Effects of Canyon Topography on Strong G Ro U N D Motion by H. L. Wong and P. C. Jennings

The two-dimensional scattering and diffraction of S H waves of arbitrary angle of incidence from irregular, canyon-shaped topography is formulated in terms of an integral equation. Taking advantage of the simple boundary conditions of SH-wave problems, the method of images is applied to reduce the integral equation to one with a finite integral, which can readily be solved numerically by available methods. The method is first applied to the analytically solved case of a cylindrical canyon to verify its accuracy, and then to two idealized cross sections based upon Pacoima Canyon to investigate the effects of topography in a more realistic case. The results of the harmonic analysis include wave amplification patterns and transfer functions for different wavelengths and for different angles of incidence. The study also includes analysis of transient motions. With the N76°W component of the Pacoima Dam accelerogram specified to occur at one point in the cross section, the effects of different angles of incidence upon the required input motion and upon the motion at several other points in the cross section were examined by calculating accelerograms and response spectra. The effects of canyon-shaped topography are seen most prominently in the amplification patterns and transfer functions for harmonic response, wherein shielding and focusing can cause variations up to a factor of six for wavelengths comparable to, or shorter than, the canyon width. In the case of transient motions, the accelerograms at different points show significant differences, but not as large as seen in the harmonic analysis. The response spectra show the smallest differences; significant effects are confined to the higher frequencies. INTRODUCTION A detailed understanding of the effects of topographic features upon strong ground motion would be of obvious value to earthquake engineering and seismology, but it has not yet been achieved because of the scarcity of measured shaking and the difficulties in obtaining theoretical solutions to realistic problems. In recent years, however, significant progress has been made in the development of solutions to some simple problems. One of the most extensive studies of topographic effects on strong ground motion is that done by Riemer et al. (1974). In that study, a three-dimensional, finite element method was used to model Pacoima Canyon and Pacoima Dam in an effort to examine, under simplified conditions of wave propagation, the effects of topography and dam-foundation interaction upon the strong motion accelerogram recorded near one abutment during the San Fernando earthquake. Most other studies of topographic effects have been two-dimensional (e.g., Boore, 1972, 1973; Bouchon, 1973; Ttifunac, 1973; Asano, 1966; Mclvor, 1969; Wong and Trifunac, 1974). The studies by Boore (1972, 1973) use finite differences to consider transient motions; the other studies cited determined steady-state response to harmonic excitation. Bouchon (1973) used the method developed by Aki and Larner (1970) to estimate the effects of irregular surfaces under the condition of shallow slopes and incident motion with long wavelengths. Studies of 1239 1240 H. L. WONG AND P. C. JENNINGS individual, simplified geometries were accomplished by McIvor (1969), by Boore (1973), and by Asano (1966), who studied a corrugated surface and by Trifunac (1973) and Wong and Trifunac (1974). The latter two studies treat scattering and diffraction of SH waves by cylindrical and elliptical discontinuities, respectively, using separation of variables. The purpose of the present study is to introduce a method for calculating the twodimensional scattering of incident SH motions by an arbitrarily-shaped canyon. As shown schematically in Figure 1, the method uses images to satisfy the boundary conditions on the surface of the ha!f-space away from the portion of arbitrary surface. The accuracy of the approach is illustrated first by comparing the results with the exact solution for a cylindrical canyon. The method is then applied to two idealized cross Ca) ur . . . . . . . . . ~ F I 0u , y (b) D C D ui~----~ " ~ "iT A = C + . O A A D Z (c) E F / ' ~ . x _ F D u i / ~ 7]E ~ H _ F F D D u i ~ ' w TI" FIG. 1. (a) The symmetrical formulation of the imageproblem. (b) Themethod of images applied to elevated topographies. (c) Alternative approach for topographies totally above the half-space. sections taken from Pacoima Canyon to obtain additional insight into effects of topography on strong