**6. Rapid full-wave CMT inversion**

In Southern California, preliminary 3D earth structure models are already available, and efficient numerical methods have been developed for 3D anelastic wave-propagation simulations. We develop an algorithm to utilize these capabilities for rapid full-wave centroid moment tensor (CMT) inversions. The procedure relies on the use of receiver Green tensors (RGTs), the spatial-temporal displacements produced by the three orthogonal unit impulsive point forces acting at the receivers. Once we have source parameters of earthquakes, a nearreal time full-wave ground motion map, that considers both source and wave-propagation effects in a 3D structure model, may also available for earthquake early warning purposes.

138 Earthquake Engineering

generate ground shaking maps.

additional warning time for some sites.

magnitude errors will decrease [2].

structures, for example, basin amplification effects.

**6. Rapid full-wave CMT inversion** 

**5. ElarmS: Earthquake alarms systems for California** 

In Southern California, an earthquake-prone area, many cities are under earthquake risks, hence earthquake early warning systems are becoming an important role in earthquake disaster mitigation [2]. Allen [2] developed earthquake early warning systems called Earthquake Alarms Systems (ElarmS) for California. In the ElarmS methodology, three steps are designed for rapid estimations of earthquake source parameters and prediction of peak ground motions [1, 2]. First, using the time information of first-arrived signal to locate earthquakes and estimate the warning time. Second, using frequency information of first four seconds of P-wave to estimate magnitudes of earthquakes. Third, using attenuation relations and the earthquake source information, an estimated location and magnitude, to

In the ElarmS, the arrival times of P-waves are used to rapidly locate earthquake locations. The possible areas of an earthquake location could be inferred by using the information of the first two or three stations trigged by an earthquake. To locate a more accurate earthquake location, including, longitude, latitude, depth and origin time, the first arrival time form four stations are required. A grid search algorithm is used to find an optimal earthquake location that has minimum arrival time misfits. The warning time, the remaining time before the peak ground motion arrived, can be estimated by using the predicted Swave arrival times of sites. Peak ground motions are usually caused by S-wave or surface wave, so use predicted S-wave arrival times as peak ground motion times may provide

The magnitude, which represents the released energy of an earthquake, is an important parameter in earthquake early warning systems. The rapid magnitude estimation method of an earthquake by using the frequency information of the first four seconds of P-wave is adopted in the ElarmS [1, 2]. Basically, the magnitude estimations take two procedures. The first step is finding the maximum predominant period within the first 4 seconds of the vertical component P-wave waveforms, and then use linear relations to scale the maximum predominant period value to an estimated earthquake magnitude [1, 2]. As the number of the maximum predominant period value from different receivers increases, the average

When the location and magnitude of an earthquake is available, attenuation relations can be used to estimate ground motions of sites and then generate a ground motion prediction map for whole California. In the ElarmS, the attenuation relations are based on the recordings of earthquakes with magnitude larger than 3.0 in California [2]. However, the empirical attenuation relations used in the ElarmS do not account effects of wave propagation in 3D

In Southern California, preliminary 3D earth structure models are already available, and efficient numerical methods have been developed for 3D anelastic wave-propagation simulations. We develop an algorithm to utilize these capabilities for rapid full-wave centroid In our CMT inversion algorithm, the RGTs are computed in our updated 3D seismic structure model for Southern California using the full-wave method that allows us to account for 3D path effects in our source inversion. The efficiency of forward synthetic calculations could be improved by storing RGTs and using reciprocity between stations and any spatial grid point in our model. In our current model, we will use three component broadband waveforms below 0.2 Hz to invert source parameters. Based on Kikuchi and Kanamori's [23]source inversion method, any moment tensor can be expressed as linear combination of 6 elementary moment tensors. In our current coordinate (*x*=east, *y*=north, *z*=up), the moment tensor can be expressed as below:

$$\begin{aligned} \mathbf{M1} &= \begin{bmatrix} & & & \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}; \quad \mathbf{M2} = \begin{bmatrix} & & & & \\ -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}; \quad \mathbf{M3} = \begin{bmatrix} & & & & \\ 0 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix}. \\\ \mathbf{M4} &= \begin{bmatrix} & & & \\ 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0 \end{bmatrix}; \quad \mathbf{M5} = \begin{bmatrix} & & & & \\ 0 & 0 & 0 & -1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{8}$$

There are two main advantages of using this method. First, different subsets of 6 elementary moment tensors could represent different source parameter assumptions such as M1~M6 could recover general moment tensors and M1~M5 could represent pure-deviatoric moment tensors [23]. From efficiency point of view, we only need to generate synthetic waveforms of 6 elementary moment tensors at grid points close to initial sources locations for receivers to invert an optimal CMT solution.

