Soil-Structure Interaction (SSI) have been studied the last decades, and proper analysis for the linear elastic case in frequency domain has been established success- fully. However, SSI is rarely considered in the seismic design of building structures. Regardless of its importance as a signi cant source of exibility and energy dis- sipation, buildings are analyzed using a rigid base assumption, and the design is based on a response spectrum analysis, for which not only the soil, but also time are totally ignored. In a rst attempt to improve and to incentivize time domain analyzes compatible with standard nite element packages for the engineering com- munity, the state-of-practice introduces two major simpli cations to transform the frequency domain analysis into a time domain analysis: (a) it assumes the frequency at which the impedance value should be read is the exible-base frequency, and (b) it also assumes that the foundation input motion preserves the phase of the free eld motion. Upon these simpli cations, the following questions may arise: How does NIST recommendations perform in overall against a full nite element model? Are the embedment e ects for shallow foundation not important so that the phase angle can be neglected? What is the best dimensionless frequency to estimate the soil impedance? Is it possible to make a better estimation of the dimensionless fre- quency to increase the NIST accuracy? In this study, we attempt to address these questions by using an inverse problem formulation. Keywords| Soil-structure interaction, System identi cation, Ensemble Kalman Inver- Corresponding author: domniki@caltech.edu 1 sion, Period Elongation, Radiation Damping, Inverse problem, NIST, Reduced-Order Model, Mathematical modeling. 1 Introduction Accurate quanti cation of dynamic soil-structure interaction (SSI) e ects is critical in design of earthquake resistant structures. If SSI e ects are poorly estimated or even neglected, then an unsafe or overly conservative designs is produced because of an over- or under-estimation of the critical response of the structure [1,2]. In general, SSI analyses are carried out by means of either the direct or the substructure methods [3{7]. In the direct method, the near- eld soil and the structure are modeled explicitly and arti cial boundary conditions, which ideally introduce transparency to both incoming and outgoing waves, are employed to truncate the semi-in nite extent of the far- eld soil [8{10]. The substructure method [4,11,12] , on the other hand, divides the SSI problem into two sub-systems: the structure subsystem and the soil subsystem represented by impedance functions [13{15]. These functions are complex-valued, the real part is the representation of the soil sti ness and inertia while the imaginary part is the representation of geometric damping within the soil. To excite the model properly, one needs to modify the free eld motion (FFM) to map the ground motion in-coherency e ects along the interface of the structure foundation and the soil. Despite the computational eciency of the substructure method, one can only take full advantage of the method accuracy in frequency domain analysis which is only suitable for linear elastic problems. This is mainly because the soil impedance functions are nonlinear function of frequency and, in time domain analysis, they need to be convoluted with the so-called foundation input motion (FIM) to map the correct tractions along the structure foundation and the soil. To make the use of the substructure method practical in time domain analysis, researchers and practitioners have proposed di erent models to approximate the soil impedance func- tions [16{18]. Among them, and for building structures with shallow foundation, the Na- tional Institute of Standards and Technology (NIST) under the project entitled Improved Procedures for Characterizing and Modeling Soil-Structure Interaction for Performance- Based Seismic Engineering recommends using single-valued functions at a representative frequency, which can be modeled as constant-valued springs and dashpots along the soil- structure interface (see Figure 1d). The frequency at which the soil impedance is read is computed using an iterative method proposed by Bielak and Veletsos [1, 19]. In this method, one starts assuming that the period of the interaction is the xed-base building period (i.e., T). This value and geometry of