THE SIZE OF AN EARTHQUAKE

Transcription

1 university of ljubljana, faculty of mathematics and physics, department of phyisics SEMINAR THE SIZE OF AN EARTHQUAKE Matic Smrekar Mentor: dr. Jure Bajc Ljubljana, March 2008

2 Abstract In the following text we try to show the physical description of earthquakes. We will see that the knowledge of physics and mathematics combined with geological observations can provide us many hidden details about the source and the cause of earthquakes. First we will try to find the description connected to energy release of the earthquake. At the end we will try to find also the geometry of the source.

3 Contents 1 Earthquakes 2 2 Source parameters Seismic Intensity Seismic Intensity scale Scales of Magnitude Seismic energy Seismic moment Stress drop Focal Geometry 8 4 Moment tensors Isotropic and CLVD moment tensors Stereographic fault plane representation 12 6 Conclusion 15 1

4 1 Earthquakes Earthquakes are probably obvious phenomena to many people. Because of their destructive nature, people try to understand the processes causing it. By understanding these processes we hope to avoid the catastrphic damage caused, when earthquakes occur. And if possible, it would be very useful to have a method for predicting the timing of these events. Through the history man probably tried many ways to describe the cause of earthquakes. Today it seems that describing them by combination of methods of physics and geology gave us the best results so far. 2 Source parameters 2.1 Seismic Intensity Before invetion of first measuring instruments people tried to quantify the size of an earthquake by using the description of the effects on buildings and surrounding nature. These effects describes the seismic intensity scale. It is not a physically defined quantity. It is based on the description of the damage. Seismic intensity can be measured in every city or village, where the effects of an earthquake are observable. Usually the geologists determine seismic intensity for as many cities as possible and make an intensity map. From that map they approximately calculate the point of the epicenter of an earthquake. From the intensity distribution and it s maximal size an estimate of the energy, released by the earthquake, can be made. 2.2 Seismic Intensity scale I. Not felt. (a) Not felt. II. Scarcely felt. (a) Felt by a few. III. Weak. (a) Felt indoors by a few. (b) Hanging objects swing. IV. Largely observed. (a) Felt indoors by many, outdoors by a few. (b) Dooes and glasses rattle, furniture shakes. V. Strong. (a) Felt indoors by most, utdoors by a few, strong shaking, people awake. (b) Objects sing, some fall down, doors open or shut, window panes break. (c) Grade 1 damage to a few buildings 1. VI. Slightly damaging. (a) Felt by most indoors and many outdoors, many people frightened. (b) Small objects fall, furniture shifts, glassware breaks, animals frightened. (c) Grade 1 damage to many buildings, grade 2 damage to a few. VII. Damaging (a) Most people frightened, find it difficult to stand. (b) Furniture shifted and overturned, objects fall, water splashes. (c) Many buildings of class B and a few of C suffer grade 2 damage, many buildings of Class A suffer grade 4 damage, especially to their upper parts. 1 Type of structure: masonry (five classes from rubble stone to reinforced brick), reinforcet concrete (RC) (four classes from RC without antiseismic design (ASD) to RC with a high level of ASD), and wood. Vulnerability (classes A-F): A, rubble stone, adobe; B stone, unreinforced brick; C, brick with RC floors, RC without ASD; D, reinforced brick, RC with minimum ASD; E, RC with moderate ASD; and F, RC with high ASD. Classification of damage: grades (1-5): 1, negligible to sight (no structural damage); 2. moderate (slight structural, moderate nonsreuctural); 3, substantial to heavy damage (moderate structural, heavy nonstructural); 4, very heavy (heavy structural, very heavy nonstructural); and 5, destruction (very heavy structural, near or total collapse). 2

