This work studies the feasibility of imaging a coupled /uniFB02uid-solid system by using the elastodynamic and acoustic waves initiated from the top surface of a computational domain. We consider a one- dimensional system, where a /uniFB02uid layer is surrounded by two solid layers. The bottom solid layer is truncated by using a wave-absorbing boundary condition (WABC). We measure the wave responses on a sensor located on the top surface, and the measured signal contains information about the underlying physical system. By using the measured wave responses, we identify the elastic moduli of the solid layers and the depths of the interfaces between the solid and /uniFB02uid layers. We employ a multi-level Genetic Algorithm (GA) combined with a frequency-continuation scheme to invert for the values of sought-for parameters. The numerical results show the following /uniFB01ndings. First, the depths of solid-/uniFB02uid interfaces and elastic moduli can be reconstructed by the presented method. Second, the frequency-continuation scheme improves the convergence of the estimated values of parameters toward their targeted values. Lastly, a preliminary inversion, using an all-solid model, can be employed to identify if a /uniFB02uid layer is presented in the model by showing one layer with a very large value of Young’s modulus (with a similar value to that of the bulk modulus of water) and the value of mass density being similar to that of water. Then, the primary GA inversion method, based on a /uniFB02uid-solid model, can be utilized to adjust the soil characteristics and /uniFB01ne-tune the locations of the /uniFB02uid layer. If this work is extended to a /three.pnumD setting, it can be instrumental to /uniFB01nding unknown locations of /uniFB02uid-/uniFB01lled voids in geological formations that can lead to ground instability and/or collapse (e.g., natural/anthropogenic sinkhole, urban cave-in subsidence, etc.). /one.pnum Introduction Detecting underground cavities (e.g., karstic cavities, caves, tunnels, etc.) is a challenging task for geotech- nical engineering projects due to the geological/hydrogeological complexity of the subsurface environ- ment. Geological hazards, such as collapsing soil or urban ground collapse due to subsurface voids, could induce signi/uniFB01cant damage to infrastructures. Such hazard is one of the major issues for land planning, in- frastructure operation and maintenance, and disaster management. Cavity evolution that occurs due to hydrogeological process may cause sinkhole collapse. For example, Florida’s karst environment involves active groundwater recharge to the Floridan Aquifer that makes overburden soils to be eroded; thus, as shown in Fig. /one.pnum, a cavity could gradually grow, leading to a sudden collapse (cover-collapse type) or grad- ual subsidence (cover-subsidence type) (Beck, /one.pnum/nine.pnum/eight.pnum/six.pnum; Tihansky, /one.pnum/nine.pnum/nine.pnum/nine.pnum; Xiao et al., /two.pnum/zero.pnum/one.pnum/six.pnum; Kim et al., /two.pnum/zero.pnum/two.pnum/zero.pnum; Nam et al., /two.pnum/zero.pnum/two.pnum/zero.pnum). These large underground water-/uniFB01lled cavities hidden below building and transporta- tion infrastructures should be pre-detected so that prevention and mitigation measures are applied before catastrophic collapse or excessive ground settlement takes place. Cone penetration test (CPT) and standard penetration test (SPT) have been employed for detecting and 1 CavityCavity initiates on top of bedrock and grows upward in the cover soil layer 100 ft CavityAt a shallow soluble bedrock area, cavities often exist in upper bedrock (due to dissolution)Figure /one.pnum: Schematic diagram of water-/uniFB01lled subsurface cavity (common in Florida’s karst environment): (a) cavity in either cover soils layer and (b) cavity within near surface of limestone bedrock. characterizing subsurface cavities in a cover soil layer, referred to as a raveled zone in the karst areas (Bloomberg et al., /one.pnum/nine.pnum/eight.pnum/eight.pnum; Shamet et al., /two.pnum/zero.pnum/one.pnum/seven.pnum; Nam et al., /two.pnum/zero.pnum/one.pnum/eight.pnum; Nam and Shamet, /two.pnum/zero.pnum/two.pnum/zero.pnum). Although SPT and CPT provide a continuous subsurface pro/uniFB01le—revealing soil type, resistance, and stratigraphy—, these inva- sive methods only collect point-wise pro/uniFB01les of the subsurface environment. As such, it is time-consuming and labor-intensive to apply CPT and SPT to wide areas multiple times. Numerous geophysical studies over the past decades have introduced nondestructive evaluation (NDE) methods for detecting and estimating the geometry of subsurface cavities. For example, ground-penetrating radar (GPR), micro-gravimetry, and resistivity imaging and their combinations can characterize an under- ground cavity based on its size and depth position (Cardarelli et al., /two.pnum/zero.pnum/zero.pnum/three.pnum; Mochales et al., /two.pnum/zero.pnum/zero.pnum/eight.pnum; Fehdi et al., /two.pnum/zero.pnum/one.pnum/four.pnum; Ki/uniFB02u