VARIATION OF THE INTENSITY OF EARTHQUAKE MOTION BENEATH THE GROUND SURFACE

Robert E. Kayen¹ and James K. Mitchell²

Sixth U.S. National Conference on Earthquake Engineering
FINAL PROGRAM

THEME: Seismic Design and Mitigation for the Third Millennium

May 31 - June 4, 1998
Sheraton Seattle Hotel
Seattle, Washington

CONTENTS:

Abstract
Introduction
Field Data
Numerical Site Response Analysis
Results
Conclusions
References

ABSTRACT

An analysis of Arias intensity profiles with depth of soil burial was developed principally through a parametric analysis of synthetic seismograms. Numerical site-response methods were used to generate synthetic seismograms for the surface and depth nodes of soil columns representative of a variety of site conditions. Seismograms from rock motion records recorded during strike-slip, normal, and shallow thrust faulting events ranging between 5.25 M 5 7.6 were propagated through these soil column-types and output synthetic acceleration-time histories were computed for a suite of depth nodes in the upper 100 meters of soil. By integrating these synthetic records, the Arias intensities at the layer nodes were computed, and then normalized by their corresponding surface Arias intensity. This normalization process allows for a unified evaluation of the variation of earthquake shaking intensity below ground surface over a broad range of earthquake magnitudes and site geologic conditions by collapsing the results to a common reference point of unity at the ground surface.

The typical (mean) Arias intensity profile of the soil column is of a rapid falloff of intensity to 58% of the surface value at 6m depth, decreasing to 46% at 10m. Below 10 meters, the Arias intensity remains relatively constant. These results compare favorably with field data from a downhole array in Southern California. The parametric study shows that the reduction of Arias intensity with depth is less for larger magnitude earthquakes, with typically longer predominant periods, than for lower magnitude earthquakes with shorter predominant periods. Local site geology had a strong influence on the variation of Arias intensity with depth. Sites underlain with deposits having monotonically increasing shear modulus profiles saw earthquake shaking intensities fall-off consistently with depth. In contrast, sites underlain by soft, low velocity soil and capped with higher velocity surface layers (e.g. densified fill) had a pronounced amplification of intensity beneath the layer interface. These findings indicate that careful site-assessment is critical for estimating the magnitude of earthquake shaking intensity within the ground, and that the use of surface intensities, or simple chart solutions in landslide and liquefaction studies may result in large effors in the earthquake engineering analysis.

INTRODUCTION

Several new and promising methods for assessing geotechnical hazards during earthquakes have been proposed based on the parameter Arias intensity. A landslide potential assessment methodology, developed by Wilson and Keefer (1985), uses surface measured Arias intensity and a modified Newmark-approach to determine the potential for landslide displacement of hillslopes during earthquakes. The method of Wilson and Keefer uses the surface measured Arias intensity, and should be amenable to modification by incorporating a depth correction factor to better estimate the intensity at depth. A liquefaction potential assessment methodology developed by Kayen and Mitchell (1997), uses Arias intensity estimated at depth and field standard- and conepenetration tests (SPT & CPT) to determine the initial liquefaction potential of soil (excess pore water pressure generation to a level equal to the effective overburden stress) during earthquakes.

A previously developed measure of the relative intensity of motion below ground (Seed and Idriss, 1971) is the dimensionless shear stress (_max) depth reduction factor, r_d. This parameter is equal to the ratio of the peak-average acceleration (a_max) over the depth of interest to the corresponding acceleration at the surface.

The parameter r_d measures the attenuation of peak shear stress with depth due to the non-elastic behavior of soil. It was found that r_d fell from a value of unity at the surface to typically between 0.3-to-0.7 at a soil depth of 30 meters (~100; Figure 1). Seed and Idriss proposed that an estimation of the peak seismic shear stress-induced at depth in a soil mass could be determined, on the basis of r_d, by correcting the surface measured value with the following equation:

where h is the total overburden stress of the soil and g is the gravitational acceleration.

Figure 1: The range of values of rd for different soil profiles (Seed and Idriss, 1971). Click on figure for larger image (24K).

A dimensionless depth correction factor, r_b, for the parameter Arias intensity was presented by Kayen (1993) and Kayen and Mitchell (1997). Arias intensity (Arias, 1970) is the sum of the energy absorbed by an evenly-spaced population of idealized undamped simple oscillators under the excitation of an earthquake motion. For one horizontal component of motion, Arias intensity, I_a, is calculated as follows:

where, t₀ is the duration of earthquake shaking, and a_x (t) is the transient acceleration. The Arias intensity integral has the dimensional units of velocity.