the foundation allow to compute the dimen- sionless frequency a0=!B Vsas shown in Figure 1b. Such frequency enables to compute the values of the lumped translational and rotational spring and dashpot coecients as represented in Figure 1c. Provided with the lumped soil impedance coecients, one can compute the period elongation of the system using 2 ~T T=s 1 +k kxx+kh2 k; (1) where the variable ~Tis the exible-base period, Tis the xed-base period, his the rst- modal height, kis the xed building sti ness, kxxandkare the lumped translational and rotational springs coecients that account for the exibility of the surrounding soil. The process is repeated until there is no variation of the period elongation. Finally, provided with the exible-base period, one can read from the impedance function the lumped soil springkxx;kas well as the lumped soil dashpots cxx;cand { if necessary { distribute them along the foundation perimeter as shown schematically in Figure 1d. Figure 1: NIST procedure to compute frequency independant foundation soil spring and dashpot coecients. (a) The real SSI problem. (b) Simpli ed SSI model using lumped translational and rotaional spring and dashpots and the xed rst modal information. (c) Frequency dependant impedance function. (d) Distributed soil spring and dashpot alog the foundation perimeter. Furthermore, for modifying the FFM due to the foundation embedment, the NIST rec- ommends using zero-phase transfer functions HuandH, which reduce the translational and introduce the rotatational motion, respectively. Hu=8 >< >:cosD! Vs ;ifD! Vs<1:1 0:45;otherwise; (2) H=8 >< >:0:26 1cosD! Vs ;ifD! Vs< 2 0:26;otherwise; (3) 3 where the variable Drepresents the foundation embedment, Vsthe shear wave velocity, and !the frequency. This mapping to determine the FIMs are shown schematically in Figure 2 where the FFM signal in Figure 2a is rst transformed in Fourier space (dashed blue line), then the Fourier coecients are scaled by the translational and rotational transfer functions as shown in Figure 2b (in solid black line), and the results are transformed back to the real space (using the inverse Fourier transform) to obtain the FIM as shown in Figure 2c in red solid lines. Figure 2: NIST procedure to compute foundation input motion. (a) Free eld motion time series, here a Ricker pulse is considered. (b) Transfer function in black-solid line and FFM Fourier transform function of the Ricker pulse in dashed-blue line. (c) Translational and rotational foundation input motion. It is evident that the NIST recommendations make two important simpli cations for us- ing the substructure method in time domain analysis: (a) the frequency at which the impedance value should be read is the exible-base frequency, and (b) it assumes that the FIM preserves the phase of the FFM and it does not change as a function of foun- dation embedment ratio D=B . Upon these simpli cations, the following questions may arise: 1. How does the NIST recommendations and the proposed model perform in overall? 2. Are the embedment e ects for shallow foundation not important so that the phase angle can be neglected? 3. What is the best dimensionless frequency a0to read the soil impedance? 4. Is it possible to make a better estimation of the dimensionless frequency a0to increase the NIST accuracy? The main objective of this study is to address the above listed questions, and to the best of our knowledge, this is the rst comprehensive study in this regard. In the rest 4 of this manuscript, we provide details of the approach we used to assess the performance and optimality of the NIST recipe for modeling SSI e ects for building structures. This calls for (1) computing the ground truth (reference) solution, and searching the parameter space of the reduced order model to nd the optimal impedance function values or the frequency at which one needs to read them. For the former, we model the problem using the direct method and for the latter we use a Bayesian approach based on the ensemble Kalman ltering; details of each are provided in x2{x4. Then, inx5, we use the developed framework to comprehensively assess the predictive capability of the NIST recommendations for a series of building structures embedded in elastic half-space. We summarize the ndings of this study in x6. 