et al., /two.pnum/zero.pnum/one.pnum/six.pnum). Namely, GPR is ine/uniFB03cient in highly heterogeneous media—such as back/uniFB01lls, moist clays with high soil conductivity, and saturated soils below the groundwater table. Micro-gravimetric method is e/uniFB00ective in detecting shallow cavities (Butler, /one.pnum/nine.pnum/eight.pnum/four.pnum; Bishop et al., /one.pnum/nine.pnum/nine.pnum/seven.pnum), but its density contrast cannot be obtained clearly if the size of a void is relatively small compared to its depth. An electrical resis- tivity method has been applied to detect air-/uniFB01lled or water-/uniFB01lled cavities (Van Schoor, /two.pnum/zero.pnum/zero.pnum/two.pnum; Coşkun, /two.pnum/zero.pnum/one.pnum/two.pnum), but it requires a large areal space for surveying and cannot detect small-scale irregularities in the geologic interfaces since it measures averaged resistivity values and is easily ruined by noise sources, such as piping, power lines, and house structures. On the other hand, elastodynamic imaging has been widely used in site characterization (Roesset et al., /one.pnum/nine.pnum/nine.pnum/zero.pnum; Brown et al., /two.pnum/zero.pnum/zero.pnum/two.pnum; Kallivokas et al., /two.pnum/zero.pnum/one.pnum/three.pnum). It hinges on elastodynamic waves that are initiated by wave sources on the ground surface and/or borehole sources, and, then, are re/uniFB02ected/refracted due to material heterogeneity within a medium. The resulting wave motions can be measured on the ground sur- face or at borehole seismic arrays. The inverse modeling, associated with elastic wave propagation anal- yses, provides promising results of identifying the material properties of the media. Inverse modeling— employing a partial di/uniFB00erential equation (PDE)-constrained optimization method and the /uniFB01nite or spectral element method (FEM or SEM)—has been utilized to infer the spatial distribution of material properties of host media in /uniFB01ne resolution (Kang and Kallivokas, /two.pnum/zero.pnum/one.pnum/one.pnum; Fathi, /two.pnum/zero.pnum/one.pnum/five.pnum; Fathi et al., /two.pnum/zero.pnum/one.pnum/five.pnuma). Speci/uniFB01cally, for instances, the geotechnical site characterization methodologies have been investigated in a soil do- main that is truncated by Perfectly-Matched-Layers (PML), where waves decay and are not re/uniFB02ected o/uniFB00 surrounding boundaries (Kang and Kallivokas, /two.pnum/zero.pnum/one.pnum/zero.pnumb; Kucukcoban and Kallivokas, /two.pnum/zero.pnum/one.pnum/one.pnum, /two.pnum/zero.pnum/one.pnum/three.pnum; Fathi et al., /two.pnum/zero.pnum/one.pnum/five.pnumb; Mashayekh et al., /two.pnum/zero.pnum/one.pnum/eight.pnum; Poul and Zerva, /two.pnum/zero.pnum/one.pnum/eight.pnum), by using state-adjoint-control equation-based full- waveform inversion (FWI) approaches (Kallivokas et al., /two.pnum/zero.pnum/one.pnum/three.pnum; Kang and Kallivokas, /two.pnum/zero.pnum/one.pnum/zero.pnuma, /two.pnum/zero.pnum/one.pnum/one.pnum; Fathi et al., /two.pnum/zero.pnum/one.pnum/five.pnuma, /two.pnum/zero.pnum/one.pnum/six.pnum; Pakravan et al., /two.pnum/zero.pnum/one.pnum/six.pnum; Kucukcoban et al., /two.pnum/zero.pnum/one.pnum/nine.pnum). The Gauss-Newton-based FWI method, which is based on a gradient vector and a Hessian matrix, had been studied for characterizing the material pro/uniFB01les in a truncated two-dimensional solid domain (Tran and McVay, /two.pnum/zero.pnum/one.pnum/two.pnum; Pakravan et al., /two.pnum/zero.pnum/one.pnum/six.pnum) for the appli- /two.pnum cation of the site characterization. The PDE-constrained optimization has been also used in consideration of the boundary element method (BEM), of which computational e/uniFB03ciency is much greater than the FEM due to the reduction of the dimensionality, to detect the geometries of wave-scattering objects in host media. Namely, there have been studies on inverse scattering algorithms using BEM wave solvers, hinging on the moving boundary concept and the total derivative (Petryk and Mroz, /one.pnum/nine.pnum/eight.pnum/six.pnum) that allows for comput- ing the derivative of an objective functional with respect to the geometry variables of scattering objects (Guzina et al., /two.pnum/zero.pnum/zero.pnum/three.pnum; Jeong and Kallivokas, /two.pnum/zero.pnum/zero.pnum/eight.pnum; Jeong et al., /two.pnum/zero.pnum/zero.pnum/nine.pnum). The extended /uniFB01nite element method (XFEM) has been used as a wave solver of the studies to identify cracks or air-/uniFB01lled voids in solid media because it allows for avoiding expensive remeshing process in inverse iterations while using the /uniFB02exibility of the FEM for heterogeneous materials (Jung et al., /two.pnum/zero.pnum/one.pnum/three.pnum; Jung and Taciroglu, /two.pnum/zero.pnum/one.pnum/four.pnum, /two.pnum/zero.pnum/one.pnum/six.pnum). In addition, a frequency-continuation scheme was devised to help the inversion solver tackle the multiplicity of solutions of subsurface imaging problems (Bunks et al., /one.pnum/nine.pnum/nine.pnum/five.pnum; Na