The depth of burial correction parameter, r_b is determined by integration of the Arias intensity at depth from either field measurements or numerical studies and normalization of that measure by the respective intensity at the surface.

As with the parameter r_d, the normalization of r_b allows us to evaluate the depth-dependency of Arias intensity over a broad range of earthquake motion and site conditions by collapsing the profiles to a common reference value of unity at the ground surface. The r_b parameter is somewhat analogous to r_d and can be calculated from either one- or two-component horizontal Arias Intensity.

The usefulness of this parameter is is that it serves to correct the surface measured Arias intensity to an estimate of the Arias intensity at depth in the soil mass. Kayen (1993), and Kayen and Mitchell (1997), proposed that the single-component Arias intensity at depth in the soil,, I_ab, can be estimated by the following equation:

and that the sum of the orthogonal-horizontal Arias intensities at the surface, I_h, could be corrected with r_b to estimate the total horizontal Arias intensity at depth, I_hb.

Arias intensity has certain advantages over peak ground acceleration (a_max) for assessment of damage potential below the ground surface. These are that (1) Arias intensity is computed from the acceleration record over the entire duration of recorded motion, thereby incorporating all the amplitude cycles and a measure of shaking duration, whereas a_max utilizes a single value that is independent of the duration of motion; (2) Arias intensity incorporates the severity of motions over the full range of recorded frequency, whereas, a_max is often associated with high-frequency motion; and (3) the breakdown of soil structure that results in landsliding and liquefaction is fundamentally more dependent upon input energy than on a single level of acceleration (Liang and others, 1995).

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FIELD DATA

On November 23 and 24, 1987, two earthquakes occurred in the region triggering recording instruments at the Wildlife array. The first event, the Elmore Ranch earthquake (M=6.2), struck approximately 23 kilometers to the west of the Wildlife Liquefaction Array, and three-component acceleration time histories were recorded at the surface and at 7.5 meters, although no excess pore pressures were measured. The values of peak acceleration; r_d; Arias intensities; and r_b are presented in Table 1. The plots in figure 2 present the time-histories for the north-south records for (a) surface acceleration; (b) acceleration at 7.5 meters; (c) cumulative Arias intensities for surface and 7.5 meters and (d) r_b. It can be seen that the r_b ratio in figure 2d is somewhat erratic during the first seven seconds of the event, prior to the arrival of the S-waves, during the period when the intensities at the surface and at depth are quite low. After approximately 7 seconds (shear wave arrival), the intensity ratio plateaus at approximately 0.28 to 13 seconds, jumps to 0.33 and then climbs somewhat steadily to 0.38. The most noteworthy aspect of both of these records is the generally flat response of rb during the principal portion of strong shaking (figure 2d).

Figure 2: Analysis of the surface and 7.5m depth 360° records from the Elmore Ranch (EREQ) and Superstition Hills (SHEQ) Earthquakes: a) EREQ surface accelerogram; b) EREQ -7.5m accelerogram; c) EREQ computed surface and -7.5m Arias intensities; d) EREQ rb time-history; e) SHEQ surface accelerogram; f) SHEQ -7.5m accelerogram; g) SHEQ computed surface and -7.5m Arias intensities; h) SHEQ rb time-history. Click on figure for larger image (76K).

On 24 November 1987, the following day the Superstition Hills earthquake (M=6.6), centered 31 kilometers west-southwest of the site, triggered recording instruments at the Wildlife site. Corresponding time-histories for the Superstition Hills earthquake are plotted in Figure 2e-2h for the north-south component. During this event, pore pressures began to develop in the soil at approximately 13.6 seconds, and between 20 seconds and 40 seconds the soil fully liquefied.

Plotting the Arias intensity ratio, r_b, as a time-history for the Superstition Hills event, in the same manner as those for the Elmore Ranch Earthquake, shows a similar, though somewhat elevated r_b response as a result of liquefaction. The rapid rise in r_b to a value of approximately 0.44-0.46 at 20 seconds probably indicates a partial decoupling of the surface layer from the soil beneath the liquefied layer. The liquefied zone above the buried accelerometer appears to have attenuated transmission of shear waves to the surface accelerometer, resulting in an elevation of r_b. However, both the Elmore Ranch and Superstition Hills events indicate that r_b drops precipitously from 1.0, at the surface, to between 0.33-0.38 and 0.44-0.46, respectively, at 7.5 meters depth.

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NUMERICAL SITE RESPONSE ANALYSIS

Figure 3.—A simple box-car-model of the influence of amplification at the surface on the instantaneous variation of rb. Click on figure for larger image (48K).