2 Direct modeling of the problem Figure 3: Reduced in nite half-space model using absorbing dashpots elements. For direct modeling of the building structure response on elastic half-space { see Figure 1a { we use the procedure shown in Figure 3. We truncate the semi-in nite half-space along the red line shown in Figure 3a, and introduce Lysmer dashpots accompanied with prescribed nodal forces to arti cially approximate an interface transparent to both incoming and outgoing waves [8, 20]. The Lysmer dashpot coecient shown in Figure 3b for absorbing outgoing waves are computed as: 5 cs=sVsthli; (4) cp=sVpthli: (5) Similarly, the introduced nodal forces, shown in Figure 3c and d, for translation of the vertically propagating incoming shear waves within the domain of interest are Fb i= 2cs_ug(t); (6) Fs i=cp_us i(t); (7) Fr i=cs_ur i(t): (8) In Equation (4) and (5) the variable th= 1:0 [m] is the out-of-plane thickness of the truncated soil domain,  liis either  xior ziand represent the tributary i-th side length of the element which are di erent for corner and inner boundary nodes, Vsis the shear wave velocity of the soil, and Vpis the compressive wave velocity in the soil. It worth to point out that the boundary conditions enforced in this manner can almost perfectly absorbs body waves with angles of incidence greater than 30degrees with respect to the vertical axis, and will lose their e ectiveness for lower angles of incidence or for surface waves. Moreover, in Equation (6)-(8) the signals _ ug(t) and _us i(t) represent the incident and the total free eld motion for which a Ricker-wavelet is prescribed at the bottom of the model and given in Equation (9) as: _ug(t) = 12 (tt0)2 exp  (tt0)2 ; (9) where = (f0)2,f0is the characteristic frequency, t0is the time position where the velocity will become maximum. In all simulations, we consider a characteristic frequency f0= 2:0 [Hz], and a peak time velocity t0= 1:0 [s]. The node velocity _ us i(t) and _ur i(t) needed to compute the horizontal and vertical forces on the left and right boundary are _us i(t) = _ug zi Vs+t + _ugzi2H Vs+t ; (10) and, _ur i(t) = _ugzi2H Vs+t _ug zi Vs+t ; (11) whereziis the vertical coordinate of the i-th node measured with respect to the bottom of the soil domain, His the total vertical height of the truncated soil domain, and sign in Equation (8) represents the left (+) or right ( ) vertical boundary forces respectively. We verify our implementation by reproducing the free eld motion in a homogeneous half-space with length Lx= 350 [m] and height Lz= 200 [m]. We consider linear elastic soil-element withs= 2000 [kg=m3],s= 0:25,th= 1:0 [m], for which plane-strain conditions are 6 (a) (b) Figure 4: Synthetic seismogram for the soil-wave propagation problem (a) normalized horizontal component ^ vx(t), and (b) normalized vertical component ^ vy(t) of the velocity recorded on the soil surface. enforced. The sv-wave velocity is set to be Vs= 225 [m=s], leading to a wave-length of n= Vs=f0= 112:5 [m]. The results of the normalized velocity eld ^ v(^x;^y) =v(^x;^y)=maxjvj as a function of the normalized coordinates ^ x=x=n, ^y=y=nare displayed in Figure 4 for both components of the velocity eld. This gure makes evident that the implemented absorbing boundary interface works perfectly for this case. Here, one can observe that there is no re ection coming in from the boundaries, and as expected no velocities in the vertical direction are developed throughout the simulation. 3 Reduced order modeling of the problem For modeling the problem using the substructure method, we consider the reduced order model (ROM) shown in Figure 5c. The ROM is constructed in such a way that it preserves the modal information of the xed base building. To this end, we employ frame elements with three degree of freedom per node to represent the structure's geometry as shown in Figure 5a. We assume that each oor acts as a rigid diaphragm, so that the buildings mass can be lumped at the oor levels as shown in Figure 5b. We then use static condensation [21, 22] on the xed-base building to compute the Ms;Cs;Ks2Rnnmatrices where n is the number of oors; therefore, the vector x2Rnrepresents the horizontal degree of freedom of the building, since there is only one translational degree of freedom on each oor. In addition, we assume a rigid rectangular foundation of half-width Band depthD sitting on an elastic half-space, for which two additional degree of freedom u;2Rappear because of the exibility of the soil. The foundation has a total mass mf, and a