and Kallivokas, /two.pnum/zero.pnum/zero.pnum/eight.pnum; Kang and Kallivokas, /two.pnum/zero.pnum/one.pnum/zero.pnuma; Fathi et al., /two.pnum/zero.pnum/one.pnum/five.pnuma). That is, when the convergence rate of the inversion solver is decreased due to a lo- cal minimum of an objective functional that is comprised of measurement data corresponding to a given dominant frequency of excitation, another set of measurement data that are induced by excitation of a di/uniFB00erent dominant frequency are employed. Such a frequency shift can change the curvature of the ob- jective functional so that a minimizer can escape a local minimum. In general, one may consider increasing frequency from one inversion set to another because it has been reported that using a lower frequency leads to the inversion result of a lower resolution (due to a larger wavelength), then, one can /uniFB01ne-tune it using a higher frequency (due to a smaller wavelength) (Na and Kallivokas, /two.pnum/zero.pnum/zero.pnum/eight.pnum). Despite such recent developments in elastodynamic wave imaging techniques, few papers have demon- strated the feasibility of identifying the properties of a solid system that includes /uniFB02uid-/uniFB01lled voids. There have been a /uniFB01nite di/uniFB00erence method (FDM)-based FWI approaches, which employed all solid elements in an entire computational domain and detect the areas with smaller values of shear wave speeds Vs(e.g., around /one.pnum/zero.pnum/zero.pnum m/s) than those of typical soils and rocks to indicate air-/uniFB01lled voids (Tran and McVay, /two.pnum/zero.pnum/one.pnum/two.pnum; Tran et al., /two.pnum/zero.pnum/one.pnum/three.pnum; Mirzanejad et al., /two.pnum/zero.pnum/two.pnum/zero.pnum). However, those works have not explicitly modeled the inter- faces between air voids and the solid domain. Thus, the authors (henceforth, we) are concerned that, /uniFB01rst, modeling a /uniFB02uid domain as solid elements with a small value of Vscould introduce numerical error in for- ward wave solutions. Second, using a small value of Vsrequires a small size of a solid element for wave simulations. However, the aforementioned FDM works did not use such a small-sized element to model a void so that it could su/uniFB00er from the additional error of wave solutions. To address this issue, we suggest to explicitly use /uniFB02uid elements to model voids /uniFB01lled with water (or air) so that accurate /uniFB02uid-solid coupling (Lloyd et al., /two.pnum/zero.pnum/one.pnum/six.pnum) should be incorporated into the wave solver. Then, by using such a wave solver, inverse modeling could accurately identify the interfaces between /uniFB02uid and solids as well as the material pro/uniFB01les of solids. As a prototype work of our suggested method, this paper investigates the feasibility to estimate the interfaces between /uniFB02uid and solid layers and the material properties of solid layers in a multi-layered system by using acoustic and elastodynamic waves generated from the ground surface. Such investigation will prove the feasibility of detecting a water-/uniFB01lled void and identifying its location and geometry in soils, to which overlaying ground surface could potentially sink. This work uses the FEM to study the wave responses of the coupled /uniFB02uid-solid media in a prototype one- dimensional setting. The solid formation of a semi-in/uniFB01nite extent is truncated by using an absorbing bound- ary, and a proper /uniFB02uid-solid interface condition is considered. The inverse problem is cast into a minimiza- tion problem, where the value of an error between the synthetic measured wave response due to target pro/uniFB01les and the computed counterpart due to guessed pro/uniFB01les is minimized. The solution multiplicity of the minimization process is addressed by using the multi-level Genetic Algorithm (GA) in this work. Under this new scheme, we suggest the following steps: ( i) The optimizer identi/uniFB01es an inversion solution around a local minimum after a number of GA iterations in the /uniFB01rst-level GA; ( ii) Then, we can introduce a mis/uniFB01t function for a di/uniFB00erent measurement signal induced by a pulse signal of a di/uniFB00erent central frequency in the next-level GA; ( iii) By doing so, the GA optimizer is able to escape the local minimum of the previous- level GA; (iv) The same technique is employed to address the solution multiplicity of the next-level GA. The numerical results show that our numerical simulations, using the multiple-frequency-level GA, accurately detect the interfaces between /uniFB02uid and solid layers and the material properties of solid layers. We also demonstrate that a preliminary GA inversion can tell whether the subsurface media include a /uniFB02uid layer or not. Namely, the preliminary inversion can indicate the existence of a /uniFB02uid layer by detecting a very large value of Young’s modulus (with a similar value to that of the bulk modulus of water) and a mass density with a similar value to that of water. Once the existence