ground motion. Both steadY-state harmonic response and transient response, using the N75°W component of the Pacoima Dam accelerogram, are included in the analysis. METHOD OF ANALYSIS If only two-dimensional SH-waves are considered, the anti-plane displacement, u, satisfies the scalar-wave equation ~2u ~2u 1 ~2u + (1) ~X 2 ~y2 f12 ~t2 EFFECTS OF CANYON TOPOGRAPHY ON STRONG GROUND MOTION 1241 where fl = (#/p)1/2 is the velocity of shear-wave propagation, # is the shear modulus, and p is the density of the medium. For harmonic waves of the form u exp (ioot), the time dependence of (1) may be separated, leaving the Helmholtz equation ~2 u ~2u ~ F ~ v ~ + ~ u = 0 (~ = ,o//~). (2) q ~x 2 The Green's function G*(k I r0) of equation (2) for outgoing waves in an infinite fullspace is the Hankel function of the second kind and zeroth order (Morse and Feshbach, 1953) G*(r [ro) = (i/4)Ho(Z)(~:lr-rol) (3) which satisfies the equation ~-~+ ~-y~+ ~2)G*(r I ro) -",~(r-r0) (4) where r = r(x, y) and ro = r(x0, Y0) are the position vectors of the observation point and source point, respectively. In analytical studies olf local effects on surface motions, the basic geometrical configuration is the half-space. Due to the simple boundary conditions involved in the SH-wave problem considered here, the method of images can be applied to determine the Green's function for a point in a half-space. The symmetry about the x-axis (Figure la) produces the required condition of zero stresses, p(~u./~y) = 0 at the plane surface on either side of the surface F, the irregular interface to be studied. The boundary conditions on F require continuous displacements and stresses across the interface. The Green's function for a point in a half-space is found from suverposition of the Green's functions for the full-space for the point of interest and its image point, G(x, y ]Xo, Yo) = i/4[Ho(2)(x[(X-Xo) 2 +(Y-yo)2] ~/2) + Ho(2)(K[(X--Xo) 2 q-(y-l-yo)Z]a/2)]. (5) Using G(r I ro), Weber's integral formula for determination of the values of u and au/On o (no is the unit normal at r o On F) at the interface F is e"(r°) 7 1 u(r ) I u(ro)~G(r I ro)_G(r ]ro) dso = u°(r), r on F (6) where u°(r) is the total wave field for a half-space with the irregular surface F absent. It is noted from equation (6) that the method of images has made it possible to obtain a finite integral equation, a feature which has computational advantages. It should be noted also that the boundary values at the irregular surface, u(ro) and Ou(ro)/On o in equation (6), are not necessarily independent of each other. The displacement u(r) outside of the boundary F is related to the boundary values by _ 0u(ro)7 u(r) = u° ( i ) i u(r°) ~G(r [ r0) G(r I ro) J dso. (7) • j r an A detailed derivation of equations (6) and (7) may be found in Mow and Pao (1971). Although only concave ~topography is considered here, the method can be extended to study convex topography:as illustrated schematically in Figure 1, b and c. In this case the problem is first separated into an exterior problem II and an interior problem I, Figure lb. The exterior problem II has a concave geometry and the method of images can again be applied as outlined above. The conditions at each of the surfaces B, C, and D are 1242 H. L. WONG AND P. C. JENNINGS stress-free requirements; while at surface A, the corresponding displacements and stresses must be equated for I and II, a continuity requirement. Therefore, the two problems are solved with coupled boundary conditions. Another case of interest is illustrated in Figure lc in which the surface topography is totally above the half-space. For this case, surfaces E and F are stress-free and the continuity conditions for displacements and stresses are required for surface D. Problem II now consists of a half-space with line sources along the surface D while the interior problem I is analyzed by the method of images. NUMERICAL APPROXIMATION The integral equation, (6) can be solved in dosed form only when G(r I ro) can be expanded into a series of orthogonal eigenfunctions, an approach suitable for only a few simple geometries. For irregularly shaped boundaries, equation (6) may be solved numerically by replacing the integral by a finite sum, and finding the unknown values of u(ro) and 8u(ro)/Sn o at N discrete locations on the surface. The imaginary part of the kernel G(r [ ro) is singular at r = to, however, so to avoid numerical problems, a mathematical limit must be taken for this portion of the integration. The other parts of the integral can be treated directly. After the boundary values u(r0) and 