For centroid location <sup>1</sup> **x** and centroid time *t*1, the synthetic seismograms of 6 elementary moment tensors could be defined as:

$$\begin{array}{ccccc} S\_{mi}^{\tau} \text{(t:x}\_{1}, t\_{1}) & \text{ & m:1\sim6} \\ \end{array} \tag{9}$$

where *r* is receiver, *m* is index of 6 moment tensor, *i* is component index. The synthetic seismogram can be expressed as:

$$\mathbf{u}\_i^r(t) = \sum\_{m=1}^6 a\_m S\_{mi}^r(t; \mathbf{x}\_1, t\_1) \tag{10}$$

From inversion point of view, the phases have less structure heterogeneous effects can reduce the nonlinear effects caused by complex 3D structure such as body wave phases that propagate through relative simple deep structure and surface wave phases that propagate along free surface and average out the heterogeneity. We apply our seismic waveform segmentation algorithm that is based on continuous wavelet transforms and a topological watershed method to observed seismograms and then select the first (potential body wave) and biggest (potential surface wave) time-localized waveforms to invert source parameters.

Full-Wave Ground Motion Forecast for Southern California 141

1 1

*n*

*N N*

*N n nn n n n*

0 0

*P NCC P NCC H P H*

various purposes, such as to account for different signal-to-noise ratios in observed seismograms

0 0 ( ) *<sup>n</sup> nq q* | ( ).

can be used for rejecting problematic observations. In practice, we only accept observations with

A very low <sup>0</sup>( ) *<sup>n</sup> P NCC* indicates that the *n*th observed waveform cannot be fit well by any solutions in our sample space. This may be due to instrumentation problems or unusually

In the second step, we apply the same algorithm on another measurements, time-shifts between the observed and synthetic waveforms when the NCC is the maximum in allowed time-shift range. The last step is applying the same processes to the amplitude ratio measurements. By using the Bayesian approach, we can obtain the probability density functions of source parameters that contain uncertainties information rather than a single best solution. Our optimal source parameter solution is the one with highest probability. In Figure 4, examples of the marginal probabilities for some of the source parameters are

For earthquake early warning purposes, there are few approaches to make our CMT inversion method toward (near) real time and then use optimal CMTs for generating full-wave peak ground motion maps. To save some time in generating synthetic seismograms, we can store synthetic seismograms of 6 elementary moment tensors rather than extract them from RGTs. Destructive or larger earthquakes tend to occur in existing fault zones or regions where earthquakes occurred. Based on the assumption above, rather than store synthetic seismograms of all grid points in our model, we can store the synthetic seismograms of grid points near fault zones or high seismicity regions. Another possibility is to save time of inversion by using other efficient inversion algorithms. Full-wave ground motion prediction maps could be generated based on the synthetic seismograms of optimal CMT solutions.

1 1

and to avoid the solution to be dominated by a cluster of closely spaced seismic stations.

*q*

*N N*

*n n q*

*P H NCC*

( )exp (1 ) 1 exp( 2 )

*P NCC*

in front of (1-*NCCn*) in equation (14) can be used as a weighting factor for

*n*


*P NCC P NCC H P H* (16)

0 0 ( ) *<sup>n</sup> P NCC Q* (17)

(15)

0 1

*n nq q*

0

*<sup>n</sup> <sup>N</sup> <sup>n</sup>*

0

We note that the *<sup>n</sup>*

where

1

The probability for individual measurements

high noise levels in the observed waveform data.

shown for the 3 September 2002 Yorba Linda earthquake.


