固体地球物理学导论7

1、第七章,第七章 地球内部的地震波场,地震与介质的弹性性质 地震波及其特征 地震体波的传播 地震面波及其特征 地球的自由振荡 天然地震,第七章,地震波场在地球物理学中占有重要地位，当今在研究地球内部，地震活动的机制、资源与能源的地震勘探以及海、陆工程建设中均主要依赖于人工源或天然源激发的地震波场效应。 震源(包括人工源或天然源)激励出来的各种类型的地震波，在地球内部各围层介质中或沿其表面传播，依据这些波动的走时，频率和振幅特性或波的频散，可以推测地球内部各圈层介质的速度分布和结构。 根据地震台站纪录的地震事件，可推断震源的参数(震源深度、震中位置、发震时刻、地震震级和震中距离等)和震源机制，并进

2、一步了解产生这种机制的应力状态。如果发生的地震足够大，则地球作为一个整体会激发出各种振型的振荡，并可通过它来探讨地球内部的性质。,地震学研究及其意义,第七章,7.1 地震与介质的弹性性质 7.1.1 地震震源及地震波 Earthquake sources Physically, earthquake sources are the abrupt release of the potential elastic energy stored in rocks over a period ranging from a few years to thousands of years. Only a s

3、mall part of the energy converts heat to the surround rocks near the source, the most of energy is radiated away as elastic waves. In fact, rocks at an earthquake source generate plastic deformation but elastic deformation at the moment of earthquakes occurring. Up to now, no earthquake that locates

4、 deeper than 670 km has been observed.,地震震源,第七章,Seismic waves Seismic waves are the elastic wave from the source. They are of various types. While traveling through the earth, these waves are influenced by the properties of the media they pass. We are able to understand and analyze the influence by

5、applying mathematical and physical methods. In addition, we can determine the earths structures: the crust, the mantle, the outer core, the inner core, as well as the lateral changes near the surface.,地震波,第七章,Receiving The main task of receiving is to record the seismic waves at desired positions to

6、 study earthquakes. It includes the sampling technology and the receiving system consisting of seismographs(地震仪), geophones(检波器), and other instruments. The quality of recorded data is important to analysis and interpretation of the seismic waves and determination of earthquake sources.,地震波的接收,第七章,7

7、.1.2 板块构造与地震 At present the theory of plate tectonics is accepted by most of geoscientists, even though many of its details are still unclear or controversial. We can use a simplified dynamic model to describe the movement of continent. When the material in the mantle is heated, it expands and becom

8、es lighter. In spite of its high viscosity（粘性）, it rises more or less vertically in some places, especially under the oceanic ridges. With its losing pressure and heat during traveling upward, the material is forced to travel horizontally. They drag the lithosphere motion. The results of continent c

9、olliding form mountain chains (Himalayas) , and the results of their separating form ocean rifts (East Africa). So major earthquakes often cause near their collided boundaries. In the region of oceanic ridges, where new lithosphere is growing, small earthquakes occur frequently.,板块构造与地震,第七章,板块构造与地震分

10、布图,第七章,7.1.3 岩石弹性性质基本概念 (1)形变 A material occurs deformation(形变） under a force act on itself. If it recovers as the force disappears, it is called elastic material. The shape change is called as elastic deformation. Otherwise, it is called as non-elastic deformation. Whether elastic deformation occur

11、s depends on the magnitude of acting force, the acting period as well as the surrounding temperature. For most materials in the earth, this elastic property only exists in a short period.,弹性概念形变,第七章,(2)应力 Stress tensor（张量） Definition: Here stress means a force acts on unit area of a body against the

12、 elastic deformation caused by the action of an external force. Describe any stress needs consider two factors, direction and outer normal(法向）of a face. We generally express it by pst. Here s means the direction of the force and t the outer normal direction on a face. In three dimension orthogonal c

13、oordinate system, we can define stress p as (pxx pxy pxz pyx pyy pyz pzx pzy pzz).,弹性概念应力,第七章,The stresses are symmetrical（对称的）, i.e. only six components of the stress tensor p are independent because pxy= pyx , pyz= pzy , pzx= pxz For a cubic body in x-y-z coordinate system, when the face edges of

