Proc. Third ASCE Specialty Conf. on Geotechnical Earthquake Engineering and Soil Dynamics,
2-6 Aug, 1998, Seattle Washington.
ASCE, Geotechnical Special Publication No. 75,  Vol. 1, pp.  116-127.
 
QUICK REFERENCE LIQUEFACTION OPPORTUNITY MAPS
FOR A METROPOLITAN AREA
Maria I. Todorovska
 
Univ. of Southern California, Civil Eng. Dept., Los Angeles, CA 90089-2531
 
 
Figure 1
Figure 2
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Figure 4
Figure 5
Figure 6
 
 

ABSTRACT

A series of probabilistic liquefaction opportunity maps are presented for metropolitan Los Angeles, for 50 years exposure. An empirical model for liquefaction occurrence is used, based on seismic wave energy. The site susceptibility is described via the corrected SPT value, , and the overburden pressure, s 0. For given s 0, the model gives the critical for which the site will liquefy. This value is treated as a random variable. The energy of shaking is described via a functional of the Fourier spectrum of strong motion velocity, considering the effects of deep sediments on the amplitudes of motion. The mapped quantity is the logarithm of the expected number of liquefaction occurrences, mLiq. It is shown how these maps can be used to plot the average number, average return period and probability of liquefaction occurrences versus .
 

INTRODUCTION

Liquefaction of water saturated sands is believed to occur when the pore pressure approaches the confining pressure. It can initiate movement of large blocks of soil (lateral spreading), causing extensive damage to man made structures (Youd, 1991). The likelihood that an earthquake will liquefy a site depends on many site characteristics (e.g. mean grain size, percentage of fines...; Blazquez et al., 1980) and on the regional geology, which influences the amplification and attenuation of strong motion amplitudes. However, such detail is not available for most of the sites known to have liquefied. Therefore, empirical regression models for liquefaction occurrence have been limited to simple site characterization, via the corrected (for overburden pressure) SPT value, , the overburden pressure, s 0, and characterization of the strong shaking by the earthquake magnitude and site to source distance. The empirical model employed by the hazard methodology in this paper was derived based on data at 91 sites which liquefied or did not liquefy from an earthquake. These data were used by Davis and Berrill (1982) and Berrill and Davis (1985), who related the liquefaction occurrence to seismic wave energy expressed directly in terms of magnitude and distance. The same data set was used later by Trifunac (1995), who proposed several empirical energy functionals for determination of liquefaction criteria, in terms of various ground motion parameters (e.g., peak velocity and duration of shaking, and Fourier amplitude spectrum of velocity). The "en" model of Trifunac (1995), used in this paper, evaluates the ground shaking energy via Fourier spectra of ground velocity, extended to long periods and to high frequencies beyond the recording range of most accelerographs. The empirical regression models of Davis and Berrill (1982), Berrill and Davis (1985) and Trifunac (1995) represent the best fit to the data that separates (in least squares sense) the occurrences from the no occurrences of liquefaction.

Todorovska (1996) showed how the empirical models of Trifunac (1995) can be incorporated in probabilistic seismic hazard models, and presented a sensitivity analysis of the results for a simple, single fault model. Todorovska and Trifunac (1998) applied the methodology to a realistic multi-fault setting in southern California, and addressed the issue of cut-off criteria for contribution to the hazard from distant earthquakes at sites with small , and the issue of the influence of the geologic structure (e.g., sedimentary basins) on the hazard estimates, via amplification of the amplitudes of ground shaking. They used a magnitude dependent cut-off distance for contribution to the hazard based on minimum energy of ground motion, calibrated by the same data used to derive the liquefaction criteria models. They showed example maps of the average return period of liquefaction opportunity for the Los Angeles metropolitan area, first assuming rock geologic site condition everywhere, and then considering the actual thickness of the sediments. Liquefaction opportunity maps are hazard maps evaluated for assumed uniform liquefaction susceptibility throughout the area (e.g., =10 and s 0=40 kPa). The actual hazard can be evaluated by overlaying opportunity with susceptibility maps. While Todorovska and Trifunac (1998) presented a small set of results and focussed on demonstrating the validity of the methodology in a realistic seismic environment, this paper presents a series of liquefaction opportunity maps for the Los Angeles metropolitan area, for various combinations of values of and s 0, (eighteen maps, for s 0=20, 40 and 80 kPa and for =5, 10, 15, 20, 25 and 30 counts per foot, 1 foot = 30.48 cm), showing variability of the hazard in the area on and s 0,, and practical applicability.
 