of a /uniFB02uid layer is detected, a primary GA inversion, using the /uniFB02uid-solid model, can further estimate the locations of a /uniFB02uid layer and the material properties /three.pnum of surrounding solids. /two.pnum Problem De/uniFB01nition We aim to identify the depths of the interfaces between a /uniFB02uid layer and its surrounding solid layers and the sti/uniFB00ness of the solid layers by using a dynamic test, which generates elastodynamic waves into the subsurface media from the top surface. Absorbing boundary conditionSensor measuring vibrationsFluid layerGround wave sourceControl parameters of the presented inverse modeling Figure /two.pnum: Problem schematic. We consider a /one.pnumD /uniFB02uid-solid model, which consists of three di/uniFB00erent layers, i.e., two solid layers and one /uniFB02uid layer (see Fig. /two.pnum). The governing di/uniFB00erential equations for the compressional waves in the three layers are given by: @ @x Es1@us1 @x =s1@2us1 @t2; 0xLs1; (/one.pnum) @2P @x2=1 c2 f@2P @t2; Ls1xLf; (/two.pnum) @ @x Es2@us2 @x =s2@2us2 @t2; LfxLs2; (/three.pnum) whereLs1,Lf, andLs2denote, respectively, the depth of the interface between the top solid layer and the /uniFB02uid layer, that between the /uniFB02uid and the bottom solid layer, and the depth of the truncation wave absorbing boundary. Also, us1(x;t)andus2(x;t)denote the displacement /uniFB01elds of the wave motions of solid particles in the top and bottom solid layers. Pdenotes the /uniFB02uid pressure /uniFB01eld of the wave motions in the /uniFB02uid. The wave speeds of the three layers are given by cs1=q Es1 s1,cf=qf f, andcs2=q Es2 s2, respectively, where E,, andare the modulus of elasticity, the density, and the bulk modulus. The problem of interest also includes the following boundary conditions: Es1@us1 @x=f(t); x = 0; (/four.pnum) @us2 @x=1 cs2@us2 @t; x =Ls2; (/five.pnum) A pulse signal f(t)is applied at the top of the solid layer at x= 0 (per (/four.pnum)). The depth of the bottom solid layer is truncated by using the absorbing boundary condition (per (/five.pnum)). The solid and /uniFB02uid layers are /four.pnum coupled via the following interface conditions: Es1@us1 @x=P; x =Ls1; (/six.pnum) @P @x=f@2us1 @t2; x =Ls1; (/seven.pnum) Es2@us2 @x=P; x =Lf; (/eight.pnum) @P @x=f@2us2 @t2; x =Lf: (/nine.pnum) The system is initially at rest: us1=us2=P= 0; t = 0; (/one.pnum/zero.pnum) @us1 @t=@us2 @t=@P @t= 0; t = 0: (/one.pnum/one.pnum) /three.pnum Finite element modeling to compute the wave responses. To derive the weak form, the governing wave equations in the strong form (/one.pnum)-(/three.pnum) are multiplied by the test functions v1(x),w(x), andv2(x). By integrating them by parts and using the boundary and interface conditions of the strong form, the following weak forms are obtained: ZLs1 0 Es1@v1 @x@us1 @x+s1v@2us1 @t2 dx=v(0)f(t)v(Ls1)P(Ls1); (/one.pnum/two.pnum) ZLf Ls1 f@w @x@P @x+fw@2P @t2 dx=ff w(Ls1)@2us1 @t2(Ls1)w(Lf)@2us2 @t2(Lf) ;(/one.pnum/three.pnum) ZLs2 Lf Es2@v2 @x@us2 @x+s2v2@2us2 @t2 dx=v2(Lf)P(Lf)Es2rs2 Es2v2(Ls2)@us2 @t(Ls2): (/one.pnum/four.pnum) Then, we approximate the functions as: v1(x) =vT 1 (x); us1(x;t) = (x)Tus1(t); (/one.pnum/five.pnum) w(x) =wT (x); P (x;t) = (x)TP(t); (/one.pnum/six.pnum) v2(x) =vT 2 (x); us2(x;t) = (x)Tus2(t); (/one.pnum/seven.pnum) where (x), (x), and (x)denote vectors of global basis functions constructed by local shape functions, andus1,P, andus2are the vectors of nodal solutions. Introducing the /uniFB01nite element approximations (/one.pnum/five.pnum)- (/one.pnum/seven.pnum) into the weak forms (/one.pnum/two.pnum)-(/one.pnum/four.pnum) provides the following discrete form: Ks1us1+Ms1@2us1 @t2=fL1P(Ls1); (/one.pnum/eight.pnum) KfP+Mf@2P @t2=L2ff@2us1 @t2L3ff@2us2 @t2; (/one.pnum/nine.pnum) Ks2us2+Ms2@2us2 @t2=p s2Es2L5@us2 @t+L4P; (/two.pnum/zero.pnum) /five.pnum where the matrices are de/uniFB01ned as: Ks1=ZLs1 0Es1@ (x) @x@ (x)T @xdx; Ms1=ZLs1 0s1 (x) (x)T; (/two.pnum/one.pnum) Kf=ZLf Ls1f@ (x) @x@ (x)T @xdx; Mf=ZLf Ls1f (x) (x)T; (/two.pnum/two.pnum) Ks2=ZLs2 LfEs2@ (x) @x@ (x)T @xdx; Ms2=ZLs2 Lfs2 (x) (x)Tdx; (/two.pnum/three.pnum) L1= (Ls1) T(Ls1) =2 640 0 ......... 1 03 75;L2= (Ls1) T(Ls1) =2 640 1 ......... 0 03 75; (/two.pnum/four.pnum) L3= (Lf) T(Lf) =2 640 0 ......... 1 03 75;L4= (Lf) T(Lf) =2 640 1 ......... 0 03 75; (/two.pnum/five.pnum) L5= (Ls2) T(Ls2) =2 640 0 ......... 