*P H NCC*

1

(14)

In source inversion, we applied a multi-scale grid-searching algorithm based on Bayesian inference to find an optimal solution [Figure 4]. We consider a random vector *H* composed of 6 source parameters: the longitude, latitude and depth of the centroid location **r**S, and the strike, dip and rake of the focal mechanism. We assume a uniform prior probability *P*0(*H*) over a sample space Ω0, which is defined as a sub-grid in our modeling volume centered around the initial hypocenter location provided by the seismic network with grid spacing in three orthogonal directions given by a vector **θ0** and a focal mechanism space with the ranges given by 0°≤ strike ≤360°, 0°≤ dip ≤90° and -90°≤ rake ≤90° and with angular intervals in strike, dip and rake specified by a vector 0.

We apply Bayesian inference in three steps sequentially. In the first step, the likelihood function is defined in terms of waveform similarity between synthetic and observed seismograms. We quantify waveform similarity using a normalized correlation coefficient (NCC) defined as

$$\text{NCC}\_{n} = \max\_{\Delta t} \left[ \underbrace{\stackrel{\boldsymbol{t}\_{\star}^{1}}{\operatorname{\mathbf{s}}\_{n}} \text{( $t$ )} \text{s}\_{n} (\boldsymbol{t} - \Delta t) dt}\_{\boldsymbol{t}\_{n}^{0}} \right/ \left. \sqrt{\underset{\boldsymbol{t}\_{n}^{0}}{\operatorname{\mathbf{s}}\_{n}} \text{2}^{\dagger} \text{( $t$ )} \text{d}t} \right]\_{\boldsymbol{t}\_{n}^{0}} \text{s}^{2} (\boldsymbol{t} - \Delta t) dt}. \tag{11}$$

where *n* is the observation index, ( ) *ns t* and ( ) *ns t* are the filtered observed seismogram and the corresponding synthetic seismogram, 0 1 , *n n t t* is the time window for selecting a certain phase on the seismograms for cross-correlation (Figure 4b). We allow a certain time-shift *t* between the observed and synthetic waveforms. To prevent possible cycle-skipping errors, we restrict *t* to be less than half of the shortest period. We assume a truncated exponential distribution for the conditional probability

$$P(\text{NCC}\_n \mid H\_q) = \frac{\lambda\_n \exp\left[-\lambda\_n (1 - \text{NCC}\_n)\right]}{1 - \exp(-2\lambda\_n)}, \text{ -1} \sphericalangle \text{CCC}\_n \leq 1, \ H\_q \in \Omega\_0 \,\,\,\tag{12}$$

where *<sup>n</sup>* is the decay rate. Assuming the NCC observations are independent, the likelihood function can be expressed as

$$L\_0\left(H\left|\bigcap\_{n=1}^N \mathrm{NCC}\_n\right.\right) = \exp\left[-\sum\_{n=1}^N \lambda\_n (1 - \mathrm{NCC}\_n)\right] \prod\_{n=1}^N \left\{\lambda\_n \left[1 - \exp(-2\lambda\_n)\right]^{-1}\right\}\tag{13}$$

where *N* is the total number of NCC observations. The posterior probability for the first step can then be expressed as

#### Full-Wave Ground Motion Forecast for Southern California 141

$$P\_0\left(H \mid \bigcap\_{n=1}^N \text{NCC}\_n\right) = \frac{P\_0(H) \exp\left[-\sum\_{n=1}^N \lambda\_n (1 - \text{NCC}\_n)\right] \prod\_{n=1}^N \left\{\lambda\_n \left[1 - \exp(-2\lambda\_n)\right]^{-1}\right\}}{P\_0\left(\bigcap\_{n=1}^N \text{NCC}\_n\right)}\tag{14}$$

where

140 Earthquake Engineering

(NCC) defined as

where *<sup>n</sup>* 

in strike, dip and rake specified by a vector

exponential distribution for the conditional probability

likelihood function can be expressed as

*L H NCC*

0

can then be expressed as

propagate through relative simple deep structure and surface wave phases that propagate along free surface and average out the heterogeneity. We apply our seismic waveform segmentation algorithm that is based on continuous wavelet transforms and a topological watershed method to observed seismograms and then select the first (potential body wave) and biggest (potential surface wave) time-localized waveforms to invert source parameters.

