EARTHQUAKESA SELF-ORGANIZED

CRITICALITY ??

Vinod K Gaur Indian Institute of Astrophysics Bangalore 560 034 vgaur@iiap.res.in

EARTHQUAKE KINEMATICS WELL UNDERSTOOD SLIP ON A RUPTURE PLANE OR FAULT CAN BE SHOWN TO REPRODUCE

GROUNDMOTION RECORDS AT SEISMOGRAPH SITES ON THE EARTH’S SURFACE

BUT PHYSICAL PROCESSES RESPONSIBLE FOR ITS DYNAMICS STILL OBSCURE

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLY

ENERGY DISSIPATED IN BURSTS AT ALL LENGTH SCALES AND TIME (INTERMITTENTLY)

AFTERSHOCK SEQUENCE IS SATISFIED BY N(t) = [K/(c + t)p]: OMORI’S LAW

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLY

ENERGY DISSIPATED IN BURSTS AT ALL LENGTH SCALES AND TIME (INTERMITTENTLY)

If the magnitudes associated with hypocenters occurring inany random box over the globe are sampled for long

enough, the distribution will look like:log10N(M≥m) ∝ – bm , N(S ≥ s) ∝ s – b : AN EMPIRICAL POWER LAW

EARTHQUAKE MAGNITUDES M ≥ m AND THEIR FREQUENCIES, N(m): Log N(m) = a - b m

Left: Global earthquakes of M≥4Right: Earthquakes in southern California

AFTERSHOCK SEQUENCE IS SATISFIED BY N(t) = [K/(c + t)p]: OMORI’S LAW

K Richards-Dinger et al. Nature 467, 583-586 (2010) doi:10.1038/nature09402

OMORI’S LAW GOVERNING RECENT(2009) JAPANESE MW ~6.6 AFTERSHOCKS

N(t) = [K/(c + t)p] REQUIRE THAT c IS VERY SMALL AND THAT p = CLOSE TO UNITYTHUS, N(t) = [Kt-1] ANOTHER POWER LAW

A TWO DIMENSIONAL MODEL OF AN EARTHQUAKE : AN ARRAY OF SPRING COUPLED EQUAL BLOCKS PULLED

ACROSS A FRICTIONAL SURFACE

FREQUENCY-MAGNITUDE DISTRIBUTION OF SLIDER FREQUENCY-MAGNITUDE DISTRIBUTION OF SLIDER BBLOCKS AND SAND-SLIDES YIELDED BY BBLOCKS AND SAND-SLIDES YIELDED BY COMPUTER MODELS, FOLLOW A POWER LAW COMPUTER MODELS, FOLLOW A POWER LAW

EARTHQUAKE PHENOMENOLOGY*EARTHQUAKES ARE SUDDEN SLIPS ON FRACTURED PLANES IN THE ROCK MASSES

OF THE EARTH’S BRITTLE OUTER LITHOSPHERE. THEY RELIEVE CENTURIES OF ACCUMULATED STRAIN WITHIN A FEW TENS TO HUNDREDS OF SECONDS.

THUS, TWO VASTLY DIFFERENT TIME SCALES ARE INVOLVED .

*DESPITE VAST RANGE OF SPACE-TIME SCALES, EARTHQUAKES EXHIBIT SOME STATISTICALLY SIGNIFICANT GENERIC PATTERNS:

1. magnitude distribution in time (Gutenberg-Richter law) Log N(M>m) = a - b m, or, P>(E) = 10 –bm,

where b is statistically equal to unity but this universality is not easily established

2. Decay of aftershock numbers with time (Omori-Utsu law): N(t) = [K/(c + t)p], where c is found to be very small and p close to unity

3. Earthquake epicentres cluster both in space (Long Correlation Scales) and time

4. Fracture planes in the earth’s lithosphere have been found to have a fractal distribution in common with several other fragmentation products

5. slider blocks Model experiments also show self similar distribution both in space and time.

IS THERE SOME UNIFIED FRAMEWORK IN WHICH ALL THE IS THERE SOME UNIFIED FRAMEWORK IN WHICH ALL THE ABOVE FEATURES COULD BE UNDERSTOODABOVE FEATURES COULD BE UNDERSTOOD

HALLMARK OF SELF ORGANIZED CRITICALITY?