0 13 75; (/two.pnum/six.pnum) and the force vector is f= (0)f(t). The discrete equations (/one.pnum/eight.pnum)-(/two.pnum/zero.pnum) lead to the following coupled discrete form: 2 4Ks1L1 0 0 Kf0 0L4Ks23 5 |{z} K2 4us1 P us23 5+2 640 0 0 0 0 0 0 0Es2q s2 Es2L53 75 |{z} C2 4@us1 @t@P @t@us2 @t3 5 +2 4Ms1 0 0 ffL2MfffL3 0 0 M s23 5 | {z } M2 64@2us1 @t2 @2P @t2 @2us2 @t23 75=2 4f 0 03 5; (/two.pnum/seven.pnum) which, at a discrete time tn, can be written as: M qn+C_ qn+Kqn=Fn; (/two.pnum/eight.pnum) where qn= [uT s1;PT;uT s2]Tis the vector containing the nodal solutions of solid displacements and acous- tic pressures at the n-th time step, and _ qnandqndenote the /uniFB01rst and second-order derivatives of qnwith respect to time. We solve the second-order ordinary di/uniFB00erential equation (/two.pnum/eight.pnum) by using the Newmark’s unconditionally-stable time integration scheme (Newmark, /one.pnum/nine.pnum/five.pnum/nine.pnum). /four.pnum Inverse Modeling We aim to identify the four control parameters ( Es1,Es2,Ls1, andLf) by using measured wave responses initiated by a known excitation signal f(t). Here, the values of the other variables ( Ls2,s1,s2,f, and f) are set to be known during the inversion process, and we also know the excitation signal f(t). We formulate this problem into a minimization problem to identify estimated control parameters that could lead to the minimum of an objective functional: L=ZT 0(um(0;t)u(0;t))2dt: (/two.pnum/nine.pnum) In (/two.pnum/nine.pnum),um(0;t)denotes a wave signal recorded by the sensor at x= 0 due to a target pro/uniFB01le of the control parameters, and u(0;t)denote computed counterpart at the same measurement location due to a guessed pro/uniFB01le. In this computational study, um(0;t)is synthetically obtained by running the presented /six.pnum FEM solver using a target pro/uniFB01le as an input. To prevent an inverse crime from taking place, we use smaller values of element sizes when um(0;t)is computed than when u(0;t)is computed. This work uses GA, which is a heuristic algorithm to reconstruct the control parameters that are the /uniFB01ttest to the objective of the problem. In each generation of GA, there are individuals, each of which has a set of di/uniFB00erent values of control parameters. In this work, GA runs the FEM wave solver, for the estimated values of control parameters of each individual, to compute u(0;t)and evaluate the mis/uniFB01t (/two.pnum/nine.pnum). Then, GA evalu- ates the /uniFB01tness of each individual by using the value of its mis/uniFB01t. Then, GA weeds out less-/uniFB01tting individuals when it creates the next generation of individuals, each of which is characterized by its own set of estimated parameters. In this process, by virtue of the mutation process, where GA learns the better-/uniFB01tting charac- teristics of control parameters, GA gives rise to new individuals in the next generation. The GA calculates the sensitivity of the /uniFB01tness function with respect to the changes of the values of the control parameters and determines how a next generation of individuals is created. Namely, GA repeats the tasks of weeding out, selecting the best individual, and mutating at each generation. Therefore, our inverse modeling solver could identify targeted values of the parameters by using the iterative process of GA, where the values of the estimated parameters of the best-/uniFB01t individual evolve over the progress of generations. In the overall GA process under this problem, we impose constraints on the allowable ranges of each control parameter. For instance, we set Ls1(the depth of upper face of a /uniFB02uid layer) is always smaller than Lf(the depth of the lower face) such that the thickness of the /uniFB02uid layer is always positive. We also set the values of elastic moduli Es1andEs2are always positive. It is known that, in the minimization problem of a small number (e.g., less than /two.pnum/zero.pnum) of control parame- ters, GA is likely to /uniFB01nd a set of control parameters that are close to the global minimum (Jeong et al., /two.pnum/zero.pnum/one.pnum/seven.pnum). Since there are only four control parameters, we originally hypothesized that the GA is a suitable solution approach to /uniFB01nding the global minimum solution of this inverse problem. However, the numeri- cal experiments show that the GA su/uniFB00ers from the solution multiplicity, and, thus, we test a new multiple frequency-level GA approach. Namely, after the /uniFB01rst level of GA is /uniFB01nished, our inversion solver uses the reconstructed values from the /uniFB01rst level to de/uniFB01ne the upper and lower limits of the control parameters in the next GA level, which uses a pulse signal of a di/uniFB00erent central frequency from its predecessor, creating a new synthetic um. In the /uniFB01nal-level GA, the /uniFB01nal best-/uniFB01t estimated control parameters are obtained as the inversion solution. /five.pnum Numerical Experiments This section shows a set of numerical experiments, studying the performance of the presented multi- level GA-based parameter-estimation method. In the /uniFB01rst two examples, three targeted /uniFB02uid-solid models, which di/uniFB00er from each other in terms of Ls1,Lf, andLs2, are considered as follows. Model /one.pnum has a total lengthLs2of /six.pnum/zero.pnum m and its /uniFB02uid is located between xofLs1= /two.pnum/zero.pnum m andxofLf= /two.pnum/five.pnum m. The corresponding geometry information of Model /two.pnum are Ls2of /six.pnum/zero.pnum m,Ls1of /four.pnum/zero.pnum m, and Lfof /four.pnum/five.pnum m, and those of Model /three.pnum are Ls2of /one.pnum/two.pnum/zero.pnum m,Ls1of /five.pnum/zero.pnum m, and Lfof /eight.pnum/zero.pnum m.Ls1andLfare unknown parameters in the inversion. In all the models, the mass density ( ) of the solid layers is /two.pnum/zero.pnum/zero.pnum/zero.pnum kg/m3; their Young’s moduli are Es1= 2/one.pnum/zero.pnum8 Pa andEs2= 5/one.pnum/zero.pnum8Pa; and the bulk modulus ( kf) and mass density ( f) of the /uniFB02uid layer are /two.pnum./three.pnum/four.pnum /one.pnum/zero.pnum9 N/m2and /one.pnum/zero.pnum/two.pnum/one.pnum kg/m3, respectively. While Es1andEs2are set to be unknown, ,kf, andfare set to be known in the inversion. In the presented numerical experiments, a three-level GA-based inversion method is used with a frequency- continuation scheme. Namely, in each GA set, a dynamic force with a di/uniFB00erent frequency content is used as follows. •The upper and lower bounds of each control parameter are set for the /uniFB01rst-level GA, and its last- generation leads to the best-/uniFB01t individual. •Then, in the second and third-level GA, the upper and lower limits of the four control parameters are updated with respect to the /uniFB01nal-reconstructed values of the parameters obtained in the previous- level GA. Namely, in the second-level GA, the values of the estimated Ls1andLfare bounded using the same upper and lower limits as the /uniFB01rst GA level, while the values of the estimated Es1andEs2 are bounded by using /five.pnum/zero.pnum% derivations of their /uniFB01nal-reconstructed values obtained during the /uniFB01rst GA level. •In the third GA level, the values of estimated Ls1andLfare bounded by using /five.pnum% derivations /seven.pnum of their reconstructed values from the second GA level, while /one.pnum/zero.pnum% derivations of the values of estimatedEs1andEs2that are reconstructed from second-level GA level are used as their bounds in the last GA level. •The number of generations (GN) and population size (PS) are both /five.pnum/zero.pnum in all three GA levels. In order to avoid an inverse crime, to compute uminduced by targeted control parameters, the /uniFB02uid-solid domain is discretized by using an element size of /zero.pnum./one.pnum m, while an element size of /zero.pnum./two.pnum m is used for computing udue to estimated control parameters. In the forward and inverse modeling, the time step is /zero.pnum./zero.pnum/zero.pnum/one.pnum s, and the total observation duration Tis set as /one.pnum s. In the following, we present three examples of numerical experiments. The /uniFB01rst one tests the inversion performance of the three-level GA approach by using a dynamic force of a Ricker pulse signal with its central frequency of /five.pnum, /one.pnum/zero.pnum, and /one.pnum/five.pnum Hz, respectively, in each level. In the second example, we examine the inversion performance by using a decreasing frequency-continuation counterpart of /one.pnum/five.pnum, /one.pnum/zero.pnum, and /five.pnum Hz. The last example investigates the utilization of preliminary inversion using all-solid layers for detecting the presence of a /uniFB02uid- /uniFB01lled cavity by identifying Eandof all the solid layers, among which one layer shows a very large value of E(with a value being similar to that of the bulk modulus of water) and of the value of the mass density of water. Once the preliminary inversion is completed, our optimizer uses the presented three-level GA approach based on the /uniFB02uid-solid wave model as a primary inversion. To analyze the inversion results, the error between each target control parameter ( Ls1,Lf,Es1, andEs2) and its estimated counterpart of the /uniFB01ttest individual at each generation is calculated as: E=jA targeted valueAn estimated value j jA targeted valuej100 [%]: (/three.pnum/zero.pnum) We also calculate an averaged error of all the control parameters of the best-/uniFB01t individual at each genera- tion as: E=P4 k=1Ek 4[%]: (/three.pnum/one.pnum) whereEkis the error (/three.pnum/zero.pnum) of the reconstruction of the k-th control parameter. /five.pnum./one.pnum Exemplary forward wave responses Prior to the study of the inversion performance, we show an exemplary forward wave response in the computational domain induced by a pulse loading at the top surface. We consider the /uniFB02uid-solid model /three.pnum withLs1= 50 m,Lf= 80 m,Ls2= 120 m,Es1=/two.pnum/one.pnum/zero.pnum8N/m2, andEs2=/five.pnum/one.pnum/zero.pnum8N/m2. The time-dependent value of the dynamic force applied at the top surface is a Ricker wavelet with its central frequency of /two.pnum/zero.pnum Hz and its peak amplitude of /one.pnum/zero.pnum/zero.pnum/zero.pnum N/m2. The wave responses over space and time are shown in Fig. /three.pnum. In this seismogram, we show the stress /uniFB01eld in the solid layers and the acoustic pressure in the /uniFB02uid layer. The seismogram shows the following behaviors. (i) The elastic wave propagates throughout the /uniFB01rst solid layer from the top surface and is transmitted via the /uniFB01rst solid-/uniFB02uid interface at x= 50 m to the /uniFB02uid layer (please see the wave response around x=/five.pnum/zero.pnum