PostScript le created Decemb er time minutes Why do es theoretical physics fail to explain and predict earthquake o ccurrence Yan Y Kagan Department of Earth and Space Sciences University of California Los Angeles California USA Abstract Several reasons for the failure can b e prop osed The multidimensional character of seismicity time space and earthquake fo cal mech anism need to b e mo deled The latter is a symmetric secondrank tensor of a sp ecial kind The intrinsic randomness of earthquake o ccurrence necessitati ng the use of sto chas tic p oint pro cesses and appropriate complex statistical techniques The scaleinvariant or fractal prop erties of earthquake pro cesses the theory of ran dom stable or heavytailed variables is signicantly more dicult than that of Gaussian variables and is only now b eing develop ed Earthquake pro cess theory should b e capa ble of b eing renormalized Statistical distributions of earthquake sizes earthquake temp oral interactions spa tial patterns and fo cal mechanisms are largely universal The values of ma jor parame ters are similar for earthquakes in various tectonic zones The universality of these dis tributions will enable a b etter foundation for earthquake pro cess theory The quality of current earthquake data statistical analysis is low Since little or no study of random and systematic errors is p erformed most published statistical results are artifacts During earthquake rupture propagation fo cal mechanisms sometimes undergo large D rotations These rotations require noncommutative algebra eg quaternions and gauge theory for accurate mo dels of earthquake o ccurrence These phenomenological and theoretical diculties are not limited to earthquakes any fracture of brittle materials tensile or shear would encounter similar problems Index terms Keywords Earthquakes seismicity earthquake size time space and fo cal mechanisms scaleinvariance stochastic point process statistical anal ysis quaternions earthquake forecasting Copyright by the American Geophysical Union X KAGAN THEORETICAL PHYSICS AND EARTHQUAKE OCCURRENCE the simplest o ccurrences such as a glass breaking Although Intro duction computer simulations of brittle fracture for example see OBrien and Ho dgins are b ecoming more realistic The diculties of seismic analysis are obvious Earth they cannot yet provide a scientically faithful representa quake pro cesses are inherently multidimens ion al Kagan tion Brittle fracture is a more dicult scientic problem b Kagan and VereJones in addition to than turbulence and while the latter attracted rstclass the origin time D lo cations and measures of size for mathematicians and physicists no suchinterest has b een each earthquake the orientation of the rupture surface shown in the mathematical theory of fracture and largescale and its displacement requires for its representation ei deformation of solids ther secondrank tensors or quaternions see more b elow One sees the degree of diculty in assessing the eects Earthquake o ccurrence is characterized by extreme ran of brittle fracture by lo oking at the investigation results of domness the sto chastic nature of seismicity is not re the space shuttle Columbia disaster Columbia Accident In ducible bymorenumerous or more accurate measurements vestigation Rep ort p The only way to test the Even a cursory insp ection of seismological datasets sug p ossible cause of the accident a breach on the edge of the gests that earthquake o ccurrence as well as earthquake shuttle wing was to conduct a fullscale exp eriment No fault geometry are scaleinvariant or fractal Mandelbrot realistic computation of the breachwas p ossible Kagan b Kagan and VereJones Tur Belowwe rst review the seismological background in cotte Sornette see also httpwwwesi formation necessary for further discussion as well as basic topicscomearthquakesi ntervi ewsYanYKagan html mo dels of earthquake o ccurrence Section Short Sec Adequate mathematical and statistical techniques have tion describ es the available earthquake catalogs In Sec only recently b ecome available for analyzing fractal temp o tions evidence for the scaleinvarian ce of earthquake pro ral spatial and tensor patterns of p oint pro cess data gen cess is presented in particular marginal distributions for erally and earthquake data in particular Such metho ds are the multidimensi on al earthquake pro cess Fractal distribu still in the development stage Moreoveritisonlyinthe tions of earthquake size time intervals spatial patterns past years that the quality precision and completeness fo cal mechanism and stress are discussed Section de of earthquake datasets and the pro cessing p ower of mo dern scrib es several multidimensio nal sto chastic mo dels used to approximate earthquake