*THAT A LARGE, INTERACTIVE, OPEN SYSTEM SELF DRIVES ITSELF TO A CRITICAL STATE OF MARGINAL

STABILITY: AN ATTRACTOR

*IN THIS STATE, THERE IS NO CORRELATION BETWEEN THE RESPONSE OF THE SYSTEM TO A PERTURBATION AND DETAILS OF

THE PERTURBATION. AS IN EQUILIBRIUM SYSTEMS AT CRITICAL POINTS (PHASE TRANSITIONS), ∴ NO CHARACTERISTIC SCALES

FLUCTUATIONS AT ALL SCALES POSSIBLE, CORRELATION LENGTHS & TIMES GO TO INFINITY

*SINCE A POWER LAW DISTRIBUTION IS THE ONLY ONE THAT HAS NO CHARACTERISTIC SCALE (SCALE INVARIANT)

POWER LAW IS THE HALL MARK OF SOCPOWER LAW IS THE HALL MARK OF SOC

,

*

THE LURE OF SOC

• THAT SIMPLE LOCAL RULES OPERATING IN EXTENDED INTERACTIVE SYSTEMS LEAD TO EMERGENCE OF COMPLEX STRUCTURES AND PHENOMENA SUCH AS THOSE EXHIBITING SCALE INVARIANCE

• FROM A STUDY OF COMPLEX STRUCTURES AND PHENOMENA MAY WE DISCERN THE SIMPLE OPERATING RULES

AMBIGUITIESAMBIGUITIESSCALE INVARIANCE, SELF SIMILARITY & FRACTALSSCALE INVARIANCE, SELF SIMILARITY & FRACTALS

• STRONG ANALOGIES BETWEEN SCALE INVARIANT PROCESSES AND FRACTAL GEOMETRIES WHICH ARE SELF SIMILAR AT DIFFERENT SCALES.

• ALSO, SOLUTIONS OF SCALE INVARIANT PROCESSES ARE SELF SIMILAR AT DIFFERENT SPATIO-TEMPORAL SCALES AND OFTEN PRODUCE FRACTAL STATISTICS AS THE RENORMALIZED GROUP METHOD (EXPLICITLY SCALE INVARIANT)

• BUT WHILST FRACTALS ARE SELF SIMILAR THEY ARE NOT FULLY SCALE INVARIANT (EXCEPT FOR A DISCREET SET OF DILATATIONS)

SEARCH FOR A COHERENT FRAMEWORK TO EXPLAIN EARTHQUAKE PHENOMENOLOGY

APPROACHES ASSUME THAT THE MAIN SHOCK , AFTERSHOCKS AND THE FRACTAL

STATISTICS OF EARTHQUAKE OCCURRENCE ARE ALL DRIVEN

BY THE SAME LAW AND LOOK FOR SCALING FUNCTIONS THAT MAY

REDUCE THE MULTIDIMENSIONAL SPACE-TIME RELATIONSHIPS TO A UNIVARIATE DPENDENCE

Guteberg-Richter Law: log10N(M≥m) ∝ – bm ,

m ∝ log s(energy) : N(S ≥ s) ∝ s – b

m ∝ log cM(seismic moment, c~ 1.5) and M ∝ A 3/2 (L3)

N(S ≥ s) ∝ (L-2b)

suggesting a fractal dimension of distributed seismicity df= 2b

COMBINING THE ENERGY (MAGNITUDE), RUPTURE AREA (LENGTH2)

AND THE FRACTAL DISTRIBUTION OF SEISMICITY), SUGGESTS A

SCALING FUNCT ION: X ∝ s – bLdf

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLY

ENERGY DISSIPATED IN BURSTS AT ALL LENGTH SCALES AND TIME (INTERMITTENTLY)

The distribution P S ,L (T) of inter-occurrence times T with magnitude greater than S. The solid circles, squares, and triangles denote cutoffs m = 2, 3, and 4, respectively. The color coding represents the linear size of cells: L = 0.25° (black), 0.5° (red), 1° (green), 2° (blue), and 4° (orange) covering Southern California. For T < 40 s, earthquakes overlap, and individual earthquakes cannot be resolved. This result causes the deficit for small T, so only intervals T > 38 s are considered. Bak et al., 2002, Phys.Rev. Lett.

First-return (inter-occurrence) time distributions

L = 1° and m = 2, 3, and 4. PS,L= 1 °(T) follows a power law PS ,L = 1 °(T) ∝ T −αg1(TS-b)

THE OMORI LAW REGION INCREASES WITH m

First-return (interoccurrence) time distributions for L = 0.25° (black), 0.5°(red), 1°(green), 2°(blue), and 4°(orange) and m = 2 (s = 100).

The function follows a power law PS =100,L(T) ∝ T −α g2(TLdf)

THE OMORI LAW REGION DECREASES WITH INCREASING THE OMORI LAW REGION DECREASES WITH INCREASING LL

The data in Figure 3 with T > 38 s, replotted with TαPS ,L(T) as a function of the variable

x = cTS−b Ldf, c = 10−4. The choice of the Omori Law exponent α ≈ 1, Gutenberg–Richter value b ≈ 1, and fractal dimension df ≈ 1.2 allow all data to collapse onto a single, unique curve f(x).

f(x) is constant for x < 1, corresponding to a correlated, Omori Law regime But decays fast for x > 1, associated with uncorrelated events