m andt=/zero.pnum./two.pnum s). While the wave enters into the /uniFB02uid layer, it re/uniFB02ects o/uniFB00 the interface back to the solid layer at the same time. ( ii) The acoustic pressure wave in the /uniFB02uid layer is transmitted into the second solid layer through the second solid-/uniFB02uid interface at x= 80 m and re/uniFB02ects o/uniFB00 the interface back to the /uniFB02uid layer (please see the wave response around x=/eight.pnum/zero.pnum m andt=/zero.pnum./two.pnum s). (iii) The stress wave in the second solid layer is transmitted through the absorbing boundary at x= 120 m without any re/uniFB02ection (please see the wave response around x= /one.pnum/two.pnum/zero.pnum m and t=/zero.pnum./three.pnum s). The system repeats ( i) to (iii) until the amplitude of the wave fades away. In the presented inverse modeling, the sensor on the top surface records the response signal in the time domain. The amplitudes and timings of the recorded signal may provide the inversion solver with infor- mation about the material properties of solid layers and the locations of /uniFB02uid-solid interfaces. In other words, the timings of particular parts of the signal could indicate, primarily, the locations ( Ls1andLf) of the /uniFB02uid-solid interfaces, while the amplitudes of particular parts of the signal could indicate, primarily, the material properties ( Es1andEs2) in the solid layers. /eight.pnum Wave reﬂected back to a solid layerWave reﬂected back to a ﬂuid layerFigure /three.pnum: The seismogram of the stress /uniFB01eld [N =m2] in the solid layers ( 0x50and80x) and the acoustic pressure [N =m2] in the /uniFB02uid layer ( 50< x < 80) induced by a dynamic loading at the top surface (x= 0). The solid lines at x= 50 and /eight.pnum/zero.pnum m indicate the /uniFB02uid-solid interfaces. The arrows indicate the waves that are re/uniFB02ected from the interfaces and propagating backward. The acoustic waves that are re/uniFB02ected from the second interface are more visible in the time of /zero.pnum./five.pnum/five.pnum s than in /zero.pnum./two.pnum/five.pnum s in this plot. /five.pnum./two.pnum Example /one.pnum - Investigating the inversion performance by increasing the dominant frequency of excitation for each GA level In this example, we study the performance of the presented multi-level GA for identifying targeted control parameters by increasing the frequency of an excitational Ricker signal for each GA level. That is, a dynamic pulse of its dominant frequency /five.pnum Hz is used in the /uniFB01rst-level GA, and, then, /one.pnum/zero.pnum and /two.pnum/zero.pnum Hz are used in the next levels. The control parameters to be identi/uniFB01ed are Ls1,Lf,Es1, andEs2. We considered Cases /one.pnum-/three.pnum, which use the /uniFB02uid-solid models /one.pnum, /two.pnum, and /three.pnum, respectively. •In Case /one.pnum, the targeted values of Ls1andLfare /two.pnum/zero.pnum and /two.pnum/five.pnum m, respectively, while the values of their estimated counterparts, during the /uniFB01rst-level GA level, are bounded as /one.pnum/zero.pnum Ls1/two.pnum/two.pnum m and /two.pnum/three.pnum Lf/three.pnum/five.pnum m. Please note that their upper and lower limits are changed for the second and third GA levels as previously discussed. •In Case /two.pnum, the targeted values of Ls1andLfare set to be /four.pnum/zero.pnum and /four.pnum/five.pnum m, respectively, while the values of their estimated counterparts, during the /uniFB01rst GA level, are bounded as /three.pnum/zero.pnum Ls1/four.pnum/two.pnum m and /four.pnum/three.pnum Lf/five.pnum/five.pnum m. •In Case /three.pnum, the estimated values of Ls1andLfbounded as /four.pnum/zero.pnumLs1/six.pnum/five.pnum m and /six.pnum/six.pnumLf/nine.pnum/zero.pnum m in the /uniFB01rst-level GA, and their targeted counterparts are /five.pnum/zero.pnum and /eight.pnum/zero.pnum m, respectively. For all cases, the targeted values of Young’s moduli are set to be Es1= 2/one.pnum/zero.pnum8Pa andEs2= 5/one.pnum/zero.pnum8Pa, while their estimated values for both moduli during the /uniFB01rst GA level are bounded as /one.pnum /one.pnum/zero.pnum8PaE /seven.pnum./five.pnum/one.pnum/zero.pnum8Pa. Their upper and lower limits are changed for the second and third GA levels as previously discussed. Table /one.pnum presents the reconstructed control parameters of the /uniFB01ttest individual that is obtained at the last generation of each GA level per each case. In the table, we also present the average error E, (/three.pnum/one.pnum), of all the parameters reconstructed at the end of each GA level and the error E, (/three.pnum/zero.pnum), of each control parameter of the /uniFB01ttest individual at the end of the last-level GA. In all Cases /one.pnum to /three.pnum, the value of Eat the end of the last-level GA is smaller than that at the end of the /uniFB01rst-level GA, and the /uniFB01nal values of Ein all the cases /one.pnum to /three.pnum are smaller than /one.pnum%. Fig. /four.pnum shows that the average error for the best-/uniFB01t individual at each