

10-2

10-1

100

grid

“grown”

“critical”

size

Cum.Prob.

All producePower laws

HOT

SOC

d=1

dd=1d

• HOT decreases with dimension.• SOC increases with dimension.

SOC and HOT have very different power laws.

1

d 1

10

d

• HOT yields compact events of nontrivial size.• SOC has infinitesimal, fractal events.

HOT

SOC

sizeinfinitesimal large

A HOT forest fire abstraction…

Burnt regions are 2-d

Fire suppression mechanisms must stop a 1-d front.

Optimal strategies must tradeoff resources with risk.

Generalized “coding” problems

Fires

Web

Data compressionOptimizing d-1 dimensional cuts in d dimensional spaces.

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

Cumulative

Frequency

Decimated dataLog (base 10)

Forest fires1000 km2

(Malamud)

WWW filesMbytes

(Crovella)

Data compression

(Huffman)

(codewords, files, fires)

Los Alamos fire

d=0d=1

d=2

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

FrequencyFires

Web filesCodewords

Cumulative

Log (base 10)

-1/2

-1

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

WWWDC

Data + Model/Theory

Forest fire

SOC = .15

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

FF

WWWDC

Data + PLR HOT Model

HOT

SOC

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

SOC and HOT are extremely different.

SOC HOT & Data

Max event size Infinitesimal LargeLarge event shape Fractal Compact

Slope Small LargeDimension d d-1 1/d

SOC and HOT are extremely different.

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?

Robust, yet fragile?

Extreme robustness and extreme hypersensitivity.

Small flaws

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

If probability of sparks changes.

disaster

Tradeoffs?

Sensitivity to:

sparks

flawsassumed

p(i,j)

Critical percolation and SOC forest fire models

HOT forest fire models

Optimized

• SOC & HOT have completely different characteristics.• SOC vs HOT story is consistent across different models.

Characteristic Critical HOT

Densities Low HighYields Low HighRobustness Generic Robust, yet fragile

Events/structure Generic, fractal Structured, stylizedself-similar self-dissimilar

External behavior Complex Nominally simpleInternally Simple Complex

Statistics Power laws Power lawsonly at criticality at all densities


Densities Low HighYields Low HighRobustness Generic Robust, yet fragile.




Characteristics

Toy models

?• Power systems• Computers• Internet• Software • Ecosystems • Extinction• Turbulence

Examples/Applications

• Power systems• Computers• Internet• Software • Ecosystems • Extinction• Turbulence

But when we look in detail at any of these examples...

…they have all the HOT features...


Densities Low HighYields Low HighRobustness Generic Robust, yet fragile.




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!

Tong Zhou

HOT and evolution : mutation and natural selection in a community

Barriers to cascading failure: an abstraction of biological mechanisms for robustness

• 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

Genotype (heritable traits): lattice layoutPhenotype (characteristics which can be observed in the environment): cell sizes and probabilitiesFitness (based on performance in the organisms lifetime): Yield

Fast mutators (hares)Slow mutators (tortoises)

Hares: -noisy patterns -lack protection for rare events

hares tortoises

(Primitive) Punctuated Equilibrium:

Hares win in theshort run.

But face episodicextinction due to rare events (nicheprotects 50).

Tortoises takeover, and diversity increases.

Until hares win again.

Tortoise population exhibits power lawsHares have excess large events

Convergent Evolution: Species which evolve in spatiallyseparate, but otherwise similar habitats develop similarphenotypic traits. They are not genetically close, but havedeveloped similar adaptations to their environmental niches.

Our analogy: different runs with the same P(i,j) evolve towardsphenotypically similar, genotypically dissimilar lattice populations

The five great extinctions are associated with a rate ofdisappearance of species well in excess of the background, asdeduced from the fossil record.

Paleontologists attribute these to rare disturbances, such asmeteor impacts. Robust, yet Fragile!

