O n estimation of the displacement in an earthquake source and of source dimensions V . K E Y L I S - B O R O K 1. - INTRODTJCTION. The study of an earthquakes mechanism is usually fìnished by the determination of a fault piane and the direction of displacement along it. In many aspects of geodynamics and seismology it would be very important to estimate also the dimensions of a source and cliiefly — the magnitude of the displacement in it. This problem can be reduced to the comparing of fields of sfaticai stress before an earthquake and after it. The difference of energies of these two fields depends upon source properties in question; lience it is possible to determine dimensions of the source and the displacement in it if the forni of the source and the difference of the energies be known. The energy difference can be found from observations on the seismic waves energy. The idea of such an approach belongs to L. Knopoff. He solved the two — dimensionai problem which. deals with the study of very strong earthquakes with long faults. The analogica! three — dimensionai problem is considered in this paper; it is necessary for studying of more numerous weaker earthquakes and investigation of some contiguous problema. 2. - T H E O R Y . The problem can be formally stated as following. Before an earthquake a homogeneous field of tangentional tractions V exist xxy — p, with corresponding displacement u = cy, v = ex, c M being arbitrary. W e shall cali this field an initial field. In the mo- ment of the earthquake tangentional tractions vanisli along the fault piane, x* + z2 < 62, y = 0. 2 0 6 V . K E Y L I S - B O R O K A zone of crusMng or intence plastic fìows can appear around the fault piane. This zone must be included in the source model because the tangential tractions in it deerease or vanish; that is why we shall consider the source as an ellipsoid. x2 -f- z2 y2 ~b2 a2 0 < a < b [1] Now we shall examine an final field of stress which is defined by two conditions: 1 displacements and stress tend to initial ones at infinite distances; 2 the source boundary is free from stress except the hydro- static pressure. W e shall suppose the final field to appear after the earthquake with the source [1]. I neglect here plastic fiows and an inhomogeneity of the initial field which can really exist. Besides, the source boundary in the above meaning can be really not sharp. Thus idealized problem in its main part can be reduced to this one which is already solved by Neuber in the theory of stress concen- tration. I omit here ealculations which are not complicated principally and shall give the final result. The Neuber's solution is obtained in elhptical coordinates: x = chu sin v cos w, y = sliu cos v, z = chu sin v cos w; it can be represented by the formulas: Normal stresses. V , G an = - ~ sh 2 u sin v cos v cos iv + p - -- q (T shu — 2 chu) + + | CÌl2 Uq +•1 2 1>2 chu — 2 2 eh2 u0 2 eh2 u0 chu eh u eh2 u0 chu ci? u sin v cos v cos w [2a] V 7 „ • c cr„ = — sii 2 u sin v cos v cos w + p — • sh2 u„ o \ 2 2 ( Ch2 UQ T chu h F 1 \ chu •— q (T sh2 u — 2 chu) + •—chu | sin v cos v cos w [2b] 2 pG h2 1 — a chu eh2 u0 eh3 u sin v cos v cos w . [2c] ON E S T I M A T I O N OF T H E D I S P L A C E M E N T I N A N E A R T H Q U A K E SOURCE, ETC. 2 0 7 Tangentional stresses. v i , eh 2 u sin2 v\ V „ rm = 7j2 [ch*u j cos w + — G a \ shu q (shu — T sh2 u) 2 shu — qT + ( eli2 u0 — —— 1- cos2 v (Tch 2u—2 shu) q A ) Gii U . , a\ shu 2 shu , , , — cos2 V ( eh2 w0 — 0 H M — eli2 m„) cos w [3a] V V T = , - s/rasini;sinw + •- G li li qs + |C/Ì2 - sin » sin w [3b] V V xuw = - - c/mcos»sinw -j——•G h h q (thu — T cliu) + 2 shu eh2 u0 eli3 u [3c] Displacements. . . . V C òU„ = — p A„ sin v cos v cos w ÒUV (Av + Av cos2 ®) 2 [A, h V òTJm = --- G Aw cos ® sin w . 2 11 [4a] [4b] [4c] Energy-difference. òW = -P V i l i " , a [i \ b R • A„ sli2 u\ — + sc7f.2 Uq ) — A v T s h u 0 + s(Avsh2u0— Avch2u0) -f- Av eh 2 u[ — - + s sh2 Uq A„j cliu 11 , 2 71 41 ) TRI ^ + [5] bere e,,, r„„ are the stress components; óJ7„ are the differences be- tween the displacement in the initial and final fields; òVI' is the stress 2 energy difference (the last terni — n thu0 in [5] represents the energy O inside [1]). _ A + 2 fi T = are e tg (shu) ; li — ] sh2 u + cos2 v ; 2 0 8 V . K E Y L I S - B O R O K = T shu — 1 ; thu0 G _ eh2 u0 a b ' q = 3 sh2 u0 + 1 N(u0) ' A„ = (3 sh2 u + 1) (2 sste + T) — s/iw jV = q (sshu + T) — 2 s/tw0 . c/i.2 M„ eh2 u + a (sshu — T ) 0.2 0.f 0.6 Oi (.0 A Fiff. 1 The maximal tangentional stress, according to Neuber is: Tmax — txy I * = y = 0 — V « i - f " 9 - jST" ShUn I t is interesting to note, that Tmax is always more than ^ 2p (fig. 1). The displaeements change quaintly from point to point. P o r the measure of displacement we shall take V — the maximal one on the ?