14、the body are parallel to coordinate planes, pxx , pyy , pzz are normal stresses and pxy , pxz , pyz , are shear(剪切） stresses.,弹性概念应力 (续),第七章,Pressure At a given point the sum of the normal stresses on any three orthogonal(直角的）planes is a constant (a scalar). The pressure P is defined as P = - (pxx+

15、pyy+ pzz)/3 This is a general definition of the “pressure”. In the special case of a liquid at rest, pxx= pyy= pzz = - P, this is the hydrostatic pressure. In geology, lithostatic pressure is often estimated by using P=gh where is the density, g is the acceleration of gravity, and h is the depth. Bu

16、t it is not always correct near the surface.,弹性概念压强,第七章,(3) Strain（应变） tensor Definition: In general, the relative change in the length or in the shape of an object acted by forces is called as strain. This kind of length or shape changes should be recovered after the forces disappear - Elastic defo

17、rmation. Linear strain A rod is 50cm long initially in the direction x-axis. When a force or forces are applied to it, its length increases to 50.2cm in the x-direction. The relative change in length is (50.2-50)/50.,弹性概念应变,第七章,To measure the relative change, we define u as the length change in x, x

18、 as the original length of the rod. On any point inner the rod, the linear strain can be defined as exx = u/x u/x,第七章,In XYZ-coordinate system, in same way we can obtain eyy = v/y, ezz = w/z Here exx , eyy ,ezz are normal strains. shear strain,Y f X,第七章,Suppose that the graph shown is as a result of

19、 external forces, the cross section of the body is deformed to the rhombus(菱形) shown by dashed lines, and in the procession all points move parallel to the x-axis. The area of the cross section has not changed, but the shape has. The angle is a measure of this distortion, called shear strain. Here t

20、an = u/ y At the limit while y0 = u/y Consider the variation at another direction = v/x,第七章,The shear strain in the x-y plane is defined as exy = (+)/2 = (u/y+v/x)/2 For three dimension orthogonal coordinates, we also have eyz = (v/z+w/y)/2 exz = (u/z+w/x)/2 Their symmetry gives exy= eyx , exz= ezx

21、, eyz= ezy .,第七章,Dilatation (体膨胀) The sum of normal strains is defined as dilatation = exx+ eyy+ezz The dilatation is a measure of the relative change in volume. For a homogeneous bulk applied by external forces, the relative change in volume is V/(xyz). V=x(1+exx)y(1+eyy)z(1+ezz) -xyz Hence V/(xyz)

22、=exx+eyy+ezz+exxezz+eyyezz+exx eyy +exxeyyezz exx+ eyy+ezz,第七章,(4)Elastic modulus and equations Suppose a body is homogeneous and isotropic, i.e. its properties are independent of both spatial coordinates and directions. Hookes law tell us the stresses in the body are linear combinations of the stra

23、ins. For instance, pxy=aexx+beyy+cezz+dexy+fexz+geyz (a,b,c,d,f,g are constants) According to elasticity theory, we have pxy=2exy , pxz=2exz , pyz=2eyz where is the modulus of rigidity or shear modulus.,第七章,The shear stresses are proportional to the shear strains. On the other hands, the relations o

24、f the normal stresses and normal strains are, e.g., pxx=+2exx where is another elastic modulus. and are called Lam elastic constants. They are difficult to measure directly. For this reason, they are often computed from other elastic parameters.,第七章,All the relations to describing stresses and strai

25、ns can be written in pij=ij+2eij where ij is Kronecker delta-a function. i and j represent x, y or z. when i=j the value of ij is 1, otherwise, 0.,第七章,Other elastic modulass Youngs modulus, E Youngs modulus measures the resistance to extension in a direction. It is defined as E=pxx/exx (if the force

26、 only applied in x-direction) Since pxx=+2exx pyy=+2eyy=0 pzz=+2ezz=0,第七章,To add them Pxx=3+2=(3+2) By symmetry, eyy=ezz =exx+2 eyy , +2eyy=0 so E=(3+2)/ (+),第七章,Incompressibility, This parameter measures the resistance to a change in volume under pressure. It also called bulk modulus. = - dP/d sinc