Arias intensities at depth are expected to be somewhat less than predicted by the simple model, above. This is due to the effects of damping of the soil column, destructive interference of the upward and downward pulses, and the deformable (non-elastic) behavior of soils. To account for damping, interference and deformation of the soil column, numerical simulation using site response methods was used to determine the influence of varying soil properties and earthquake records on the attenuation of strong motion with depth.

The reduction of peak acceleration with depth in the soil column has been investigated in detail by Seed and Idriss (1971), Iwasaki et al. (1978), Imai et al. (1981), and Golesorkhi (1989). A parametric study, somewhat more complex than those referenced above, can be performed using numerical techniques to assess the reduction of Arias Intensity with depth: This study requires the integration of synthetic acceleration time histories computed for nodes in the site response model. Using the site response program SHAKE (Schnabel et al., 1972), a variety of recorded ground motions were propagated through four representative soil columns: loose sand (D_r=45%); dense sand (D_r=75%); fill overlying a cohesive soil with a monotonically increasing shear wave velocity profile with depth, and fill overlying soft mud with a shear wave velocity inversion (Figure 4a). The thicknesses of each of these soil columns were varied between 30 and 90 meters (approximately 100, 200, and 300 feet).

Figure 4: Steps used in the computational analysis of synthetic seismograms: A) model soil column for SHAKE analysis; B) propagate recorded motion up or down through SHAKE profile nodes and compute synthetic seismograms for each node; C) integrate Arias intensity time-histories from synthetic seismograms; and D) normalize the downhole intensity by the surface intensity for each shake solution. Click on figure for larger image (132K).

The procedural steps used to compute the variation in intensity from synthetic seismograms required that the soil column be modeled with an initial shear modulus profile, and that each representative layer have an appropriate strain dependent-shear modulus reduction backbone curve (Vucetic and Dobry, 1991). Earthquake motions were typically propagated from the bottom rock node upwards through the soil column profile. The program SHAKE was modified, through the options settings, to output acceleration-time histories for each node of the soil column model (figure 4b) and Arias intensities were integrated from the records using equation 3 (figure 4c). Finally, equation 4 was used to normalize the computed intensities by the surface value (Figure 4d).

Strong motion records from earthquakes ranging in magnitude between 5.2 - 7.6 were used to represent a variety of motions that could cause landslide or liquefaction damage to a site.

The eight seismograms are rock motion records during strike-slip and shallow thrust-type faulting events. The records can be sub-divided into three general magnitude classes: 1) M_w=5.0-6.0 (San Francisco 3/22/1957; Coyote Lake, 8/6/1979); 2) M_w=6.0-7.0 (Borrego Mountain, 4/8/68; Helena, 10/31/35; Parkfield, 6/28/1966; San Fernando, 2/9/71); and 3) M_w>7.0 (Tabas, Iran 9/16/78; Kern County, 7/21/52). Predominant periods of the suite of motions range from 0.085 to 0.44 seconds; peak accelerations range from 0.041g to 0.386g; and Arias intensities range from 0.0275 to 1.912 m/s (Table 2)

Synthetic soil columns representative of sand at relative densities of 45% and 75% were constructed using shear modulus relationships of Hardin and Drnevich (1972) and Seed and others (1984b). For cohesionless soil, the soil column shear modulus profile was approximated by the equation below of Seed and others (1984b).

In the above equation G is the shear modulus, K₂ is a density dependent factor, and _m' is the mean normal stress in psf. In accordance with Golesorkhi (1989), for a sand of a relative density of 45%, K₂ was taken as 40, and for a relative density of 75%, K₂ was taken as 61. The above parameters yield the shear wave velocity profiles shown in Figure 4a. For the shallower 30 and 60 meter soil columns, the profiles were truncated at the appropriate depths and underlain by an elastic base-rock material with a shear wave velocity of 2560 m/s (8000 fps). Shear wave velocity profiles at a number of San Francisco bayshore sites aided in the development of two profiles representative of fill overlying cohesive deposits (Kayen, 1993).

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RESULTS

For this first class, it was found that the value of r_b decreased rapidly within the first 10-15 meters (Figure 5). Below these depths, the shaking intensity tends to stabilize with depth. Within the upper 15 meters of the profiles, in the depth range of most interest for landsliding and liquefaction studies, there is a strong uniformity in the r_b profiles subjected to the same earthquake record. Although nine soil profiles (3 thicknesses • 3 soil types) are presented in figure 5, the variation among r_b profiles is modest for a given earthquake record. On the other hand, there is a clear difference in response of the nine soil column profiles when taken as a function of the input motions. For example, the profiles for the 1957 San Francisco earthquake attenuate to an average r_b value of 0.32 at a depth of 10 meters feet, whereas, the Kern County, Tabas, and Borrego Mountain earthquakes attenuate to only about 0.64 at similar depth (figure 5).

