earthquakes a self-organized critical phenomena - Indian Institute of ...

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLYENERGY DISSIPATED IN BURSTS AT ALL LENGTH SCALES AND TIME (INTERMITTENTLY)

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLYENERGY 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 longenough, the distribution will look like:log 10N(M≥m) ∝ – bm , N(S ≥ s) ∝ s – b : AN EMPIRICAL POWER LAW

EARTHQUAKE MAGNITUDES M ≥ m AND THEIRFREQUENCIES, N(m): Log N(m) = a - b mLeft: Global earthquakes of M≥4Right: Earthquakes in southern California

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

OMORI’S LAW GOVERNING RECENT(2009) JAPANESEMW ~6.6 AFTERSHOCKSN(t) = [K/(c + t) p ] REQUIRE THAT c IS VERY SMALL AND THAT p = CLOSE TO UNITYTHUS, N(t) = [Kt -1 ] ANOTHER POWER LAWK Richards-Dinger et al. Nature 467, 583-586 (2010) doi:10.1038/nature09402

A TWO DIMENSIONAL MODEL OF AN EARTHQUAKE :AN ARRAY OF SPRING COUPLED EQUAL BLOCKS PULLEDACROSS A FRICTIONAL SURFACEFREQUENCY-MAGNITUDE DISTRIBUTION OF SLIDERBBLOCKS AND SAND-SLIDES YIELDED BYCOMPUTER MODELS, FOLLOW A POWER LAW

EARTHQUAKE PHENOMENOLOGY*EARTHQUAKES ARE SUDDEN SLIPS ON FRACTURED PLANES IN THE ROCK MASSESOF THE EARTH’S BRITTLE OUTER LITHOSPHERE. THEY RELIEVE CENTURIES OFACCUMULATED 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 SOMESTATISTICALLY 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 established2. 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 unity3. Earthquake epicentres cluster both in space (Long Correlation Scales) and time4. Fracture planes in the earth’s lithosphere have been found to have a fractal distribution in common withseveral other fragmentation products5. slider blocks Model experiments also show self similar distribution both in space and time.IS THERE SOME UNIFIED FRAMEWORK IN WHICH ALL THEABOVE FEATURES COULD BE UNDERSTOOD

HALLMARK OF SELF ORGANIZED CRITICALITY?*THAT A LARGE, INTERACTIVE, OPEN SYSTEM SELFDRIVES ITSELF TO A CRITICAL STATE OF MARGINALSTABILITY: AN ATTRACTOR*IN THIS STATE, THERE IS NO CORRELATION BETWEEN THERESPONSE OF THE SYSTEM TO A PERTURBATION AND DETAILS OFTHE PERTURBATION. AS IN EQUILIBRIUM SYSTEMS AT CRITICALPOINTS (PHASE TRANSITIONS), ∴ NO CHARACTERISTIC SCALESFLUCTUATIONS 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 SOC

THE LURE OF SOC• THAT SIMPLE LOCAL RULES OPERATINGIN EXTENDED INTERACTIVE SYSTEMSLEAD TO EMERGENCE OF COMPLEXSTRUCTURES AND PHENOMENA SUCH ASTHOSE EXHIBITING SCALE INVARIANCE• FROM A STUDY OF COMPLEXSTRUCTURES AND PHENOMENA MAY WEDISCERN THE SIMPLE OPERATING RULES

AMBIGUITIESSCALE INVARIANCE, SELF SIMILARITY & FRACTALS• STRONG ANALOGIES BETWEEN SCALE INVARIANTPROCESSES AND FRACTAL GEOMETRIES WHICH ARE SELFSIMILAR AT DIFFERENT SCALES.• ALSO, SOLUTIONS OF SCALE INVARIANT PROCESSES ARESELF SIMILAR AT DIFFERENT SPATIO-TEMPORAL SCALESAND OFTEN PRODUCE FRACTAL STATISTICS AS THERENORMALIZED GROUP METHOD (EXPLICITLY SCALE INVARIANT)• BUT WHILST FRACTALS ARE SELF SIMILAR THEY ARE NOTFULLY SCALE INVARIANT (EXCEPT FOR A DISCREET SET OFDILATATIONS)

SEARCH FOR A COHERENT FRAMEWORK TOEXPLAIN EARTHQUAKE PHENOMENOLOGYAPPROACHESASSUME THAT THE MAIN SHOCK , AFTERSHOCKS AND THE FRACTALSTATISTICS OF EARTHQUAKE OCCURRENCE ARE ALL DRIVENBY THE SAME LAW AND LOOK FOR SCALING FUNCTIONS THAT MAYREDUCE THE MULTIDIMENSIONAL SPACE-TIME RELATIONSHIPS TO AUNIVARIATE DPENDENCEGuteberg-Richter Law: log 10N(M≥m) ∝ – bm ,m ∝ log s(energy) : N(S ≥ s) ∝ s – bm ∝ log cM(seismic moment, c~ 1.5) and M ∝ A 3/2 (L 3 )N(S ≥ s) ∝ (L -2b )suggesting a fractal dimension of distributed seismicity d f= 2bCOMBINING THE ENERGY (MAGNITUDE), RUPTURE AREA (LENGTH 2 )AND THE FRACTAL DISTRIBUTION OF SEISMICITY), SUGGESTS ASCALING FUNCT ION: X ∝ s – b L df

THE EARTHQUAKE PROCESSSYSTEM DRIVEN IN THE LONG WAVELENGTH LIMIT: ENERGY FED UNIFORMLYENERGY 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 solidcircles, squares, and triangles denote cutoffs m = 2, 3, and 4, respectively. The color codingrepresents 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 cannotbe 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 distributionsL = 1° and m = 2, 3, and 4. P S,L= 1 °(T) follows a power lawP S ,L = 1 °(T) ∝ T −α g 1(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 P S =100,L(T) ∝ T −α g 2(TL df )THE OMORI LAW REGION DECREASES WITH INCREASING L

The data in Figure 3 with T > 38 s, replotted with T α PS ,L(T) as a function of the variablex = cTS −b L df , c = 10 −4 . The choice of the Omori Law exponent α ≈ 1, Gutenberg–Richter value b ≈1, and fractal dimension d f≈ 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 regimeBut decays fast for x > 1, associated with uncorrelated events