27、e P= - (pxx+ pyy+ pzz)/3 = - (3+2)/3 = - (3+2) /3 thus = +2 /3,第七章,Poissons ratio, is the ratio of the lateral contraction to the longitudinal extension. Suppose only normal stress pxx acts on the body. Thus = - eyy/exx or = - ezz/exx Because pyy=+2eyy= (exx+eyy+ezz) +2eyy =0 By symmetry, eyy=ezz ,

28、and (exx+2eyy) +2eyy =0 Hence =/(2(+),第七章,In the earth, Poissons ratio ranges from 0.1 to 0.38 near the surface. At hydrostatic pressure equivalent to a depth of 13km it ranges from 0.23 to 0.31, except for quartzite =0.15. In the absence of any other information, it is often assumed that =0.25. Oth

29、er relations =E/(1+)/(1-2) =E/(2(1+) =E/(3(1-) /=2/(1-2),第七章,Wave velocities In a infinite , homogeneous, isotropic and elastic medium, only two kinds of waves can propagate, P-wave and S-wave. Their travel velocities Vp and Vs are given respectively by,第七章,Where is the density of the medium. The te

30、rm P-wave means “primary wave or pressure wave,” since it arrives first or it is caused by pressure. S-wave stands for “secondary wave or shear wave,” because it travel slower than P-wave or it is generated by shear strain. From the equations above, if increases the velocity Vp and Vs should decreas

31、e. In fact, it is not real in most cases. Generally the heavier materials have the higher velocities than the lighter do, because the and increase faster than . For the materials in the earth ,assume =0.25, thus =. So we have Vp=1.73Vs,第七章,在均匀各向同性介质中，质点的运动方程为 在无体应力（| f |=0）的情况下，上式变为 从场论可知，任何一个场，均有 且

32、有 则,波动方程,第七章,根据 有 讨论： 1)由于速度是位移对时间的偏导数。因此纵波和横波的速度满足波动方程； 2)由于无旋位移场的散度是无旋应变，无散位移场的旋度是无散应变。因此无旋应变或正应变与无散应变或切应变均满足波动方程； 3)由于无旋场可用标量位来表示，无散场可以用矢量位来表示，并分别设为标量位和为矢量位，即有,波动方程 (续),第七章,纵波、横波,7.2 地震波及其特征 7.2.1 地震波的类型 (1)纵波 (2)横波,第七章,瑞雷波与勒夫波面波,(3)瑞雷波 (4)勒夫波,第七章,转换波,(5)SH-wave and SV-wave For the anisotropic me

33、dia, S-waves may decomposed into two components SH- and SV-waves in their propagation. At the boundaries between different media with the differences of elastic properties, S-wave can generate polarization, i.e. the particles are restricted in a special plane. SH- and SV-wave have slight difference

34、in velocity.,第七章,地震子波,7.2.2 地震子波 Seismic waves are mechanical waves. They behave the kinds of particle vibration. Since the earthquakes have limit energy so that they only last rather limit time like pulses. We often say that seismic waves signals are wavelets and their periods are irregular.,第七章,By

35、 applying Fourier analysis we can decompose the wavelets in sine and cosine sequences. The seismic wavelets can be considered the sum of components of the simple waves with regular periods and various amplitudes. Therefore, we generally use a sine or cosine wave to discuss the feature of wave.,子波的分解

36、,第七章,A sine wave can be considered in two ways: at one point, they are periodic in time; at one instant, they are periodic in space. Suppose the motion of the particles is along the y-axis, we have y= A sin2 (x/ - t/T) where T is the period, A is the amplitude of particle vibration, and is wave leng

37、th, the distance at one instant between two crests or troughs, or between any two adjacent points having same phase. We usually use the terms wave number and frequency. The wave number is k=1/ and the frequency is f=1/T. Form the above, we know that the wave velocity can be obtained by V= /T. y=A si