5 VIII. Heavily damaging. (a) Many find it difficult to stand. (b) Furniture overturned, objects fall, tombstones displaced or overturned, waves seen on soft ground. (c) Many class C buildings suffer grade 2 damage, many class B and a few class C buildings suffer grade 3 damage, many class A and few class B buildings suffer grade 4 damage, afew class A buildings suffer grade 5 damage, a few class D buildings suffer grade 2 damage. IX. Destructive. (a) General panic, people thrown to the grownd. (b) Many monuments and columns fall or are twisted. Waves seen on soft ground. (c) Many class C buildings suffer grade 3 damage, many class B and a few class C buildings suffer grade 4 damage, many class A and a few class B buildings suffer grade 5 damage, many class D buildings suffer grade 2 damaage and a fre suffer grade 3 damage, a few class E buildings suffer grade 2 damage. X. Very destructive. (c) Many class C buildings suffer grade 4 damage, most class A, many class B and a few class C buildings suffer grade 5 damage, many class D buildings suffer grade 3 damage and a few suffer grade 4 damage, many class E buildings suffer grade 2 damage and a few suffer grade 3 damage, a few class F buildings suffer grade 2 damage. XI. Devastating. (c) Most class C buildings suffer grade 4 damage, most class B and many class C buildings suffer grade 5 damage, many class D buildings suffer grade 4 damage and a few suffer grade 5 damage, many class E buildings suffer grade 3 damage and a few suffer grade 4 damage, many class F buildings suffer grade 2 damage and a few suffer grade 3 damage. XII. Completely devastating. (c) Practically all structures above and below gorund are destroyed- 2.3 Scales of Magnitude By the invention of the seismograph seismologists were able to measure the effects of an earthquake in an objective manner. The measurements of amplitude and frequency of elastic waves propagating through the ground allows us to involve physical laws of continuous media. From here we can calculate at least the enery of an earthquake. By combining these methods with geological observations we can succesfuly describe the cause of earthquake. 3

6 FIGURE1: The map of earthqakes measured from the year 1963 to 1995 with magnitudes of m b 4. [1] First measure introduced was magnitude, which is based on the amplitude of the resulting waves recorded on a seismogram. Amplitude of waves reflects the earthquake size once the amplitudes are corrected for the decrease with distance due to geometric spreading and attenuation. This is connected to the energy released in an earthquake. Usually seismologists compare the energy released by an earthquake with the energy released in bomb explosions (in tons of TNT). As we will see, the description of earthquakes with moment tensor shows that explosions and earthquakes are caused by different mechanisms. That alows us to differ the earthquake caused by tectonic processes and the one caused by an (nuclear) explosion! The general form of a magnitude scale is: M = log(a/t) F(h, ) + C (1) where A is the maximum amplitude of waves measured on a seismogram and F(h, ) is the correction for the distance of the seismogram from the epicenter ( ) and depth of the hypocenter (h). The amplitude gives us the size of the event. The distance part of the equation gives us the correction for energy losses with the propagation of waves through the ground. The constant factor depends of the ground properties, the shape of the landfield etc. This description is similar to energy released in elastic waves. Using observations of the amplitudes of waves of earthquakes in California at seismographic stations for regional distances ( < 600km), Richter defined the magnitude in the form [7]: M L = log A log 1.67 (2) Calibration of the scale was achieved by assigning the value M L = 3 to an earthquake that, at a distance of 100 km, is recorded by a Wood-Anderson seismograph (this kind of seismograph has a period near 1 s and is a narrow band frequency instrument) with a maxmimum amplitude of A = 1 mm. This definition is applicable only to surface earthquakes at regional distances and 4

7 in its original or modified form it is today known as the local magnitude M L. It is still in use because this scale still gives good relation between the size of an earthquake and the damage on buildings (with constant factors properly calibrated for the type of seismometer, type of landscape etc.). FIGURE2: The Richter scale for local magnitude, M L. The magnitude is found from the amplitude of the largest arrival and the S-P travell time difference (S are transversal waves, P are longutudal waves). [1] If we have another type of seismometer, we have to take care of it s response. If we want to measure also deep earthquakes (up to 700 km), we have to take the body magnitude scale, m b : m b = log A T + F(,h) (3) where T is the period of the seismogram. Surface magnitude scale is defined as: M S = log A T + α log + β (4) where α and β are calibration coefficients. α depends on the propagation of the amplitude of the waves through the ground. Energy of the ground waves goes 1/r (the energy of body waves is proportional to 1/r 2, which gives the basic difference between scaling factors in m b and M S scale), so α must be α 0.5. It also depends on damping, so usually is greather than 0.5. β depends on the type of the landscape. If there are no instrumental records, magnitudes can be estimated from the epicentral intensity I 0 : M = 0.661I log h 1.4 (5) 5