moment of inertiaI0. Then, depending on the motion that one can use to excite the model, we consider two variants for forward modeling: (1) excited by FIM and (2) excited by FFM. 7 Figure 5: The reduced order modeling. (a) The column and beam element used to con- struct the The xed base FEM. (b) The xed base FEM building model where Ms;Cs;Ks are obtained using static condensation. (c) The exible-base soil-structure interaction system in which x= (x1;:::;xn) represents the relative displacement of each oor, u represents the relative deformation of the soil, and the rigid rotation of the foundation. The semi-discrete equations of motion for both cases are shown below. 2 4Ms Ms 1 Ms h 1>Msmf+ 1>Ms 1mfD 2+ h>Ms 1 h>MsmfD 2+ h>Ms 1I0+mfD2 4+ h>Ms h3 52 4x v 3 5+2 4Cs On1 On1 O1ncxxcx O1ncxc3 52 4_x _v _3 5+ 2 4Ks On1 On1 O1nkxxkx O1nkxk3 52 4x v 3 5=F(t): (12) where the external force vector for the free eld motion is F(t) =2 4Ms 1 mf+ 1>Ms 1 mfD 2+ h>Ms 13 5uFFM(t); (13) and for the foundation input motion is F(t) =2 4Ms 1 mf+ 1>Ms 1 mfD 2+ h>Ms 13 5uFIM(t)2 4Ms h mfD 2+ h>Ms h mfD2 4+ h>Ms h+I03 5FIM(t): (14) and x;_x;x2Rnare the acceleration, velocity and displacement vector for the condensed horizontal degrees of freedom of the building, also  v;_v;v2Rand ;_;2Rare the foundation horizontal and rotational acceleration, velocity and displacement. Note that 8 we de ne 12Rnthe vector of ones, this is 1= (1;1;:::; 1), and h2Rnthe vector of height, this is h= (h1+D;:::;h n+D). We nally de ne Onm2Rnmthe ma- trix of zeros. In addition, it is important to highlight that in the ROM the horizontal and vertical springs and dashpots elements are distributed uniformly over the foundation perimeter. However, the contribution can be lumped as follows for both sti ness and damping components: kxx= 2kx(D+B); kx=kxD2; k=2 3kxD3+2 3kzB3+ 2kzB2D; (15) cxx= 2cx(D+B); cx=cxD2; c=2 3cxD3+2 3czB3+ 2czB2D: (16) Distributing the spring and dashpot elements as represented in Figure 5 allows us to take the coupling e ects in the sti ness and damping matrices into account approxi- mately. It worth mentioning that the NIST recommendations assume the coupling terms are zero. 4 Ensemble Kalman inversion for parameter estima- tion In order to nd the optimal spring and dashpot coecients of the ROM, we use the Bayesian approach based on the ensemble Kalman inversion (EnKI) [23{25], which can be also considered as a particle-based derivative-free sequential optimization method. In the inversion setting, we consider the problem of nding u2Rnfromy2Rmwhere y=G(u) +: (17) The variable u2Rnconsists of all the unknown parameters that we want to estimate, the variable y2Rmconsists of the ground truth quantities of interest, and is a zero- mean Gaussian noise with covariance . The nonlinear function (a.k.a. forward model) G:Rn!Rmmaps the parameter space to the data space. In this paper, we work with one type of data-sets; the displacement time series recorded at the oors and foundation levels computed using the model described in x2. GivenNparticlesu(n) j,n= 1;:::;N within the ensemble and at each iteration j, we use the predictions G(u(n) j) by the forward model and the observation data yto update the particles for iteration j+ 1 (see Figure 6 for schematic illustration). That is, u(n) j+1=u(n) j+Cuw j+1(Cww j+1+)1(y(n) j+1G(u(n) j)) forn= 1;:::;N: (18) 9 In Equation (18), y(n) j+1can be either identical to y(the observation data) or randomly perturbed using zero-mean Gaussian noise ;Cuw j+1andCww j+1are empirical covariance matrices that can be computed at each iteration based on predictions and the ensemble mean uj+1using the following equations: Cuw j+1=1 NNX n=1(u(n) juj+1) (G(u(n) j)Gj) (19) Cww j+1=1 NNX n=1(G(u(n) j)Gj) (G(u(n) j)Gj) (20) and uj+1=1 NNX n=1u(n) j;Gj=1 NNX n=1G(un j): (21) It worth mentioning here the stopping criterion is de ned as either reaching a relative change of 0 :001 in all the parameter in two consecutive EnKI iterations or a maximum number of 500 iterations. Furthermore, the initial ensemble mean is de ned to be equal to the lumped sti ness and dashpot coecients obtained using the NIST [18] procedure. In addition, since the prior distribution of the soil parameter is unknown, we use a uniform distribution centered at the computed NIST value with lower and upper limit ten times smaller and larger to represent this uncertainty. Finally, the positiveness of the sti ness and dashpot coecients are enforced through a change of variables u2R, exp(u) :u! [0;1] on which the EnKI is applied. 