Figure 5.—Variation of Arias intensity below the ground surface computed from SHAKE solutions of soil columns with monotonically increasing shear modulus profiles. The solutions are binned into 3 magnitude classes. Click on figure for larger image (104K).

The summary statistical response for all profiles with monotonically increasing shear wave velocity (eight motions, three soil column thicknesses, and three soil column types: 72 total solutions) is plotted in figure 6. This plot presents the mean (50-percentile response), as well as the ± l (16th- and 84th-percentile response) Arias intensity depth correction profile. The mean attenuation falls nearly-linearly from 1.0 to a value of 0.58 at 6 meters, then decreases to 0.46 at 10 meters. The rate of attenuation decreases below 10 meters such that the mean value falls to 0.43 at 15 meters.

Figure 6.—Normalized Arias intensity depth-reduction profiles modeled using the ground response program SHAKE (a) and a statistical summary of the synthetic seismogram-profiles, along with the field data from the Elmore Ranch and Superstition Hills earthquakes (b). Click on figure for larger image (52K).

The results from the SHAKE runs, presented in figure 5 and 6, were binned into three magnitude classes M~5.5; M~6.5; and M~7.5 to assess the influence of earthquake magnitude on the attenuation of Arias intensity below the ground surface. Average profiles were calculated for each magnitude-bin and it was found that the lower the magnitude, the more rapid the attenuation occurs in the upper 20 meters of the soil column. For design considerations, the following equation reasonably describes the reduction of Arias intensity below the ground surface (z in meters) in terms of earthquake magnitude, and a plot of the equation for M=5.5, M=6.5, and M=7.5 is presented in figure 7:

The design equation is proposed for the upper 20 meters of ground for use in the liquefaction method proposed by Kayen and Mitchell (1997) and the landslide displacement method proposed by Wilson and Keefer (1985).

Figure 7: Design equation for the correction of Arias intensity with depth for Mw=5.5, 6.5 and 7.5 earthquakes, overlying results from SHAKE computer runs. Click on figure for larger image (48K).

In some field cases there exist an impedance inversion within the soil column. This condition is defined as a down-section reduction in seismic-shear impedance, I, across an interface, where I = V_s. An example of this type of deposit can be found along the perimeter of San Francisco Bay where compacted fill deposits overly soft Bay Mud. When an impedance inversions occurs in the soil there may be an amplification of motion beneath the base of the denser-higher velocity surface layer. Results from the SHAKE analyses of the second class of profiles, 8 meters of fill overlying a thick unit of soft mud, show normal attenuation of intensity with depth in the fill-section of the profile to a depth of 8 meters (figure 8). This attenuation is similar to that for the three soil profiles with monotonically increasing shear wave velocity.

Figure 8: Depth correction factor for fill overylying soft cohesive sediment. Click on figure for larger image (32K).

Below 8 meters there is a marked increase of the Arias intensity computed for the top of the soft mud layer. For most of the SHAKE solutions, the intensity at the top of the soft mud layer was in excess of the surface Arias intensity. This amplification is due to the reflection of upwardly propagating waves at the mud-fill interface. Similar results were found for the shear-stress depth correction factor, r_d, when computed for the same SHAKE runs (Kayen 1993). Borcherdt (pers. communication.) notes that these findings are similar to site response studies he has conducted showing that a sufficiently thick and dense fill-layer-can be used to form a foundation partial-isolation mat by reflecting upwardly propagating seismic waves back into Bay Mud deposits, thereby reducing the intensity of shaking in the surface soil layer. Yuxian (1979) reached similar conclusions regarding the influence of an impedance inversion on the intensity of earthquake shaking at the surface when he noted from observations of damage during the Haicheng earthquake "... it was found that a special site condition, i.e. a sufficiently hard layer of clayey soils over a liquefiable sand or soft layer, may reduce the ground surface motion and thus protect the structures above".

The results of SHAKE runs on soil columns with of an impedance inversion demonstrate the limitations of simplified plots or equations for the depth correction factors r_b and r_d. Where it is known that the shear modulus of a soil column increases monotonically with depth then design-plots like figures 1, 6 and 7, and design-equation 6, are useful tools for estimating the variation of the intensity of motion below the ground surface. Where complex stratigraphy or an impedance inversions occur, computational methods using programs like SHAKE are required to estimate the intensity of motion at depth, and use of simplified chart solutions can result is unconservative analysis of damage potential to soils and buried structures.