8 the formula for earthquakes, where there are no instrumental records. Today it is almost not in use. All these magnitude scales have a problem of saturation at large earthquakes. They are also totaly empirical and thus have no direct connection to the physics of earthquakes. They are not even dymensionally correct. Physically the most correct is the moment magnitude scale: M w = 2 3 log M (6) where M 0 is the scalar seismic monent that will be defined shortly. 2.4 Seismic energy Gutneberg and Richter [8] established the first empirical relations between the magnitude and the energy: loge S = 1.5M S (7) where E S in Joules is the energy propagated in seismic waves. According to this equation an earthquake of M S = 8 has a seismic energy of J. For comparison, a nuclear explosion of 5 megatons has an energy of J and is equivalent to an earthquake of magnitude 6.7. If we calculate the approximate energy of all earthquakes that happen in 1 year we obtain a value in the range J. About 90% of this energy corresponds to earthquakes with magnitudes equal to and larger than 7. FIGURE3: Comparison of frequency, magnitude, and energy release. [1] 2.5 Seismic moment Another measure of the size of an earthquake is the seismic moment M 0. It is based on idea that earthquakes are caused by shear fractures in the Earth s crust and defined as M 0 = µ us (8) 6

9 where µ is a shear modulus, u is the mean value of the slip or displacenment on the fault plane, and S is the area of the fault plane. The moment magnitude scale (M w ), that uses seismic moment as a parameter, does not get saturated. For the small earthquakes it gives us very similar results to other scales but on large earthquakes it gives us good results. The problem is that it can not be used for small earthquakes (smaller than M w = 5). So for small earthquakes we use other magnitude scales, but for intermediate and large earthquakes we use moment magnitude scale. Earthquake M s Fault area (km 2 ) Moment (dyn-cm), M 0 Moment magnitude, M w Truckee x x San Fernando x x Loma Prieta x x San Francisco x x Alaska x x Chile x x TABLE1: Source parameters for selected earthquakes. [3] 2.6 Stress drop We assume that the earthquake s slip, D, occurs on a fault with characteristic dimension L, and so causes a strain change of approximately ǫ xx = u x x D L. (9) so the stress drop averaged over the fault is approximately σ µd L (10) From seismological observations alone, the best constrained quantity is the seismic moment, so we estimate the average slip D from the seismic moment as D CM 0 µl 2 (11) where C is a factor depending on the fault s shape. Thus the stress drop is proportional to the moment and inversely proportional to the fault dimension cubed or te 3/2 power of the fault area: σ = CM 0 L 3 = CM 0 S 3/2 (12) The specific relation and values of C depend on the fault shape and the rupture direction. For example, the stress drop on a circular fault wuth a radius R is σ = 7 M 0 16 R 3 (13) strike slip in a rectangular fault with length L and width w yields σ = 2 π w 2 L In a simplified form, the total release of energy during fracturing can be expressed by M 0 (14) E = σ us (15) 7

10 (σs represents a force). On substituting into this the seismic moment, we obtain If the stress drop is total, equation (15) gives 3 Focal Geometry E = σ µ M 0 (16) E = σ 2µ M 0 (17) In modeling we assume a simple planar surface across which relative motion occured during an earthquake. Observations of faults show that this is often approximately the case. The geometry of this model is shown in figure. The fault plane is characterized by ˆn, its normal vector. The direction of motion is given by ˆd, the slip vector in the fault plane. The slip vector indicates the direction in which the upper side of the fault, known as the hanging wall block, moved with respect to the lower side, the foot wall block. Because the slip vector is in the fault plane, it is perpendicular to the normal vector. FIGURE4: Fault geometry used in earthquake studies. [1] Although the slip direction varies such that the slip angle ranges from 0 to 360, several basic fault geometries, described by special values of the slip angle, are useful to bear in mind. When the two sides of the fault slide horizontally by each other, pure strike-slip motion occurs. When λ = 0, the hanging wall moves to the right, and the motion is called left-lateral. Similarly for λ = 180, right-lateral motion occurs. To tell which is which, look across the fault and see which way the other side moved. The other basic fault geometries describe dip-slip motion. When λ = 270, the hanging wall slides downward, causing normal faulting. In the opposite case, λ = 90, and the hanging wall goes upward, yielding reverse, or thrust, faulting. Most earthquakes consist of some combination of these motions and have slip angles between these values. 8

