Predicting Site Response

Predicting Site Response

• Based on theoretical calculations– 1-D equivalent linear, non-linear– 2-D and 3-D non-linear

• Needs geotechnical site properties

THEORETICAL PREDICTION OF SITE RESPONSE

Site Response

• Observations of Site Response and Implications for Predicting Response (with a major emphasis on variability of site response)

• Methods for Predicting (Estimating) Response• Other topics

– Nonlinear response– Spatial variability– Topographic amplification

Predicting Site Response

• Empirical studies and blind prediction experiments lead to the conclusion that site-specific, earthquake-specific predictions may be very uncertain.

• Despite this pessimistic conclusion, I will now proceed to discuss site-specific, earthquake-specific predictions.

• But always keep in mind the possible uncertainties in the predictions of site response

Predicting Site Response

• Theoretical• Full resonance• Simplified (square-root impedance)

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Velocity Model

Predicting site response: given input motion and velocity model, predict surface motion

Theoretical basis to understand site effects

• Layer over a half space• Multiple layers• Effects of basins• Effects of topography• Effects of manmade structures

Site amplification due to resonance effects

• Trapped waves reverberate due to multiple reflections

• Constructive interference causes resonance, which depends on thickness of layer and elastic properties

• Can occur even if NO discontinuities in seismic impedances

Resonance effects (cont.)• Attenuation in layer modifies resonance,

especially at high frequencies• Typically see fundamental resonance and

perhaps one more peak (higher frequency peaks damped by attenuation)

Use program nrattle in site_amp collection of programs on my web site

Multiple flat layers

• Haskell (1960’s) found an exact solution for the effect of multiple layers on a vertically incident S-wave.

• Subsequently improved for greater numerical stability.

• Each layer boundary causes some amplification. Net effect can be much greater than a factor of two.

• Multiple resonances exist.

In some cases the velocity profile is more of a gradient with no significant jumps in impedance. How is the amplification computed in such situations?

The gradient can be replaced by a stack of constant velocity layers, with thicknesses and velocities chosen so that the travel time through the profile is the same as for the continuous profile (using my program site_amp is a convenient way of doing this).

Full resonance amplifications can be obtained for the model made up of a stack of constant velocity layers (using nrattle in this case; note that the amplifications approach unity for low frequencies)

There is another method for approximating the amplification, which I now discuss.

Impedance effects• Seismic Impedance (Z) = velocity * density• Conservation of energy along a ray tube due to impedances predicts an

amplification of

where ZR is the impedance at the source and Z is the average impedance at points along the ray tube

• Shear-wave velocity and density increase with increasing depth in the crust• Typical velocity and density at seismogenic depths (10km) are 3.6 km/sec

and 2.8 gm/cm**3• Typical velocity near surface depends on NEHRP site class • Rock sites range from 2.8 km/sec (ENA hard rock) to as low as 600 m/sec

(Calif. Soft rock)• Thus all seismic waves will be amplified by impedance effects as they

travel to the surface

RA Z Z

Steps in computing square-root-impedance amplifications:

Obtain velocity-depth profile:

Steps in computing square-root-impedance amplifications:

Compute travel time as a function of depth:

Steps in computing square-root-impedance amplifications:

For each depth, compute the average velocity from the surface to that depth:

( )SZV z tt z

I use z=0.4 km only as an example; the procedure is repeated for all depths

Steps in computing square-root-impedance amplifications:

• Compute the average velocity VSZ for all depths z:

Steps in computing square-root-impedance amplifications:

• For each depth z, replace the real velocity with a layer of velocity equal to the average velocity to that depth (the example here is for z=0.4 km):

Steps in computing square-root-impedance amplifications:

• Now use the square-root impedance method to compute an amplification for this new model. The amplification will only be for the frequency corresponding to the resonant frequency of the constant velocity layer

4SVfH

Recall the equation for resonance frequency of a single layer of thickness H and shear-wave velocity VS

This is a quarter-wavelength condition, as it states that resonance will occur for a period for which the wavelength (VS/f) is 4 times the layer thickness

Now replace the material down to depth z with a layer of constant velocity VSZ. Then the quarter-wavelength period for this layer will be

1 4 4SZVfz

Thus for each depth z we can replace the material above that depth with a constant velocity layer with velocity VSZ and associate a quarter-wavelength frequency f1/4λ with each depth. The next slide shows Vs, VSZ, and f1/4λ as a function of depth z.

For the example being used (z=0.4), the value is 1.45/(4*0.4)=0.9 Hz

We can also estimate the amplification for this new velocity model for each depth z, using the square-root impedance (SRZ) amplification:

where the impedance is given by:

“R” indicates a reference depth, and the overbar indicates that the impedance and density are averaged over the depth z. Because we also can associate a quarter-wavelength frequency with each depth, we then obtain an approximation of amplification as a function of frequency.

( ) ( )RA z Z Z z

SZZ V

For the example (z=0.4 km), the amplification is sqrt(3.5/1.45)=1.55, ignoring changes in density. Here is the square-root-impedance amplification for the BJ97 generic rock velocity model (which does include the density variation).