In source inversion, we applied a multi-scale grid-searching algorithm based on Bayesian inference to find an optimal solution [Figure 4]. We consider a random vector *H* composed of 6 source parameters: the longitude, latitude and depth of the centroid location **r**S, and the strike, dip and rake of the focal mechanism. We assume a uniform prior probability *P*0(*H*) over a sample space Ω0, which is defined as a sub-grid in our modeling volume centered around the initial hypocenter location provided by the seismic network with grid spacing in three orthogonal directions given by a vector **θ0** and a focal mechanism space with the ranges given by 0°≤ strike ≤360°, 0°≤ dip ≤90° and -90°≤ rake ≤90° and with angular intervals

> 0.

We apply Bayesian inference in three steps sequentially. In the first step, the likelihood function is defined in terms of waveform similarity between synthetic and observed seismograms. We quantify waveform similarity using a normalized correlation coefficient

1 1 1

*t t t*

<sup>2</sup> <sup>2</sup> max ( ) ( ) ( ) ( ) . *n n n*

*n*

0

1

*NCC*

(11)

(12)

0 0 0

*n nn <sup>n</sup> <sup>t</sup> <sup>t</sup> t t NCC s t s t t dt s t dt s t t dt*

*n n n*

where *n* is the observation index, ( ) *ns t* and ( ) *ns t* are the filtered observed seismogram and the corresponding synthetic seismogram, 0 1 , *n n t t* is the time window for selecting a certain phase on the seismograms for cross-correlation (Figure 4b). We allow a certain time-shift *t* between the observed and synthetic waveforms. To prevent possible cycle-skipping errors, we restrict *t* to be less than half of the shortest period. We assume a truncated

exp (1 ) ( |) , -1< 1, , 1 exp( 2 )


where *N* is the total number of NCC observations. The posterior probability for the first step

is the decay rate. Assuming the NCC observations are independent, the

*n n nn n*

(13)

*n q n q n NCC*

*nn n*

 

1 1 1

*N N N*

*n n n*

*P NCC H NCC H*

$$P\_0\left(\bigcap\_{n=1}^N \mathrm{NCC}\_n\right) = \sum\_q P\left(\bigcap\_{n=1}^N \mathrm{NCC}\_n \mid H\_q\right) P\_0(H\_q). \tag{15}$$

We note that the *<sup>n</sup>* in front of (1-*NCCn*) in equation (14) can be used as a weighting factor for various purposes, such as to account for different signal-to-noise ratios in observed seismograms and to avoid the solution to be dominated by a cluster of closely spaced seismic stations.

The probability for individual measurements

$$P\_0(\text{NCC}\_n) \propto \sum\_q P\left(\text{NCC}\_n \mid H\_q\right) \mathbb{P}\_0(H\_q). \tag{16}$$

can be used for rejecting problematic observations. In practice, we only accept observations with

$$P\_0(\mathcal{NCC}\_n) \ge Q\_0 \tag{17}$$

A very low <sup>0</sup>( ) *<sup>n</sup> P NCC* indicates that the *n*th observed waveform cannot be fit well by any solutions in our sample space. This may be due to instrumentation problems or unusually high noise levels in the observed waveform data.

In the second step, we apply the same algorithm on another measurements, time-shifts between the observed and synthetic waveforms when the NCC is the maximum in allowed time-shift range. The last step is applying the same processes to the amplitude ratio measurements. By using the Bayesian approach, we can obtain the probability density functions of source parameters that contain uncertainties information rather than a single best solution. Our optimal source parameter solution is the one with highest probability. In Figure 4, examples of the marginal probabilities for some of the source parameters are shown for the 3 September 2002 Yorba Linda earthquake.

For earthquake early warning purposes, there are few approaches to make our CMT inversion method toward (near) real time and then use optimal CMTs for generating full-wave peak ground motion maps. To save some time in generating synthetic seismograms, we can store synthetic seismograms of 6 elementary moment tensors rather than extract them from RGTs. Destructive or larger earthquakes tend to occur in existing fault zones or regions where earthquakes occurred. Based on the assumption above, rather than store synthetic seismograms of all grid points in our model, we can store the synthetic seismograms of grid points near fault zones or high seismicity regions. Another possibility is to save time of inversion by using other efficient inversion algorithms. Full-wave ground motion prediction maps could be generated based on the synthetic seismograms of optimal CMT solutions.