11 4 Moment tensors Earthquakes, explosions, landslides or impacts on Earth s surface can generate observable seismic waves. If they release energy into earth in the seismic wave frequency band, body forces can be derived. To get additional insight into the rupture process and to simplify inverting seismograms to estimate source parameters we use the seismic moment tensor. M = M xx M xy M xz M yx M yy M yz M zx M zy M zz (18) Earthquakes involving slip upon a fault are modeled as a double couple composed of four forces. A force couple consists of two forces acting together. One consists of a pair of forces offset in a direction normal to the force (Figure 5). The couple M xy consists of two forces of magnitude f, separated by a distance d along the y axis, that act in opposite (±x) directions. The magnitude of M xy is fd, which is given in Nm. To model a couple acting at a point, the limit is taken as d goes to zero such that the product fd stays constant. FIGURE5: Nine force couples which compose the seismic moment tensor. [1] Combining force couples of differrent orientations into the seismic monent tensor M gives a general description that can represent various seismic sources. No geophysical processes have been found that are best modeled as single couples, probably because such couples would generate large torques and thus observable rotations of the earth about differernt axes. The double and triple sets of couples used to model earthquakes and explosions, respectively, do not generate net torques. The moment tensor of an earthquake represents both its fault geometry, via different components, and its size, via the scalar moment. M for a simple slip with force couples of the same size, M 0, would look like: 9

12 M = 0 M 0 0 M = M (19) We can write the moment tensor in any orthogonal coordinate system. To see this we write the moment tensor for a double couple earthquake in an arbitrary coordinate system. The components are given by the scalar moment and the components of ˆn, the unit normal vector to the fault plane, and ˆd, the unit slip vector: or M = M 0 M ij = M 0 (n i d j + n j d i ) (20) 2n x d x n x d y + n y d x n x d z + n z d x n y d x + n x d y 2n y d y n y d z + n z d y n z d x + n x d z n z d y + n y d z 2n z d z (21) We can see that tensor is symmetric and the trace of the tensor is zero (M ii = 0). A nonzero trace implies a volume change (explosion or implosion). Such an isotropic component does not exist for a pure double couple source. The scalar moment gives the magnitude of the moment tensor M 0 = ij M2 ij 2 (22) which is analogous to the magnitude of a vector. A common use of earthquake focal mechanisms is to infer stress orientations in the Earth. A simple model predicts that the faulting occurs on planes 45 from the maximum and minimum compressive stresses. Equivalently these stress directions are halfway between the nodal planes. Thus the maximum compressive (P) and minimum compressive stress (T) axes can be found by bisecting the dilatational and compressional quadrants, respectively. Although T is called the tension axis, it is actually the minimum compressive stress, because compression occurs at depth in the earth. The intermediate stress axis known as the B or null axis is perpendicular to both the T and the P axes. This direction is also perpendicular to both the slip and the normal vectors, and is the intersection of the two nodal planes. Using the definitions of the normal and slip vectors in terms of fault strike, dip, and slip directions, we can write the moment tensor for any fault. The reverse process of finding the fault geometry corresponding to a moment tensor is more complicated. However, we need this ability for seismogram inversions that yeild the moment tensor. This can be done using some ideas from linear algebra about vector transformations, because the eigenvectors of the moment tensor are parallel to the T, P, and null axes. ˆt = ˆn + ˆd (23) ˆp = ˆn ˆd (24) ˆb = ˆn ˆd (25) The fact that P, T, and null axes are the eigenvectors of the moment tensor lets us simplify it by transforming it into the natural coordinate system whose vectors are the eigenvectors. The point of inversion is that inverting seismograms in a geographic coordinate system yields the moment tensor in that coordinate system. We then find its eigenvectors, the P, T, and null axes and use equation 21 to find the fault normal and slip vectors and hence strike, dip, and slip angles. As part of the same process the eigenvalues give the scalar moment. 10