METHODOLOGY

The "en" empirical model of Trifunac (1995) estimates the critical corrected SPT value for which liquefaction will occur as follows

                                                                 (1) where mod is in counts per foot, en is a quantity proportional to the energy of ground motion at the site (in m2/s), s0 is the overburden pressure (in kPa), and c=572 and n=2.5 are constants determined by the regression. The term en is the integral

                                                                  (2)
 
where w is the circular frequency and F(w) is the Fourier spectrum of acceleration. Integration is performed from frequencies f=w/2p=0.01 Hz to 100 Hz. Within the frequency range where the amplitude of the signal is grater than the amplitude of the average noise of recorded strong motion (0.1 to 25 Hz, Lee et al., 1982), F(w) is evaluated via empirical scaling equations of the form (Trifunac and Lee, 1990)

     log10 F(w ) = j (M, r, H, w , V, s, sL, p)                                                             (3) obtained by regression of recorded strong motion. Outside this range, F(w ) is "extended" so that it is consistent with permanent deformation (at long periods) and with the regional attenuation of high frequencies (Trifunac, 1993, 1994).

The arguments of the function j in eqn (3) are as follows: M - earthquake magnitude, r - representative source to site distance (depends on the physical distance and, close to the fault, also on the size of the rupture and on the frequency of radiation), H - hypocentral depth, s - categorical geologic site parameter (s=0 for sediments, s=2 for basement rock and s=1 for intermediate sites; Trifunac, 1990a), sL - local soil condition categorical variable (sL=0 for "rock", sL=1 for stiff soil, sL=2 for deep soil), and p - probability of no exceedance (p=0.5 for evaluation of en in eqn (2), and the uncertainty in the estimation of F(w ) is built in the uncertainty in the estimation of the critical SPT value, , for liquefaction to occur). Where the thickness of the sedimentary layer at the site, h, is available, the dependence on the categorical geologic parameter, s, can be replaced by a more accurate dependence on h. The site geology parameters s and h do not affect the liquefaction susceptibility of the site, but do affect the amplitudes of shaking, and therefore the liquefaction opportunity. Possibility of amplification of strong ground motion by wave interference in sedimentary deposits has been demonstrated via analytical and numerical models. Recording of strong shaking by relatively dense arrays showed that the ground motion amplitudes in basins attenuate less with distance (Todorovska and Trifunac, 1997a,b). The effects of site geology (via s or h) and local soil (via sL), acting simultaneously, have been incorporated in probabilistic hazard calculations and mapping since the late 1980’s for Fourier and response spectra (Lee and Trifunac, 1987; Trifunac, 1988, 1990b; Todorovska, 1995).

The site will liquefy if its susceptibility is specified by value <crit. However, there is an uncertainty in the estimation of Ncrit by the regression model, and therefore it is treated as a random variable. The data is insufficient for a rigorous statistical analysis of acceptance or rejection of various types of distribution functions for . Instead, is assumed to be normally distributed, with mean and variance obtained by analysis of the residuals,obs –mod. The residuals have mean value –1.95 and standard deviation value 5.45, which implies mean of equal to. Then, the probability that an earthquake with magnitude M and at distance r will liquefy the site equals the probability that is larger than at the site. This probability is denoted by qLiq and it is the conditional probability that the site will liquefy given that such an earthquake has occurred.

The probability that the site will liquefy during specified exposure period (e.g., 50 years), will depend on the likelihood of sufficiently large earthquakes occurring sufficiently close to the site. For practical evaluation of probabilistic seismic hazard, the earthquake magnitude scale and the possible earthquake locations are discretized. Let an index k be assigned to the magnitude sizes and index l to the locations. Then, the average number of liquefaction occurrences during exposure t is

                                                           (4) where nlk(t) is the expected number of earthquakes of size type k and at location l during exposure t. The above equation is valid for earthquake occurrence in time as a Poissonian sequence of events. Similar result is obtained for generalized Poissonian sequences, that is those which occur in time orderly, as in a Poissonian process, but with time dependent hazard rate (Todorovska, 1994; Todorovska et al., 1995).

Once the average number of liquefaction occurrences is known, the average return period of liquefaction can be calculated, as

                                                                 (5) and the probability that liquefaction will occur at the site at least once during the exposure t as                                                      (6) It is informative to present the results of the hazard calculations at a site in the form of plots of mLiq, Tave and p versus , for different values of the overburden pressure s 0, or via maps for some area. These maps may show Tave, or mLiq for specified and s 0, or corresponding to specified probability, p, of liquefaction occurrence during the exposure for all the sites on the map. Examples of such presentations, and use, are shown in what follows.