/-axis, that is in the middle of the source boundary: chu ix = —2 N (k + 2 a s ) ON E S T I M A T I O N OF T H E D I S P L A C E M E N T I N A N E A R T H Q U A K E SOURCE, E T C . 2 0 9 I n particular, it is interesting to consider the displacements at the source boundary (u = ua). If the source is thin (a = 0), we have: Ò U x = ± J L 4 " I [i TC(2-\-a) xjr p a —1 ÒTJy = — - x fi 2 + a ÒTJZ = 0 here sings " + " and " — " correspond to displacements in upper and lower sides, so that the fault does exist in our model. I t can be seen from this that in elastic case the fault piane under- goes also a rotation (óUy ^ 0) which reestablishes the sfaticai equili- brium after the forming of a fault. I t is not clear whether this rotation will talee place in fact because the reestablishment of the equilibrium can be the result of imperfect elasticity. This will not change the order of quantities studied here. However the rotation will essentially distort the seismic waves, generated after the formation of a fault; therefore in constructing of the dynamic model of a fault it is necessary to allow for tlie plasticity near the source. (Of course, even in elastic case the above formulae can not be used directly for construction of dynamic model because they represent only its zero-frequency component). 3. - INTERPRETATION. The following formulae can be of the main interest for the inter- pretation of seismic data: 8 = òb* = 71W V t* Vl3 pVS = òW t* Rn 2/3 I òW Le [6] [7] [8] where ò W is the difference of energies of the initial and final fields, V is our measure of displacement, 8 is the fault piane area and Ls, Rs are plotted on fig. 2. (Rs = R-n-3'* ; Ls = L ti 1/2 Le = LR~lh). 2 1 0 V . K E Y L I S - B O R O K B y means of these formulae one can try to determine the fault piane area S and the displacement V in the source if the energy of seis- mic waves E is known. Because of energy dissipation E is less than òW. E However the ratio —— is much more for earthquakes than for explo- oW sions and according to Byerly E and AW are of the same order. tu = 1.5 I t is better to use these formulae in logarithmic forni which corre- sponds to the accuracy of E — determination. The values of the coefficient Rs, Ls are given in fig. 2 for the case a = 1.5, which corresponds to the Poisson coefficient v = 0,25; the variation of v in reasonable limita (0.23-0.27) may be neglected. The dashed line corresponds to the case when the energy inside the source model (ellipsoid [1]) is not included in ò W and is considered as dissipated in the plastic flow hear the source. ON E S T I M A T I O N OF T H E D I S P L A C E M E N T I N A N E A R T H Q U A K E SOURCE, E T C . 2 1 1 The stress p in the initial field is not known and this is the main difficili t v in using formulas [6-8], I t may be asserted only that p is less than the strengh limit as it corresponds to an average stress in a great region. H . Benioff estimated p for Kern county earthquake of 1951 from p2 the formula E = —— IIF (H = thickness of the crust, F - area of Z fi aftershocks' zone); the formula is based on the assumption that the energy E of seismic waves was accumulated in the zone coinciding with the zone of aftershocks. He obtained p = 2.67 dyne/cm2. Since E is known for this earthquake 5.IO22 ergs) p can be estimated using [6]. The movement took place along the fault ~ 60 km long. Desi- gnating the depth of the rupture in kilometres by H we obtain S = 6H IO11 cm2. Then with fi = 5.IO11 cgs and a = 0 we get 5 • IO22 ~ 6-10n)s/al,l 5 • IO11 H is unlikely to be greater than 35 km. (the thickness of the crust in California) and less than 15 km. Than p ~ (5-9).IO7 cgs. Since the source of this earthquake is greatly elongated if would be more reasonable to use formulas for two-dimensional problems. Assuming again that II = from 15 to 35 km, we obtain from the for- mulas of Knopoff: p ~ (1.5-3.1).IO7 cgs. The obtained values of p are of the sanie order; we shall try to use them for the study of other earthquakes. In conclusion it should be noted that in a number of paper the volume V of the region of stress accumulation was estimated from the energy of earthquakes E applying tlie formula: The value v should not be mixed up with the volume of the mo- del [1]: the lattei- may be equal to zero (with a = 0); although E remains finite. One can see from fig. 2 that E relatively little depends upon the volume of the source and is determined mainly by the area of a fault and by the initial stress p. Formulas [2], [3] can be used also for the estimating of the infìuence of earthquakes on the stress field around the source. In a small region around [1] the tangentional stress diminishes near the #-axis and be- comes maximal near z-axis (where the screw-deformation occurs). This 2 1 2 V . K E Y L I S - B O R O K can stimulate the expansion of the fault or the formation of new faults along direction, perpendicular to the motion direction x. But it would be especially important to take into account non-ideal elasticity when studying the stress in the final field. 