38、n2 k(x-Vt),波数与频率,第七章,7.2.2 平面波 Assume that the wave propagates only in x-direction and the particles move only in y-z plane. This means that all particles move in phase-they form wave fronts. We say the waves are plane waves. The movement of the particles may be described by f(x,t)=A sin2k(x-Vt),平面波

39、,第七章,7.2.3 球面波 For a point source, waves propagate in all directions. If particles move in phase and they constitute spherical wave fronts. We say they are spherical wave. Practically, for the far away point sources the spherical wave can be approximately considered as the plane waves. f(r,t)=A sin2

40、k(r-Vt),球面波,第七章,Dispersion of body waves From a point source the P- and S-waves spread radially from the source along a straight line. The spherical wave fronts still dilate, so that the energy of vibrating particles on the spheres decreases continually. Assume E is the energy of a seismic source, r

41、 is the radial distant from the source to a sphere. At one instant the energy per unit area of the sphere E can be written in E=E/(4r2) At any point out of source, the energy is proportional to the inverse of the square of r and the amplitude is proportional to the inverse of r. They both decay with

42、 the increasing of the distance from the source.,球面波的扩散,第七章,Absorption and attenuation of body waves Up to now, we have assumed that rock or other materials are perfectly elastic. In fact, pure elastic material does not exist. The energy of the waves transform to heat due to the friction of vibratin

43、g particles. The energy and the amplitudes of the waves decay with the traveled distance and the frequency of the wave. A =A0 e-fr,地震波的吸收与衰减,第七章,地震波射线理论,7.3 地震体波的传播 研究地震波传播通常有两种途径，一个是依据波动方程的动力学理论，另一个是依据地震波走时的射线理论。 7.3.1 地震波射线理论 (1)费玛原理 射线理论的基础是费马原理。费马原理指出：在连续介质中，扰动沿着一条走时稳定的路径传播。若以 t 表示扰动从P点沿着一条路径传到Q

44、点所用的时间，以v(x,y,z)表示扰动的传播速度，以l表示该路径的弧长，则费马原理可以表示为,第七章,地震波射线理论 (续),换句话说，扰动沿任一射线S传播所用时间t与沿其它路径传播用时一样，即 显然，在均匀介质中，射线为直线，上式可写为 而在非均匀介质中，射线方程的积分形式可写成,第七章,Snell定律,(2)Snell定律 在均匀介质中，地震波射线是直线，在连续介质则为曲线。在非均匀介质中，当射线到达速度的不连续界面时，其方向会发生偏折，在界面上出现反射波、折射波和转换波。 Snell定律指出了入射与反射和透射射线之间的关系。Snell定律是费马原理的延伸。,第七章,Snell定律 (续

45、),如果介质中有界面存在，界面两边的弹性参数及密度各不相同。因为界面两边介质的弹常数和密度都不相同，所以弹性波的速度也不相同。一部分弹性被能量穿过界面，产生透射；另一部分弹性被能量由界面反射回来。纵波经过界面时产生纵波反射与透射，还可以转换成横被的反射与透射。 假设界面是一个平面，当一个单纯的纵波P1入射到界面时，便有四个不同的波同时产生，P1 P1和P1 S1表示反射的纵波和横波（SV），P1P2和P1S2表示透射纵波和横波（SV）。 横波的质点运动可有两个方向，质点运动与界面垂直的称为SV波，质点运动与界面平行的称为SH波。入射的SV波在界面上同样可以产生上述四种波，而SH波因质点运动在与

46、界面平行的面上，所以没有纵波产生。,第七章,单一水平界面地震波走时,7.3.2 水平层状介质中的地震波 (1)匀速层状介质中体波的走时 单个水平界面 在距离振动源不同的地点设置观测仪器，接收某种地震波到达的时刻，以距离x为横轴，到达时刻t为纵轴，所得的曲线称为走时曲线(或称时距曲线)。,xc,第七章,单一水平界面地震波走时方程,假定振动源位于地面A点，地下存在一个水平界面，其深度为h，在地面B点接受到的直达波、反射波和折射走时可分别写成 直达波 反射波 折射波,第七章,多水平界面地震波走时,多层水平界面反射波 设Vk, hk,为第k层的速度，则有地震波向下传播的射线走时和距离分别为 根据Sne