Other simplified forms of liquefaction opportunity maps could be presented. For example, if the water table is typically at 3 m (10 feet) depth, and assuming that liquefaction occurs below the lowest historical water table (e.g. 5 m or 15 feet), only one overburden pressure (say 50 kPa) could be considered. Then, the contours of for different levels of probability of liquefaction could be presented. In an area with a dense grid of measured and s 0 , the contours of and s 0 could be convolved with the maps illustrated in this paper. Then, final hazard maps can be presented showing probability that a site will liquefy during the specified exposure, or the return period of liquefaction.
 

RESULTS

Results are shown for the Los Angeles metropolitan area. These were evaluated by the liquefaction assessment module of the program NEQRISK (Lee and Trifunac, 1985; Todorovska et al., 1995; Todorovska, 1996; Todorovska and Trifunac, 1996). Figure 1 shows the location of the seismic sources for the Southern California model used by Todorovska and Trifunac (1998). These source zones correspond approximately to those defined by the Working Group on California Earthquake Probabilities (1995), and have seismic moment rates consistent with those for the alternate model proposed by the Working Group (Poissonian earthquake occurrence in time, both for the "random" and for the "characteristic" earthquake populations). The model earthquake rates versus magnitude, for the entire region, were compared against the rates for the Working group alternate model and against the observed rate, and show close agreement with the observed rates. Further details about the seismicity model can be found in Todorovska and Trifunac (1998).

The Los Angeles basin area is shown by the shaded rectangle in Figure 1, and a more detailed view is in Figure 2. The shaded areas outline the hills and mountains. These are also shown with the hazard maps, to help visualize the locations of rock and sediments, and interpret the patterns in the presented hazard maps. The actual thickness of sediments, h, used for the grid of points where the hazard was evaluated, can be found in Todorovska and Trifunac, 1998).

The results were evaluated at a grid of 5’ by 5’, with coordinates corresponding to the ticks on the coordinate axes in Figure 2, and at the King Harbor site, shown by the triangle (this site liquefied during the 1994 Northridge earthquake; Kerwin and Stone, 1997). The results were then interpolated and smoothed to get the contour maps. Figure 3, Figure 4 and Figure 5 show such maps for s 0 = 20, 40 and 80 kPa, and each for =5 through 30 counts per foot. The quantity plotted is the log10 mLiq. Once this quantity is known, the average return period and the probability of liquefaction occurrence can be calculated, by eqns (5) and (6). It is seen from Figure 3, Figure 4 and Figure 5 that the shape and the amplitudes of the contours are affected by the proximity to the local faults and by the local geology (thickness of sediments). The liquefaction opportunity is largest in the central Los Angeles basin, and is small in the hills and mountains. It is also large for the small values of (=5) and s 0 (=20 kPa). Similar maps for rock geology (h=0) would show smaller variability in the shapes of the contours and prominent influence of the San Andreas fault (no. 6 and 13 in Figure 1) on the liquefaction opportunity estimates (Todorovska and Trifunac, 1998).

Figure 6 illustrates results for King Harbor. On the left, log10 mLiq and log10 Tave are shown versus , and on the right, the probability of occurrence, p. Plots like this one can be drawn approximately for any site on the map, using the information from the series of maps in Figure 3, Figure 4 and Figure 5. The procedure is as follows. From the maps in Figure 3, log10 mLiq is read and tabulated for each , which will give the curves for s 0=20 kPa. This procedure is then repeated for the maps in Figure 4 and Figure 5. The tabulated values of log10mLiq are used to evaluate Tave and p by eqns (5) and (6). The points read are shown in Figure 6, in the table and also in the plots, by the open circles.

The accuracy of the contour maps can be improved by evaluating the hazard at a denser grid of points, and by more detailed modeling of the geometry of the faults (Todorovska and Lee, 1995). The accuracy of reading the maps can be improved by drawing maps on a large scale and for smaller intervals of log10 mLiq. Such maps will agree qualitatively with those presented in this paper, but will offer far greater detail locally. Also, the information on the seismic moment rates and maximum magnitudes can be updated as required, and time dependent earthquake occurrence rates can be used (Todorovska, 1994; Todorovska et al., 1995). The mapped values can also be stored in a file, and used to construct plots like the one in Figure 6. Liquefaction hazard maps can be constructed by overlaying liquefaction opportunity maps with liquefaction susceptibility maps (Tinsley et al., 1985).
 

DISCUSSION AND CONCLUSIONS

An empirical, energy based, model for direct scaling of liquefaction probability was used to illustrate mapping liquefaction opportunity for the Los Angeles metropolitan area, for 50 years exposure. A series of maps like those presented in this paper will help visualize the spatial distribution of the liquefaction opportunity, and the influence of the local faults and geologic structure. Such maps can also be used to construct detailed curves describing the average return period and probability of occurrence of liquefaction versus the corrected SPT value at the site, , for any site on the map. It is recommended that, for actual use, such maps be calculated at a much denser grid of points.
 

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