CONCLUSIONS

A soil column with a shear modulus inversion below a high velocity surface layer (crust) will have a complex Arias intensity and shear stress profile due to downward reflection at the base of the surficial layer. It was found that simple chart solutions for these parameters were not useful and could lead to unconservative solutions for such profiles by underestimating the intensity of ground motion at certain depths. For complex soil stratigraphies, or soil profiles that have known shear modulus inversions, computational site response methods are required to reasonably estimate intensity profiles.

REFERENCES

Golesorkhi, R. (1989) Factors Influencing the Computational Determination of Earthquake-Induced Shear Stresses in Sandy Soils, Ph.D. Thesis, Department of Civil Engineering, University of California at Berkeley.

Hardin, B.O. and Drenevich, V.P. (1972) Shear Modulus and Damping of Soils:, Measurement and Parameter Effects, Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 98, No. SM6, Proc. Paper 8977.

Holzer, T.L., Youd, T.L., and Hanks, T.C. (1989) Dynamics of Liquefaction During the 1987 Superstition Hills California Earthquake, Science, Vol. 244, pp. 56-59.

Imai, T., Tonouchi, K., and Kanemori, T. (1981) The Simple Evaluation Method of Shear Stress Generated By Earthquakes in Soil Ground, Bureau of Practical Geological Investigation, Report No. 3, pp. 39-58.

Iwasaki, T., Tatsuoka, F., Tokida, K.I., and Yasuda, S. (1978) A Practical Method for Assessing Soil Liquefaction Potential Based on Case Studies At Various Sites In Japan, Proc. of the Second International Conference on Microzonation for Safer Construction-Research and Application, Vol. II, San Francisco, California, pp. 885-896.

Kayen, R.E., 1993, Accelerogram-Energy Approach for Prediction of Earthquake-Induced Ground Liquefaction: Ph.D. Dissertation, Berkeley, University of California, 289 pp.

Kayen, R.E., Mitchell, J.K., and Holzer, T.L., 1994, Ground motion characteristics and their relation to soil liquefaction at the Wildlife Liquefaction Array,. Imperial Valley. California: Proceedings of the Fifth U.S.- Japan Workshop on Earthquake Resistant Design of Lifeline Facilities and Countermeasures for Soil Liquefaction, T.D. ORourke and M. Hamada, eds. Technical Report NCEER-94-0026, p. 267-283.

Kayen, R.E. and Mitchell, J.K. (1997) Assessment of liquefaction potential during earthquakes by Arias intensity: American Society of Civil Engineers Journal of Geotechnical and Geoenvironmental Engineering Vol. 123, No. 12, p. 1162-1174.

Liang, L. Figueroa, J.L., and Saada, A.S. (1995) Liquefaction under Random Loading: Unit Energy Approach, Journal of Geotechnical Engineering, ASCE, Vol. 121, No. 11, pp. 776-781.

Schnabel, P., Lysmer, J., and Seed, H.B. (1972) SHAKE: A Computer Program for Earthquake Ground Response. Earthquake Engineering Research Center Report No. UCB/EERC-72-12.

Seed, H.B., and Idriss, I.M. (1971) Simplified Procedure for Evaluating: Soil Liquefaction Potential, Journal of the Soil Mechanics and Foundations Division, ASCE, 97:SM9, pp. 1249-1273.

Seed, H.B., and Idriss (1982) Ground Motions and Soil Liquefaction During Earthquakes, Earthquake Engineering Research Institute, Berkeley California.

Seed, H.B., Wong, R.T., Idriss, I.M., Tokimatsu, K. (1984) Moduli and Damping Factors Dynamic Analyses of Cohesionless Soils, Earthquake Engineering Research Center Report No. UCB/EERC-84/14.

Sun, J.I., Golesorkhi, R., and Seed, H.B. (1988) Dynamic Moduli and Damning Ratios for Cohesive Soils , Earthquake Engineering Research Center Report No. UCB/EERC-88/15.

Vucetic, M. and Dobry, R. (1991) Effect of Soil plasticity on Cyclic Response, Journal of Geotechnical Engineering, ASCE, Vol. 17, No. 1, 89-107.

Wilson, R.C. and Keefer, D.K. (1985) Predicting, Areal Limits of Earthquake-Induced Landsliding, in Ziony, J.I., ed. Evaluating the Earthquake Hazards in the Los Angeles Region: U.S. Geological Survey Professional Paper 1360.

Yuxian, H. (1979) Some engineering features of the 1976 Tangshan Earthquake, 2nd U.S. National Conference on Earthquake Engineering, August 22-24, 1979, Stanford University, EERI, Oakland, CA.

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