Punctuated Equilibrium (left) vs. Gradualism (right): PE: rapid, burstsof change (horizontal lines), followed by extended periodsof relative stability (verticallines), followed by extinction.Our analogy: aftera transient periodof rapid evolutionlattices havebarrier patterns,which are relativelystable until extinction

Large extinctionevents are typicallyfollowed by increased diversity.The recovery periodis the time lapsebetween the peak extinction rate, andthe maximum rate of origination of new species.

Our analogy: extinction of the hares is arefollowed by diversification ofboth families

The current mass extinction is frequently attributed to overpopulation and causes which can be attributed to humans, such as deforestation

Our analogy: large eventscan be due to rare disturbances, especiallyif they are not not part of the evolutionaryhistory of the (vulnerable)species.

Robust, yet Fragile!

UniformSparks

SkewedSparks

Evolution by natural selection in coupled communities with different environments:

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 otherin each habitat, with a small chance of migration from one habitatto the other.

Efficient barrier patterns develop in the uniform habitat. Afteran extinction in the skewed habitat, uniform lattices invade, andsubsequently lose their lower right barriers: a successful strategyin 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 invadinglattices adapt to their new environment. This is followed by a suddendecline when the lattices lose a barrier. This adaptation is beneficial for common events, but fatal for rare events.

HOT

Disturbance

Evolution and extinction

Specialization

density

fitness

• 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.

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).

• 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

4 data sets

10

-410

-310

-210

-110

010

110

0

101

102

103

104

All four data sets are fit with the PLR model with α=1/2.

Size (1000 km2)

Rankorder

Forest fires dynamics

IntensityFrequency

Extent

WeatherSpark sources

Flora and fauna

TopographySoil type

Climate/season

Los Padres National ForestMax Moritz

Yellow: lightning (at high altitudes in ponderosa pines)

Red: human ignitions(near roads)

Ignition and vegetation patterns in Los Padres National Forest

Brown: chaperalPink: Pinon Juniper

Santa Monica Mountains Max Moritz and Marco Morais

SAMO Fire History

Fires 1991-1995

Fires 1930-1990

Fires are compact regions of nontrivial area.

10-4

10-3

10-2

10-1

100

101

100

101

102

103

104

4 Science data sets+LPNF + HFIREs (SA=2)

PLR

SM

Rescaling data for frequency and large size cutoff gives excellent agreement, except for the SM data set

Cumulative P(size)

size

We are developing a realistic fire spread model HFIREs:

GIS data for Landscape images

Models for Fuel SuccessionRegrowth modeled by 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)

Data: typical five year period HFIREs simulations: typical five year period

Fire scar shapes are compact

10-4

10-3

10-2

10-1

100

101

100

101

102

103

104

LPNF

PLR

HFIRE SA=2,RoS=.033 m/s,FC=46 yr

Excellent agreement between data, HFIREs and the PLR HOT model

10-4

10-3

10-2

10-1

100

101

100

101

102

103

104

4 Science +LPNF + Hfire (SA=2)

PLR

SM

• small: incomplete• large: short catalog, or aggressive human intervention (inhomogeneous)

SM discrepancy?

100

101

102

103

104

105

100

101

102

103

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)

SA=0, vary stopping rate

SA= 1, 2, 4, 6

SA=2, =.5

SA=4, =.5SA=6, =.3

SA=0, =.65

102

103

104

105

100

101

102

103

Increased rate of SA leads to flatter curves, shorter fire cycles

Type conversion!

What is the optimization problem?

• 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

(we have not answered this question for fires today)

• 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

Plausibility Argument:

HFIREs Simulations:

The shape of trees by Karl Niklas

• 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

Simulations of selectivepressure shaping earlyplants

Our hypothesis is that robustness in an uncertain environment is thedominant force shaping complexity in most biological, ecological, andtechnological systems

Documents

Are Forest Fires HOT? Jean Carlson, UCSB. Background Much attention has been given to “complex adaptive systems” in the last decade. Popularization of