47、ll定律，有 则,第七章,连续介质地震波走时方程,(2)垂向连续介质（横向均匀） 若 hk很小, n很大，则有地震波向下传播的射线走时和距离分别为 由此可见，在连续介质中，射线为一条曲线。,第七章,连续介质地震波射线曲率,根据曲率的定义， 射线曲率为 若速度随深度呈线性变化，即 则有 射线曲率为常数，且射线是半径为1/Pa的圆弧。,第七章,地震波的能量分配,(3)地震波的能量分配 直达波的扩散与衰减 r为震源到接收点的距离，为相应的频率衰减系数。 反射波能量分配与扩散 若不考虑吸收因素，反射波的能量取决于反射波能量的分配和扩散。其中能量分配系数反射系数为 能量扩散与射线路径有关，不考虑能量分配

48、的情况下，扩散函数为,第七章,地震波的能量分配 (续),透射波的能量分配与扩散 若不考虑吸收因素，透射波的能量取决于反射波能量的分配和扩散。其中能量分配系数透射系数为 在不考虑能量分配的情况下，且折射波接收距较大时，扩散函数为 其中xc为折射波临界距离。,第七章,球坐标中Snell定律的形式,7.3.3 球对称介质中的地震波 (1)球坐标中Snell定律的形式 假设地球由数个厚度不等的同心球壳组成，每层内波速均匀，根据Snell定律，有 根据正弦定理，有 所以,第七章,球坐标中Snell定律的形式 (续),多层情况下，球坐标中Snell定律的形式为 径向连续介质中Snell定律：,第七章,本多

49、夫定律,(2)射线参数方程 本多夫定律 假定射线PQ1的参数为P，走时为t，角距离为，相邻射线PQ2的相应数值为t+dt，+d，作Q1N垂直于PQ2，则 V* 为视速度，R为地球半径，V0为地球表面速度，有 本多夫定律,第七章,射线曲率,射线曲率与临界条件 设射线曲率半径为，则 上式表示了速度随深度变化的条件。dV/dr可以为正，也可以为负。为负时表示速度随深度而增加，射线向下弯曲；为正时表示速度随深度而减小，射线向上弯曲。,第七章,射线临界条件,当射线达到最低点时，有i=90，若=r，则有 射线临界条件 表示该处射线曲率与地球曲率相同。 当 时，无论速度随深度增加还是减小，射线曲率小于地 球

50、曲率，射线的另一端都能在地面出现；当 时，且速度 随深度减小，射线曲率大于地球曲率，射线的另一端不能在地面出 现，除非当更深处出现速度随深度增加的情况。,第七章,射线临界条件 (续),第七章,走时曲线方程,连续介质中的走时曲线方程,第七章,走时曲线方程 (续),设 rc 为最低点半径，R为地球半径，则有走时方程,第七章,近震与远震走时曲线,7.3.4 近震与远震走时曲线 由于地球表面并非一个平面，因此，观测点距震中的远近不同，研究问题的方法也有所不同。震中距小于100km叫地方震，在100km至1000km范围内叫近震。对地方震和近震而言，地面可近似看为平面，直达波可以直接切于莫氏面到达接收点

51、。如果震源O位于地壳中(即在莫氏面以上)，不难证明，下式成立 这里H为莫氏面的深度，R为地球的半径，若取H= 40km，R=6271km，则有712kmOS1424km，故取1000k皿作为远震和近震的界线。换言之，从震源出发，直达波不经过莫氏面的反射，可直接到达的区域所观测到的地震叫地方震或近震，而远于1000km，直达波不能直接到达，故称为远震,第七章,近震与远震走时曲线 (续一),7.3.4 近震与远震走时曲线 (1)近震与地方震走时曲线 首波 震中距,第七章,近震与远震走时曲线 (续二),(2)远震走时曲线 由于实际地球的速度结构复杂，不仅是由于地球各圈层的成分不同，水平方向的不均匀，