Full-Wave Ground Motion Forecast for Southern California 143

In this chapter, we compare full-wave based and non-full-wave based methods of ground motion forecast for (southern) California. There are advantages and disadvantages to different methods. Since the full-wave methods involve numerical simulations of wave propagation in 3D velocity models, the computational resource requirements are much higher than non-full-wave methods. However, numerical simulations are usually affordable in most of super computers. In general, ground motion estimations of non-full-wave methods are usually based on empirical attenuation relations. In full-wave methods, the ground motion estimations are based on numerical simulations that considered source effects, basin amplification effects and wave propagation effects in a 3D complex velocity [5, 6]. Those effects may play very important roles in ground motion estimations. For example, if a large earthquake occurs on southern San Andreas fault, the released energy will channel into Los Angeles region, one of the most populous cities in the United States, and basin effects will amplify the ground motion [22, 5]. Full-wave based ground motion forecast should able to provide more accurate and detailed ground motions and

The full-wave CMT inversion research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357. En-Jui Lee is supported by the Southern California Earthquake Center. Po Chen is supported jointly by the School of Energy Resources and the Department of Geology and Geophysics at the

[1] Allen RM, Kanamori H (2003) The potential for earthquake early warning in southern

[2] Gasparini P, Manfredi G, Zschau J (eds) (2007) Earthquake early warning systems.

[3] Petersen MD, Frankel AD, Harmsen SC, et al (2008) Documentation for the 2008 update of the United States national seismic hazard maps. U.S. Geological Survey Open-File Report [4] Frankel AD, Field EH, Petersen MD, et al (2002) Documentation for the 2002 update of

[5] Graves R, Jordan T, Callaghan S, Deelman E (2010) CyberShake: A Physics-Based Seismic Hazard Model for Southern California. Pure appl geophys 168:367–381 [6] Lee E, Chen P, Jordan T, Wang L (2011) Rapid full‐wave centroid moment tensor (CMT) inversion in a three‐dimensional earth structure model for earthquakes in Southern

the national seismic hazard maps. U.S. Geological Survey Open-File Report

California. Geophysical Journal International 186:311–330

University of Wyoming. Comments from the book editor improved our manuscript.

this will benefit cities under earthquake risks, such as Los Angeles city.

*Department of Geology and Geophysics, University of Wyoming, USA* 

**7. Conclusion** 

**Author details** 

**8. References** 

Springer, Berlin

California. Science 300:786–789

En-Jui Lee and Po Chen

**Acknowledgement** 

**Figure 4.** An example of our CMT inversion procedure. (a) The map shows epicenter of the 3 September 2002 Mw 4.3 Yorba Linda earthquake (the star), the best-fit double-couple solution (the red beachball) and stations (gray triangles) selected for this inversion. (b) Examples of the waveforms selected for in the CMT inversion. The black lines are observed seismograms and the red lines are synthetic seismograms. The black bars indicate the selected waveform segments for CMT inversion. (c) The marginal probability densities for strike, dip, rake and depth obtained after our grid-search step.

The speed of earthquake source parameters estimation and accurate ground motion predictions are both play essential role in earthquake early warning systems. In the ElarmS, earthquake locations, origin time and magnitudes could be inverted in very short time. Peak ground motion maps can be generated shortly by using empirical attenuation relations. The CMT inversion method we proposed has potential for (near) real time inversion and then solutions could be used for (near) real time full-wave peak ground motion maps. In addition, the Bayesian approach used in our CMT inversion has uncertainty of solutions and could be projected into ground motion estimations.

### **7. Conclusion**

142 Earthquake Engineering

**Figure 4.** An example of our CMT inversion procedure. (a) The map shows epicenter of the 3

could be projected into ground motion estimations.

September 2002 Mw 4.3 Yorba Linda earthquake (the star), the best-fit double-couple solution (the red beachball) and stations (gray triangles) selected for this inversion. (b) Examples of the waveforms selected for in the CMT inversion. The black lines are observed seismograms and the red lines are synthetic seismograms. The black bars indicate the selected waveform segments for CMT inversion. (c) The marginal probability densities for strike, dip, rake and depth obtained after our grid-search step.