5 ROM performance assessment 5.1 Model speci cations Three di erent topologies of buildings are considered, which are illustrated from the short- est in Figure 7a to the tallest in Figure 7c. The rst reinforced concrete building has columns of rectangular cross section 0 :900:60 [m], each of 3 :5 [m] height. Beams have rectangular cross sections 0 :800:60 [m] and 6:0 [m] length. The reinforced concrete den- sity is 2500 [ kg=m3], the reinforced concrete elasticity modulus is taken as 25 [ GPa ], and beams are subjected to a dead load of 3600 [ kg=m ]. This con guration gives a total height of 21 [m], a total mass of 6 :45105[kg], a xed-base fundamental period of 0 :507 [s], a rst-modal height of 14 :91 [m], and a rst modal mass 0 :545106[kg]. The solid core foun- dation has a half-width of 10 [ m], an equivalent half-length of 1 :0 [m], and three di erent 10 Figure 6: Parameter estimation framework based on EnKI applied to SSI (a) Full nite element model from which the "true" responses are computed (observations), and (b) Reduced order method from which the spring and dashpot coecients are estimated. embedment depths of 1 :0;2:5;5:0 [m] are considered respectively. The reinforced concrete material properties for the foundation are taken such that the density provides a foun- dation mass 50 :0103[kg], elasticity modulus 30 [ GPa ], and Poisson's ratio of 0 :25. The second and third reinforced concrete buildings only di er in terms of the column cross- sections, which for the taller buildings are 1 :000:80 [m]. Beams and material properties of building and foundation are same as above. The foundation is assumed to be a rigid diaphragm and to have perfect bonding with the surrounding soil. Table 1 summarizes some of the buildings' characteristics. Building Parameters Eigen-Analysis Period Mass High Modal Mass Modal High [s] [kg] [m] [kg] [ m] 0:50 6:4510521:00 5:4510515:0 1:00 1:4010642:00 1:1510630:0 1:50 1:9810659:50 1:4810640:0 Table 1: First xed-modal-parameters using an eigen-analysis for three buildings. In the direct method, the soil is represented as an elastic, homogeneous and semi-in nite medium with density s= 2000 [kg=m3], Poisson's ratio s= 0:20, and shear wave velocity Vs= 80;100;125;150;175;200;225;250;300;400;500 [m=s]. The soil domain is truncated as explained inx2. The time step that satis es out stability constraints is  t= 0:0025 [s], 11 Figure 7: Building con gurations employed in the dimensional analysis for the SSI prob- lem. and the simulation time is set to be tsim= 30 [s]. The spatial discretization is then taken as x= z= 0:5 [m], and the soil domain is truncated so that Lx= 350 [m], and Lz= 200 [m]. The resulting nite element mesh has approximately 280 ;000 quadrilateral elements; we employed parallel computing for the simulations using SeismoVLab [26,27], we divided the domain into seven sub-domains using the computer software OpenMPI1 [28{30]. The average time per simulation was 65 [ min] approximately. 5.2 Estimated soil impedances To estimate the constant valued soil impedances, the ROM is subjected to the FIM and the EnKI is employed to estimate the soil spring and dashpot coecients. Then, they are compared against the true impedance functions derived numerically using the method proposed by [15]. These results, named hereafter as EnKI-FIM, are displayed in Figure 8 and 9 in which the vertical axis has been normalized such that ^kxx=Gs(kxx+!cxx i), ^kx=GsB(kx+!cx i), and ^k=GsB2(k+!c i), where i=p1,Bis the half-width foundation, Gs=sV2 sis the soil shear modulus, sthe soil's density, and Vs the soil shear wave velocity. We also employ the EnKI to re-estimate the soil springs and dashpots when the system is subject to FFM; the results named EnKI-FFM hereafter. Consdiering the EnKI-FIM results as the optimal solution for the considered ROM, we assessed the accuracy of the EnKI-FFM results as well as those recommended by NIST. In Figure 10, the horizontal axis is the EnKI-FIM results while the vertical axes are the EnKI-FFM and NIST results. This comparison study delineated that 1. The embedment e ect is almost negligible as the EnKI-FIM and EnKI-FFM results are highly correlated with each other, which in turn con rms that the embedment 1The message passing interface (MPI) allows to exchange messages between multiple computers running a parallel program (single execution) across distributed memory. 