52、而且还由于物质态的变化。为了区分经过不同路径的地震波，在地震学中常用以下符号： P纵波 S横波 K在外核中的P波 I在内核中的P波 J在内核中的S波C在核幔边界上的反射 f在内外核边界上的反射 用以上符号可以表示各种通过地核的地震波，如PKIKP,PKJKP，PKP，SKP以及在地球内部各分界面上发生的反射波PCP，PCS，PfP等。,第七章,地震面波及其特征,7.4 地震面波及其特征 面波有两类：即勒夫波和瑞雷波。勒夫波的振动为水平横向(与传播方向相垂直)，它与SH波相似。瑞雷波的振动为水平纵向(与传播方向平行)和铅直方向，它的轨迹为逆进椭圆。 一个扰动在半无限的均匀介质中，不会产生勒夫波，

53、而可以产生瑞雷波，但所产生的瑞雷波没有频散。地震记录中出现勒夫波以及有频散的瑞雷波，这说明地下的介质是不均匀的或是呈层状的。 不同周期的面波，其渗透深度不同，周期愈大，其渗透深度愈大。因此利用频散曲线可以求得地球内部速度随深度的变化。 尽管目前面波的形成机制尚不清楚，但一般认为，勒夫波是SH波在层间的传播的一种形式，与SH波不同的是存在频散现象；而瑞雷波是由P波与SV波干涉的结果。,第七章,地震面波及其特征 (续),勒夫波： 瑞雷波：,第七章,面波的波动方程,7.4.1 面波的波动方程 波动方程的一般形式： 考虑简化问题，设扰动信号为一谐波。在无限半空间中，解得形式为 其中c为波速，2/k为波

54、长，U，V，W为z的函数。,第七章,面波的波动方程 (续一),(1)勒夫波频散方程 假设一个单层半空间介质，层厚为h。上层顶面z=-h，上层介质参数为1、1、和V1，下半空间参数为2、2和V2。根据勒夫（1911年）的解答，有 这里，x为波的传播方向。利用边界条件,第七章,面波的波动方程 (续二),可求得 由于这是一个多解方程，写成更一般的形式为 当n=0时，称为基模式，n=1时，称为二阶模式，。 上式表明，勒夫波具有频散特性，不同的k值对应得频率不同，所对应的速度c也不同，这里的速度为勒夫波的相速度（phase velocity）。,第七章,面波的波动方程 (续三),勒夫波具有的特点 勒夫波

55、产生在层状介质表面，且有Vs1Vs2； 勒夫波是一种SH型波，其振动方向与界面平行； 其速度c满足Vs1cVs2，存在频散现象； 勒夫波具有多模式，其中，基模式能量占优； 基级模式波长 n阶模式波长,第七章,面波的波动方程 (续四),(2)瑞雷波频散方程 由于P波的入射，会转换为P波和SV波，所以瑞雷波问题要更复杂些。对于上述介质中在x-z平面内的瑞雷波问题，1886年瑞雷已证明，可选择 当z=0时，在外应力的情况下，有,第七章,面波的波动方程 (续五),求解可得波速方程 这里 。显然，方程存在一个实根。 假设一个特定条件下，在半无限空间介质中，有2=32，则c=0.92，1=0.85，2=0

56、.39。由此，可得扰动方程 其中a为常数。,第七章,面波的波动方程 (续六),当z0时，自由面上的瑞雷波扰动为 该扰动的轨迹是一个椭圆因为k(x-ct)是随时间增大而减小的参量，椭圆是逆进的。由扰动方程还可看出，在w值随z的增加而单调下降到零。u值在z等于1.21/k左右时改变符号，在此深度上椭圆为一条垂直直线。超过这个深度，椭圆成为“前进”的，而不是“逆进”的。,第七章,面波的波动方程 (续七),瑞雷波具有的特点 瑞雷波产生在介质的自由表面； 瑞雷波是一种椭圆极化波，其振动方沿椭圆逆进（在界面附近),当离开界面一定深度时为前进； 瑞雷波的速度c满足cVs，当介质为非均匀时，有频散现象； 瑞雷波也具有多模式，其中，基模式能量占优；,第七章,面波的群速度与相速度,7.4.2 面波的群速度与相速度 (1)群速度和相速度的概念 利用面波研究地球内部构造时，主要利用它