o ccurrence They are all based on computers have b ecome sucienttoallow detailed fullscale the theory of branching pro cesses in this case the multidi investigation of earthquake o ccurrence patterns mensional structure of earthquake o ccurrence is mo deled After lo oking at recent publicatio ns on earthquake In Section we discuss the branching mo del of earth physics for example Lee et al Scholz quake rupture a physical multidimensi on al mo del based on Kanamori and Bro dsky one gets the impression that random stress interactions The mo del uses very few free knowledge of earthquake pro cess is still at a rudimentary parameters and app ears to repro duce all the fundamental level Why has progress in understanding earthquakes b een statistical prop erties of earthquake o ccurrence Section so slow Kagan a compared the seismicity description briey describ es the application of statistical mo dels to fore to another problem in physics turbulence of uids Both cast an earthquake o ccurrence The nal discussion Sec phenomena are characterized bymultidimensi ona li ty and tion summarizes the results obtained thus far and dis sto chasticity Their ma jor statistical ingredients are scale cusses problems and challenges still facing seismologists invariant and b oth have hierarchicall y organized structures Moreover the scale of selfsimilar structures in seismicity and turbulence extends over many orders of magnitude The Seismological background size of ma jor structures which control deformation patterns Earthquakes in turbulence and brittle fracture is comparable to the max imum size of the region see more in Kagan b Since this pap er is intended for seismologists physicists Yaglom p commented that turbulence status and mathematicians we briey describ e earthquakes and diers from many other complex problems whichtwentieth earthquake catalogs as primary ob jects of the statistical century physics has solved or has considered study A more complete discussion can b e found in Ka gan b pp Bolt Lee et al These problems deal with some very sp ecial and complicated Scholz Kanamori and Bro dsky As a rst ap ob jects and pro cesses relating to some extreme conditions which proximation an earthquakemay b e represented by a sudden are very far from realities of the ordinary life However turbu shear failure the app earance of a large quasiplanar dislo lence theory deals with the most ordinary and simple realities of cation lo op Aki and Richards in ro ck material the everyday life such as eg the jet of water spurting from the Fig a shows a faultplane trace on the surface of the kitchen tap Therefore the turbulence is welldeservedly often Earth Earthquake rupture starts on the faultplane at a called the last great unsolved problem of the classical physics p oint called the hyp o center the epicenter is a pro jection of the hyp o center on the Earths surface and propagates Although solving the NavierStokes equations describ with a velo city close to that of shear waves kms ing turbulent motion in uids is one of the seven The centroid is in the center of the ruptured area Its p o mathematical millenni um problems for the st century sition is determined by a seismic moment tensor inversion see httpwwwclaymathorgmillen ni um the turbu Ekstrom et al and references therein As a result of lence problem is not among the ten millenni um prob the rupture two sides of the fault surface are displaced by lems in physics presented by the UniversityofMichigan avector along the faultplane For large earthquakes such Ann Arb or see httpfeynmanphysicslsaumi ched u displacement is on the order of a few meters stringsmill enn iu mhtml or among the problems The earthquake rupture excites seismic waves which are by the National Research Councils b oard on physics and registered by seismographic stations The seismograms are astronomy Haseltine In his extensive and wide pro cessed by computer programs to obtain a summary of ranging review of current theoretical physics Penrose the earthquakes prop erties Routinely these seismogram do es not include the turbulence or NavierStokes equations inversions characterize earthquakes by their origin times in the b o ok index hyp o center centroid p ositions and secondrank symmet Like uid turbulence the brittle fracture of solids is com ric seismic moment tensors monly encountered in everyday life but so far there is no real Fig c represents b eachball the quadrup olar radiation patterns of earthquakes The fo cal plots involve painting on theory explaini ng its prop erties or predicting outcomes of KAGAN THEORETICAL PHYSICS AND EARTHQUAKE OCCURRENCE X a sphere the sense of the rst motion of the primaryP covering large areas continents or their large parts and lo cal catalogs for particular