12 Figure 8: Identi ed lumped impedance (real part) computed for soil properties s= 2000 [kg=m ], ands= 0:25 using EnKI (in discrete markers) and NIST (in solid line) for buildings with xed base period T= 0:5;1:0;1:5 [s] and foundation embedment D= 1:0;2:5;5:0 [m]. Figure 9: Identi ed lumped impedance (complex part) computed for soil properties s= 2000 [kg=m ], ands= 0:25 using EnKI (in discrete markers) and NIST (solid line) for buildings with xed base period T= 0:5;1:0;1:5 [s] and foundation embedment D= 1:0;2:5;5:0 [m]. ratios considered in this study are shallow. 2. The NIST results capture very well the total lumped horizontal kxxand rotational kspring coecients. However, the total lumped horizontal cxxand rotational c dashpot coecients are poorly estimated. In particular, the lumped rotational dash- pot coecients for the case of building with T= 1:5 [s] are the worst. This problem is due to the fact that the imaginary component of the lumped rotational impedance cbecomes small as the xed building period becomes larger, and then when such dimensionless frequency is used to compute the lumped rotational impedances gen- erates a deviation about one order of magnitude smaller when it is compared against EnKI-FIM results, see Figure 10 for more details. This suggests that the frequency 13 at which exible buildings irradiates energy due to rocking mechanism is di erent from the sway ( exible base) mechanism proposed by NIST. Figure 10: Identi ed impedance component comparison between NIST and EnKI using FIM and FFM. The horizontal axis correspond to the EnKI once the system is subject to FIM. The vertical axis represents the impedance components using NIST (in blue solid dots) and EnKI subjected to FFM (in red solid dots). The black solid/dashed lines represent a linear regression for either case. The previous analysis made clear that the exible-base period may not be the best fre- quency to use to estimate the imaginary component of the soil impedances. In particular, if the exible-base period is used to compute the lumped rotational dashpot coecient for exible buildings, the radiation due to damping is underestimated. Therefore, we would like to investigate what practical dimensionless frequency (based on the exible-base pe- riod ~T, xed-base period T, or the input signal dominant frequency f0) one should use to read the impedance functions. In this section, as before we perform similar analyses, but now we consider only the NIST recommendation using these three frequencies. As shown in Figure 11, we compare on the vertical axis the soil spring and dashpot coecients ob- tained with NIST for these three cases, against EnKI-FIM results on the horizontal axis. Once again, we notice that the horizontal and rotational spring coecients are estimated properly, no matter which frequency is used. However, a smaller variability is obtained if the xed-base period is employed. On the other hand, a similar behavior, with a large variability, for the lumped horizontal and vertical dashpot coecients is obtained. In gen- eral, the input signal dominant frequency f0as well as the xed-base period Tperforms better in estimating the lumped vertical dashpot coecient. The main di erences in time history responses comes from the fact (once again) that the lumped rotational dashpot 14 Figure 11: Impedance comparison between NIST using the exible-base period ~T, xed- base period T, or the input signal dominant frequency f0and EnKI using foundation input motion. coecient is poorly estimated. In order to show this point, we quantify the total error made in a dynamic analysis between the response of the full FEM (as described in x2) and the ROM with spring and dashpot coecients obtained using NIST for the three di erent representative frequencies and using EnKI-FIM. We measure the `2-error for each case as: ky^yk2 `2=1 NtNmNmX k=1NtX j=1 y(k) j^y(k) j2 ; (22) wherey(k) jrepresents the k-th observation (from the FEM) at j-th time step, ^ y(k) jrepresents thek-th response (from the ROM) at j-th time step, Ntcorresponds to the number of time steps, and Nmrepresents the number of observations considered. We consider a time series of 15 [ s] with time step  t= 0:01 [s] for the translation and rotation of the foundation, and the horizontal translation of each oor. Figure 12 represents the total`2-error in the vertical