13 4.1 Isotropic and CLVD moment tensors A moment tensor with a nonzero isotropic component represents a volume change. Such a triple vector dipole of three equal and orthogonal force couples is the equivalent body force system for an eyplosion or an implosion. Natural explosive or implosive sources are rare, but may be associated with fluid and gas migration linked to magmatic processes or with sudden phase transitions of metastable minerals. High-velocity impacts of meteorites could also be modeled with explosive sources. FIGURE6: Modeling an explosive source as a triple force dipole. [1] Another class of non-double-couple seismic sources are compensated linear vector dipoles (CLVDs): M = λ λ/ λ/2 (26) Although sources with large CLVD components are rare, they have been identified in several complicated tectonic enviroments. Two primary explanations have been offered for CLVD mechanisms. Especially in volcanic areas it is natural to think of an inflating magma dike, which can be modeled as a crack opening under tension. The moment tensor for this case is: M = λ λ λ + 2µ (27) The trace of the tensor is 3λ + 2µ which is positive, because the crack is opened. We can decompose the tensor into isotropic and CLVD term: λ λ λ + 2µ = E E E + 2/3µ /3µ /3µ (28) 11

14 An alternative is that CLVDs are due to near simultanious earthquakes on nearby faults of different geometries. For example consider the sum of two double couple sources with moments M 0 and 2M 0, expressed in the principal axis coordinate system: M M M M 0 = 5 Stereographic fault plane representation M M M 0 (29) Fault geometry can be found from the distribution of data on a sphere around the focus. Because plotting on a piece of papaer is easier than plotting on a sphere, a stereographic projection that transforms a hemisphere to a plane is used to plot the data. The graphic construction that does this is used to plot the data. The graphic construction that does this is a stereonet. On this net the azimuth is shown by the numbers from 0 to 360 around the circumference. The dip angles are shown by the numbers from 90 to 0 along the net s equator. The angle 90, straight down, hits the middle of the net, whereas 0, the horizontal direction, is at the edge. FIGURE7: Stereonet used to display hemisphere on a flat surface. [1] Different types of faults appear differently on a stereonet. The black and white quadrants, representing compression and dilatation, show the fault geometry. A four-quadrant checkerboard indicates pure strike-slip motion on a vertical fault plane..some basic typical cases are shown and described on the cartoons below. 12

15 FIGURE8: Focal mechanisms for earthquakes with various fault geometries. Compressional quadrants are black. The strike-slip mechanism is for pure strike-slip motion on a vertical fault plane, which could be oriented either NE-SW or NW-SE. The pure dip-slip mechanisms are for faults striking N-S. [1] 13

16 FIGURE9: Focal mechanisms for earthquakes with the same N-S striking falut plane, but with slip angles varying from pure thrust, to pure strike-slip, to pure normal faulting. [1] 14

17 FIGURE10: A selection of moment tensors and their associated focal mechanisms. The top row shows an explosion (left) and an implosion (right). The next three rows are for double couple-sources. The bottom two rows show CLVD sources which have a baseball appearance (with moment tensors transformed to the coordinate system with basis vectors pointing north, west, and up). [1] 6 Conclusion With described methods it is possible to determine the energy released by an earthquake. With combination of geological observations we can also recreate the type of the fracture causing the earthquake. So by studying earthquakes, we can get a lot of useful information on the structure of tectonic plates and processes connected to their movements. This gives us good indications of the structure of the Earth and the processes going on in its interior. On the other side, all this information can help us prevent the great damage caused by earthquakes. Since Slovenia is seismically active region it is very important for us to understand all that surrounding processes to avoid the destructive consequences. 15

18 References [1] Stein S., Wysession M. (2003), And Introduction to Seismology, Earthquakes, and Earth Structure, Blackwell Publishing. [2] Udias A. (1999), Principles of Seismology, Cambridge University Press. [3] Values from Geller (1976), Wallace et al. (1991), and Wald et al. (1993). [4] Pearce, R. G. (1977), Fault plane solutions using the relative amplitudes of P and pp, Geophys. J., 50, [5] Pearce, R. G. (1980), Fault plane solutions using the relative amplitudes of P and surface reflections: further studies, Geophys. J., 60, [6] Eakins, P. R: (1987), Faults and faulting, in C. K. Seyfert (ed.), Encyclopedia of Structural Geology and Plate Tectonics, Van Nostrand Reinhold, New York, pp, [7] Richter, C. F., (1935) An instrument earthquake magnitude scale, Bull. Seis. Soc. Am. 25, [8] Gutenberg, B. and Richter, C. F., (1956) Magnitude and energy of earthquakes, Annali di geofisica 9,