The speed of earthquake source parameters estimation and accurate ground motion predictions are both play essential role in earthquake early warning systems. In the ElarmS, earthquake locations, origin time and magnitudes could be inverted in very short time. Peak ground motion maps can be generated shortly by using empirical attenuation relations. The CMT inversion method we proposed has potential for (near) real time inversion and then solutions could be used for (near) real time full-wave peak ground motion maps. In addition, the Bayesian approach used in our CMT inversion has uncertainty of solutions and In this chapter, we compare full-wave based and non-full-wave based methods of ground motion forecast for (southern) California. There are advantages and disadvantages to different methods. Since the full-wave methods involve numerical simulations of wave propagation in 3D velocity models, the computational resource requirements are much higher than non-full-wave methods. However, numerical simulations are usually affordable in most of super computers. In general, ground motion estimations of non-full-wave methods are usually based on empirical attenuation relations. In full-wave methods, the ground motion estimations are based on numerical simulations that considered source effects, basin amplification effects and wave propagation effects in a 3D complex velocity [5, 6]. Those effects may play very important roles in ground motion estimations. For example, if a large earthquake occurs on southern San Andreas fault, the released energy will channel into Los Angeles region, one of the most populous cities in the United States, and basin effects will amplify the ground motion [22, 5]. Full-wave based ground motion forecast should able to provide more accurate and detailed ground motions and this will benefit cities under earthquake risks, such as Los Angeles city.

### **Author details**

En-Jui Lee and Po Chen *Department of Geology and Geophysics, University of Wyoming, USA* 

#### **Acknowledgement**

The full-wave CMT inversion research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357. En-Jui Lee is supported by the Southern California Earthquake Center. Po Chen is supported jointly by the School of Energy Resources and the Department of Geology and Geophysics at the University of Wyoming. Comments from the book editor improved our manuscript.

#### **8. References**


[7] Field EH, Dawson TE, Felzer KR, et al (2009) Uniform California Earthquake Rupture Forecast, Version 2 (UCERF 2). Bulletin of the Seismological Society of America 99(4):2053–2107

**Chapter 6** 

© 2012 Tyapin, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Tyapin, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

First the very definition of the soil-structure interaction (SSI) effects is discussed, because it is somewhat peculiar: every seismic structural response is caused by soil-structure interaction forces, but only in certain situations they talk about soil-structure interaction (SSI) effects. Then a brief history of this research field is given covering the last 70 years. Basic superposition of wave fields is discussed as a common basis for different approaches – direct and impedance ones, first of all. Then both approaches are described and applied to a simple 1D SSI problem enabling the exact solution. Special attention is paid to the substitution of the boundary conditions in the direct approach often used in practice. Impedance behavior is discussed separately with principal differentiation of quasi-homogeneous sites and sites with bedrock. Locking and unlocking of layered sites is discussed as one of the main wave effects. Practical tools to deal with SSI are briefly described, namely LUSH, SASSI and CLASSI.

Nowadays SSI models are linear. Nonlinearity of the soil and soil-structure contact is treated in a quasi-linear way. Special approach used in SHAKE code is discussed and illustrated. Some non-mandatory additional assumptions (rigidity of the base mat, horizontal layering of the soil, vertical propagation of seismic waves) often used in SSI, are discussed. Finally, two of the SSI effects are shown on a real world example of the NPP building. The first effect is soil flexibility; the second effect is embedment of the base mat. Recommendations to

Soil-structure interaction (SSI) analysis is a special field of earthquake engineering. It is worth starting with definition. Common sense tells us that every seismic structural response is caused by soil-structure interaction forces impacting structure (by the definition of seismic excitation). However, engineering community used to talk about soil-structure interaction

**Soil-Structure Interaction** 

Additional information is available at the end of the chapter

Combined asymptotic method (CAM) is presented.

engineers are summed in conclusions.

**2. Peculiarities of the SSI definition** 

Alexander Tyapin

**1. Introduction** 

http://dx.doi.org/10.5772/48333