axis, and the model number in the horizontal axis. This gure shows that in general the NIST model using the xed-base frequency and the signal predominant frequency have smaller errors when they are compared with the exible-base case. The latter is smaller only for the building with period T= 1:0 [s] and embedment D= 5:0 [m]. However, none of the con gurations reached the error obtained using EnKI- FIM coecients. The discrepancies are attributed mostly to the poor estimation of the rotational dashpot coecients, which generates more oscillations in the transient part, and they take longer to be damped out in the free oscillation part. Since the estimation 15 of this coecient is slightly better for the signal dominant frequency and the xed-base frequency when is compared to the EnKI, we obtain errors that are smaller. Figure 12: Total time history error measured in `2-norm for the three practical dimen- sionless frequencies mentioned in NIST. In blue dots the exible-base period, in red-dots the xed-base period, and in yellow-dots the signal-predominant frequency. 5.3 Dynamic Global Properties Provided with the soil impedances using NIST and EnKI, we are interested in computing the period elongation ratio ~T=T and foundation-damping fthat the ROM experiences once is supported on lumped spring and dashpot elements. Since the proposed ROM inx3 considers the coupling between the translation and rotational degree of freedom approximately, we employ Equations (28) and (39) derived in Appendix A to estimate the period lengthening and foundation damping. We then compare these results against the iterative method summarized in x1. Figures (13) and (14) show the period elongation and foundation damping computed using the NIST and EnKI-FIM results. As shown, a good-agreement between the proposed expressions in Appendix A and NIST expressions using the iterative method is achieved. From these gures we note that Equation (28) provides with more exible system, while Equation (39) provides with less-dissipative system when they are compared with the NIST recommendations. A very close t is however obtained for the three-buildings when the foundation aspect ratio is small (i.e.,D B= 0:10). The discrepancies must be attributed to the following facts: 16 Figure 13: Identi ed period elongation ratio ^T=T computed for soil properties s= 2000 [kg=m ], ands= 0:25 using EnKI (solid dots) and NIST (solid line) for building of xed-base period T= 0:5;1:0;1:5 [s], rst modal height h= 15;30;45 [m], founda- tion embedment D= 1:0;2:5;5:0 [m], and supported on eleven soils with shear-velocity Vs= 80;100;125;150;175;200;225;250;300;400;500 [m=s]. Figure 14: Identi ed foundation damping fcomputed for soil properties s= 2000 [kg=m ], ands= 0:25 using EnKI (solid dots) and NIST (solid line) for building of xed-base period T= 0:5;1:0;1:5 [s], rst modal height h= 15;30;45 [m], founda- tion embedment D= 1:0;2:5;5:0 [m], and supported on eleven soils with shear-velocity Vs= 80;100;125;150;175;200;225;250;300;400;500 [m=s]. 1. The estimated frequency of the interaction using the EnKI is di erent from the one obtained using the NIST recommendations. 2. The iterative method uses the rst modal information to compute the lumped spring and dashpot coecients while the EnKI-FIM estimates implicitly compensates for higher mode contribution since the soil parameters are estimated from the observa- tions of the full FEM. 3. As discussed in x3, we are explicitly considering the spring and dashpot coupling 17 terms in the sti ness and damping matrices. It can be seen in both Equations (28) and (39) that mentioned coupling in the sti ness and damping matrices generate a slight increase in the period elongation ratio as well as a slight decrease in the radiation damping. Regardless of the discrepancies presented, the results of the global behavior are very simi- lar as shown in Figures 13 and 14. It is not a surprise that the period elongation ratios are well-captured since NIST estimates well the lumped translational and rotational springs. On the other hand, it is interesting to see that the foundation damping for an equiva- lent one degree-of-freedom system is also well captured, this is due to the fact that the contribution of the lumped rotational dashpot is not important to the global foundation damping. These di erences become more pronounced when the sti er buildings rest on a deeper foundation, in which case the coupling term may become more important for the foundation damping. 