4. - COMPARISON W I T H OBSERVATIONS. Unfortunately the greater part of observational data refers to strong earthquakes with surface faults which are certainly far from the circle (fault planes are unlukly to go much deeper than the earth's crust and their horizontal length amounts to hundreds of kilometres). For the San-Francisco earthquake of 1906 the length of a fault was l = 435 km; assuming 8 = 13 we obtain with 3 = 35 km (the thickness of the crust in California) 8 = 1.5; IO14. Assuming a — 0, /1 = 3.IO11 we obtain from formula [7] V ~ 3 m. This value is of the same order that the actual one. From formula [6] 8 ~ 0.7.IO14 cm! which also agrees with actual data. I t would be interesting also to compare the analogous computations with the empirical relation between E, V and 8 communicated by Don Tocher at the Toronto meeting; however he studied the strong earth- quakes for which Knopoff's formulas are more valid. In conclusion it is interesting to compare formulas [6]; [7] with the data on the frequency of occurence of earthquakes. B. Gutenberg and C. Richter showed, that with the decrease of energy of each earthquake by 100 the number of sucli earthquakes increases only by 10; therefore the part of weak earthquakes in the total amount of annually released seismic energy is very small. However with the decrease of E by 100 the total area of faults will decrease (after [6]) only by ÌO1^ and the total movement (according to [7]) will even increase by ÌO1^. Certainly fault areas and movement values cannot be summed up in such direct way. However these estimates show that weak earthquakes can take an essential part in causing great faults or in movements along them. (The frequency of weak earthquakes is determined for short-period ones; each of them can correspond to the movement along a part of a great fault. F. Press discovered also the long-period weak shoks, and explained them as a weak movement along a large faults as a whole). The author is greatly tliankful to prof. L . Knopoff, who acqunainted him with bis paper in manuscript, and to G. Pavlova for troublesome computations. ON E S T I M A T I O N OF T H E D I S P L A C E M E N T I N A N E A R T H Q U A K E SOURCE, E T C . 2 1 3 ABSTRACT An ellipsoid of rotation — a round fault piane surrounded by a crush- ing zone — is taken as a source model. The area S of the fault piane and the displacement V along it can be estimated using formulas: H = SL £3/2 Bs ; V = — i/S Ls H /i where the energy E is known (fi is shear modulus; Rs, Ls coefficients given in fig. 2; p mean stress before the earthquake; the value p = 3.1 (P gives satisfactory results in several cases). These formulas are cornpared with the well-known fact that the frequency of occurrence of earthquakes is proportional to E°-5 — E0-6; it is shoivn that the weak numerous earth- quakes can take a considerable part in the formation of large faults and movement along tliem, though their total role in releasing seismic energy is negligibly small. RIASSUNTO Un élissoide di rotazione — un piano di faglia rotondo circondato da una zona di schiacciamento — è presa come modello fondamentale. L'area S del piano di faglia e lo spostamento V lungo questo, possono es- sere calcolati impiegando le formule: E = ; V = P l/SLs H fi dove l'energia E è nota (/i è il modulo di taglio; Rs, Ls i coefficienti dati nella figura 2; p significa la forza prima del terremoto; il valore p = 3.IO7 dà risultati soddisfacenti in diversi casi). Queste formule sono confrontate con il fatto ben noto che la frequenza con cui si verificano i terremoti è pro- porzionale a E0-' — E0-6; si mette in rilievo che numerosi terremoti deboli possono assumere una considerevole pai-te nella formazione di grandi fa- glie e dei movimenti lungo di esse, sebbene la loro importanza totale nello sviluppo di energia sismica sia trascurabile. 2 1 4 V. K E Y L I S - B O R O K REFERENCES B E N I O F F , H . , Mechanism and strain characteristics of the White Wolf fault as indicated by the aftershoek sequence. Kern County, Cai. Earthquake of 1952, Bull. 171, Cai. Depart. of Mines, 1954. B Y E R I . Y , P., Report on A ' 1 ^Iss. UGGÌ, Toronto, 1 9 5 7 . I V N O P O F F , L., Energy release in Earthquakes, « Geopliys. Jorn. », 1, 44-52, (1958). LURJE, A., Theree-dimentimal problemi in elastostatics (in Russian). M. 1955. LOVE, A., Malhematical theory of elasticity. N E U B E R , I L , Kerbspannungslehre. Berlin, 1 9 3 0 . STARR, A., Slip in a cristal and rupture in a solid due to shear, Proc. « Camb. Phil. Soc.», 2 4 , ( 1 9 2 8 ) .