8u(ro)/On o are calculated, the values of u(ro) at other points can be obtained from equation (7). Details of the method of calculation and its limitations as discussed by Banaugh and Goldsmith (1963a and b) apply here, even though their Green's function is slightly different. The programming of the problem is much simpler in the present application, however, because modern computer software allows direct arithmetical operations with complex numbers; the real and imaginary parts need not be considered separately. One of the advantages of the method is its relatively rapid convergence. The errors are of order h 2 log h, where h is the distance between the N equally spaced points (de Hoog and Weiss, 1973). For application to canyon topography, the stress is zero on the surface F as well as on the planar surface on either side. Hence, 8u(ro)/Sn o = 0 and equation (6) is simplified to _1 u ( r ) f u(ro) 8G(r° I r) ds ° = uO(r), r on r . (8) 2 r 8no The wave field u°(r) can be a superposition of S H waves from arbitrary sources plus their reflections from the boundary of the half-space, but for purposes of illustrating the effects of canyon topography upon ground motion, and for simplicity, only plane incident SH-waves were assumed. An incident harmonic plane wave of unit magnitude, making a counterclockwise angle 0 with respect to the x axis has the representation ui(r) -= exp [(io9///)(-x cos 0 + y sin 0)] (9) in which the factor exp (io)t) is understood. To obtain u°(r) it is necessary to add to U~(r) a reflected wave, ur(r), with an angle of 0. Thus u°(r) = ui(r)+ ur(r) = 2 exp [-(icox///) cos 0] cos [(my///) sin 0]. (10) With u°(r) specified, equation (8) becomes a Fredholm integral equation of the second kind for the unknown function u(r), r on F. By first parameterizing x and y as functions of~ x = x(~),y = y(~); 0 < ~ < z (11) EFFECTS OF CANYON TOPOGRAPHY ON STRONG GROUND MOTION 1243 the finite integral in equation (8) may be changed to an integral on the dummy variable 4. In the integration ds o is replaced by r,'dxV (.?l,, Calculating the integral as if u(ro) is known, by using the trapezoidal rule, the continuous integral of equation (8) discretizes into N simultaneous equations. 1_ _ 1 {xCy¢¢yCx¢¢'~qu. 2 + 4 ( N l)k x, 2 +yeZ ) J YI) 2Krc ~ aG .[(xj-x ,)Y~-(Yl-Yi)X~ '1 , . (x, . y , x, . y,) = u0(x,, y,) (~s-F) j#l l--1,2,3 . . . . . N (13) in which G(xj, yj I xl, Y3 is as defined in equation (5), and N is the number of discrete points on the boundary F. Equation (13) for the unknowns u(xt, Yl) has the form of a standard problem in linear algebra [A,j]{u(xj, y j)} = {u°(x,, y,)} (14) in which the elements of (Azj) are constants, and is therefore in a form convenient for numerical computations. To investigate the accuracy of the calculation method, the solution of equation (13) was compared to the exact solution for scattering by a circular canyon with radius a. Letting x = a cos { and y = a sin 4, the surface F becomes a semi-circular cylinder, the configuration studied by Trifunac (1973). To obtain a direct comparison of the values of u on the surface F, the dimensionless frequency q is again used here t 1 = 2a/2 = ¢oa/~zfi. (15) The variable r/is the ratio of the width of the canyon to the incident wavelength; large values indicate waves with lengths short compared to the width of the canyon. Table 1 gives values of the real and imaginary parts of u on F for q = 0.5, 2.0 and for angles of incidence of 0 ° and 60 °. Three sets of solutions calculated for different values of N are listed to indicate the rate of" convergence. Note that for low frequencies, even N = 4 yields answers of acceptable accuracy, although more points are clearly needed for higher frequencies. APPLICATION TO CANYON TOPOGRAPHY Attention is next turned to more realistic shapes in order to study the effects of topography upon strong ground motion. Two idealized cross sections based on the topography of Pacoima Canyon were selected for study. As shown in Figure 2a (Trifunac, 1973), the topography is quite complex, but in the vicinity of the dam it might be permissible to idealize the site as two-dimensional for the purposes of studying motion in the longitudinal (N76°W) direction. This assumption will be examined later in the light of the numerical results. The idealized cross sections used in the analysis (Figures 3 and 4) are taken from sections C and D of Figure 2a; section C includes the location of the accelerograph which recorded the motion during the San Fernando earthquake. Section D is 70 meters downstream from C. 1244 H. L. W O N G AND P. C. JENNINGS