5.4 Can NIST do better? To nd the optimal frequency to read the impedance functions as in NIST procedure, we use the EnKI algorithm to estimate the optimal dimensionless frequency a0, so that the mis t between the FEM model and the NIST ROM is minimized. In this analysis we consider two con gurations: (a) only one frequency a 0is considered for the lumped soil impedances, and (b) two frequencies ax 0(horizontal) and a 0(rotational) for the lumped soil impedances are estimated. The results of this analysis are shown in Figure 15. Figure 15 (a) (b) Figure 15: Optimal dimensionless frequency a 0obtained using EnKI for (a) One dimen- sionless frequency, and (b) Two dimensionless frequency. reveals two interesting results. The rst con rms the hypothesis that the dimensionless frequency that controls the response of the SSI is related to the rotational soil impedance. This can be seen since the dimensionless frequency a 0in Figure 15a and a 0in Figure 15b 18 are very similar. The second result suggests that the optimal a 0is not only a function of the building itself ( T), but also a function of the embedment ratio and the signal frequency content. In this regard, the dimensionless frequency a0B Vs~Tmay not be good enough to estimate the frequency independent impedance, a form like a0h VsT;h B;D B;Bf0 Vs;smay be more suitable to carry out this task; however, there is no clear pattern on how this dependency could be taken into account. Therefore, the iterative method to obtain the exible-base period may generate large errors specially when the soil is sti er, which is con- sistent with results presented in Figure 12. This does not mean it is a bad approximation for the engineering practice, it suggests that the frequency at which each lumped spring and dashpot may respond are completely di erent from those proposed by NIST. 6 Conclusions The state-of-practice introduces two major simpli cations to transform the frequency do- main analysis into a time domain analysis. It assumes that (1) the frequency at which the impedance value should be read is the exible-base frequency and (2) the foundation input motion is preserves the phase of the free eld motion. Both simpli cations are however adopted in order to perform time history analysis using standard nite element packages. Unfortunately, the recommendations proposed in NIST only work in linear elastic soils, and they are an over simpli cation of the substructure method (described in [3{5]), since do not consider the frequency dependancy of the excavated soil to perform a rigorous time domain analysis. In this study, we found that these recommendations could capture the global dynamic behavior of building structures relatively well. However, their performance deteriorates in capturing the response of the structure such as the oor displacement time-series, and this is mainly due to the poor approximation of the lumped rotational dashpot coecients. Consequently, the foundation of buildings with large period tends to be less dissipative since the radiation due to rocking is not properly estimated. Our results in Figure 12 show that the xed-base period generates better results, however more test should be conducted to con rm the claim. Furthermore, the reader should be aware that the conditions in which NIST recommendations were tested are the best case scenario, in which not only the impedance functions are correctly estimated, but also the foundation input motion was prescribed so that it has the right phase; thus, the discrepancies are only attributed to the impedance frequency estimation. On the other hand, we showed that the foundation input motion e ects for negligible for cases considered here with embedment ratios smaller than 0 :5. Therefore, the free eld motion can be directly applied without generating a substantial error. Finally, we notice that the selection of the dimensionless frequency a0to read the impedance values depends on other dimensionless parameters such as the frequency of the input signal and the embedment ratio. Future work should aim to evaluate the inclusion of these parameters 19 in determination of the system representative frequency. Acknowledgments The authors would like to acknowledge CONICYT - Becas Chile for the nancial support to the rst author for conducting doctoral studies at The California Institute of Technology.