Here is the square-root-impedance amplification for the BJ97 generic rock velocity model, compared to full resonance amplifications

Square-root impedance method (aka Quarter-wavelength method)

• Advantages – Simple, quick– No assumptions needed about deeper velocities

(except for source region)– Provides smooth version of amplification

• Uses– amplification at “generic” sites – First approximation/ guidance– Assessing depths that control amplification– Comparing velocity profiles

Square-root impedance method (aka Quarter-wavelength method)

• Disadvantages – Underpredicts resonance peaks (but maybe OK for deep

sediment-filled basins, for which fundamental mode is lower than frequencies of most engineering frequencies)

– Purely linear calculations

Now compare amplifications for some simple models.

All models have the same travel time to 37.5 m

Use nrattle for full resonance amps and site_amp for square-root-impedance (SRI) amps

Note shift of first resonant peak

SRI amps always underestimate fundamental mode, but provide average of higher mode amplifications

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Stringent conditions are required for perfect constructive and destructive interference (the essence of resonant response)

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The amplitude spectrum of a resonant systemrequires precise constructive interference of multiple arrivals. In practice this probably will not happen because of geologic complexity or nonlinear changes in seismic velocity, and thus observations may not show a well-defined set of resonant peaks.

Nonlinear response

• Amplitude-dependent amplification• Shift of resonant peaks to lower frequencies with

increased shaking• Loss of high frequencies in records• Gain of high frequencies in records (later cusp-like

arrivals)

Nonlinear soil behaviour:If soil is linear it obeys Hooke’s Law:

stress = G * strainwhere G=shear modulus

But real soil shows hysteretic behaviour under high strains

In hysteretic behaviour, loading and unloading curves are not the same (energy is lost). Effective slope G decreases at high strain. Since the shear-wave velocity is given by vs=sqrt(G/density), this means Vs decreases with strain.

Consequences of nonlinearity

• Vs decreases at large strains• Resonance frequency decreases, since f=Vs/4H• Amplification decreases due to damping (loss of energy) in

hysteresis loop (proportional to loop area)• Thus nonlinearity will result in lower soil amplification, and a

shift to lower frequencies• There can also be a shift of energy to higher frequencies

Nonlinear site response

• Under weak motions, to a good approximation, the stress-strain relationship in the soil is linear.

• Under strong motion, that linearity fails.• A common approximation to the effect of

nonlinearity is to decrease the velocity where the material is nonlinear, and increase the energy loss Q.

“Equivalent Linear” Soil Response (SHAKE)

• The world’s most commonly used program for approximating nonlinear soil response

• Highly recommended: Strata, free from http://nees.org/resources/strata

• Strata can compute equivalent linear amplifications both using time-domain and frequency-domain calculations.

Equivalent Linear

• Use G/Gmax and damping vs. strain to approximate nonlinear response

• Do iterative linear analysis, adjusting the modulus and the damping to be consistent with the strain in various layers

Equivalent Linear• Consequences

– Shifts resonant peaks to longer periods– Large attenuation of high frequencies

• Strengths and limitations– can use RVT to avoid needing a suite of input time

series– Fast– OK for period range of most interest– Too much attenuation at short periods (some people

make the damping vs. strain period dependent to overcome this)

Example

• Use a 1 layer case• 1940 El Centro motion as input, scaled to

0.01g and 1.0g

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Fundamental mode 0.5 sec

Linear amplification as function of period period (not frequency)

Modulus reduction and damping curves

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Effective strain vs. depth

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G/Gmax.& D values for strains obtained during soil response

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period shift

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Input and surface response spectra. Note differences at high frequency for low- and high-strain cases and shift of resonant peak to longer period for high strain case

Amplification

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A limitation of method: severe overdamping of short period motions often occurs (because strain is determined by longer periods, but the strain-dependent damping applies at all periods). Using period-dependent modulus and damping values is an attempt to retain advantages of the equivalent linear approach, without overdamping. But some cases a truly nonlinear calculation is necessary.

Truly Nonlinear• Time-domain integration, following stress-strain

relation• A number of programs, based on different

assumptions and requiring varying amounts of detail regarding the soil behavior– Pore pressure effects accounted for in some codes

Empirical evidence of soil nonlinearity

• Liquefaction• Soil-vs-rock differences in spectra and time series• Period shifts and high frequency spikes

Evidence of liquefaction

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1989 Loma Prieta earthquake

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Treasure Island (EW)

Yerba Buena Island (EW)

• At Treasure Island, relative to YBI:

• loss of high frequencies• increased amplitudes

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2001 Nisqually, Washington, earthquake (M 6.8)

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SDS (NS): unfiltered SDS: filtered from 10--20 Hz

SDN: filtered from 10--20 HzSDN (NS): unfiltered

• SDS within 200 m of SDN• liquefaction at SDS, not at SDN• Note cusps at SDS and increased amplitude at high frequencies

(after Frankel et al., 2002)

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SDS (geometric mean, horizontal components)

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Note shift of amplification peaks for the mainshock to lower frequencies and the increase in high frequency motion for the mainshock.

(after Frankel et al., 2002)

Effects of Topography

• Surface (amplification, deamplification)– Ridges, peaks– Canyons

• Buried (focusing, defocusing)– Example from Northridge earthquake (Frankel et

al.) (not shown here)

(Boore, 1972)

(Boore, 1972)

Basin Effects

• Sediment filled basins• Trapped body waves • Surface waves generated at basin edges• Increased duration• Two-dimensional models

Synthetic basin models

• Note surface waves generated at the basin edge.• Duration of shaking in the basin is greatly increased.

End