generation tends to be decreased during the presented inversion process. We note that, after about /two.pnum/zero.pnum GA iterations in the /uniFB01rst-level GA, the value ofEis converged but still shows room for improvement (i.e., the optimizer /uniFB01nds an inversion solution around a local minimum). To address such solution multiplicity in the /uniFB01rst-level GA, we use a mis/uniFB01t function foruminduced by a pulse signal of a new central frequency in the next-level GA. By doing so, the optimizer in the second-level GA is able to escape the local minimum of the /uniFB01rst-level GA. By applying the same technique in the third-level GA, we can resolve the solution multiplicity of the second-level GA. Therefore, /nine.pnum Table /one.pnum: Example /one.pnum: Reconstructed values of the control parameters for each GA level obtained by increasing the frequency of the dynamic force for each GA level. The last row of each case represents the error at the third-level GA. Cases GA Ls1LfEs1Es2 Average Error level (m) (m) (Pa) (Pa) E Case /one.pnum Target /two.pnum/zero.pnum /two.pnum/five.pnum /two.pnum 108/five.pnum108 /one.pnumst /two.pnum/zero.pnum./four.pnum /two.pnum/four.pnum./nine.pnum /two.pnum./one.pnum/three.pnum 108/five.pnum./six.pnum/six.pnum108/five.pnum./five.pnum/three.pnum% /two.pnumnd /one.pnum/nine.pnum./eight.pnum /two.pnum/four.pnum./three.pnum /one.pnum./nine.pnum/seven.pnum 108/four.pnum./nine.pnum/four.pnum108/one.pnum./six.pnum/three.pnum% /three.pnumrd /two.pnum/zero.pnum./zero.pnum /two.pnum/four.pnum./eight.pnum /two.pnum./zero.pnum/zero.pnum 108/five.pnum./zero.pnum/eight.pnum108/zero.pnum./six.pnum/zero.pnum% Individual error /zero.pnum./zero.pnum/zero.pnum% /zero.pnum./eight.pnum/zero.pnum% /zero.pnum./zero.pnum/zero.pnum% /one.pnum./six.pnum/zero.pnum% Case /two.pnum Target /four.pnum/zero.pnum /four.pnum/five.pnum /two.pnum 108/five.pnum108 /one.pnumst /four.pnum/one.pnum./seven.pnum /four.pnum/six.pnum./two.pnum /two.pnum./two.pnum/zero.pnum 108/five.pnum./eight.pnum/six.pnum108/eight.pnum./five.pnum/three.pnum% /two.pnumnd /four.pnum/zero.pnum./six.pnum /four.pnum/six.pnum./zero.pnum /two.pnum./zero.pnum/six.pnum 108/five.pnum./one.pnum/six.pnum108/two.pnum./four.pnum/eight.pnum% /three.pnumrd /three.pnum/nine.pnum./nine.pnum /four.pnum/four.pnum./seven.pnum /one.pnum./nine.pnum/eight.pnum 108/four.pnum./nine.pnum/seven.pnum108/zero.pnum./six.pnum/three.pnum% Individual error /zero.pnum./two.pnum/five.pnum% /zero.pnum./six.pnum/seven.pnum% /one.pnum./zero.pnum/zero.pnum% /zero.pnum./six.pnum/zero.pnum% Case /three.pnum Target /five.pnum/zero.pnum /eight.pnum/zero.pnum /two.pnum 108/five.pnum108 /one.pnumst /five.pnum/three.pnum./one.pnum /nine.pnum/zero.pnum./zero.pnum /two.pnum./two.pnum/eight.pnum 108/five.pnum./four.pnum/nine.pnum108/one.pnum/zero.pnum./six.pnum/three.pnum% /two.pnumnd /four.pnum/eight.pnum./nine.pnum /seven.pnum/nine.pnum./two.pnum /one.pnum./nine.pnum/one.pnum 108/five.pnum./three.pnum/three.pnum108/three.pnum./five.pnum/eight.pnum% /three.pnumrd /four.pnum/nine.pnum./nine.pnum /seven.pnum/nine.pnum./nine.pnum /one.pnum./nine.pnum/eight.pnum 108/five.pnum./zero.pnum/six.pnum108/zero.pnum./six.pnum/three.pnum% Individual error /zero.pnum./two.pnum/zero.pnum% /zero.pnum./one.pnum/two.pnum% /one.pnum./zero.pnum/zero.pnum% /one.pnum./two.pnum/zero.pnum% we suggest that it is feasible to reconstruct control parameters e/uniFB00ectively by using a multi-level GA process combined with a frequency-continuation scheme, by which the frequency is progressively increased for each GA level. 0 20 40 60 80 100 120 140 16005101520253035 Figure /four.pnum:Efor the best-/uniFB01t individual versus the GA iteration in Example /one.pnum. The vertical solid lines at the /five.pnum/one.pnumst and /one.pnum/zero.pnum/two.pnumnd GA iteration indicate the starting of a new GA level and the change in the frequency of the excitational Ricker signal. Fig. /five.pnum presents the detail of the inversion performance of Case /three.pnum. Namely, Fig. /five.pnum shows the histograms of the estimated control parameters of the entire population of the individuals during all the GA levels in Case /three.pnum. During the initial few generations of each GA level, the inversion solver explores a broader range of estimated values of control parameters. After these initial generations, the values of estimated parameters of individuals are converged. Fig. /six.pnum also shows that uminduced by the target parameters matches udue to the /uniFB01nally-reconstructed control parameters, obtained at the end of each GA set, at the sensor on the top surface when a Ricker signal with its central frequency of /five.pnum Hz, /one.pnum/zero.pnum Hz, or /two.pnum/zero.pnum Hz is employed in Case /three.pnum. /one.pnum/zero.pnum (a)(b) (c)(d)Figure /five.pnum: The histograms of (a) Ls1, (b)Lf, (c)Es1, and (d)Es2of the entire population of the individuals at all the generations in Case /three.pnum. 00.10.20.30.40.50.60.70.80.91-50510-5 00.10.20.30.40.50.60.70.80.91-20210-5 00.10.20.30.40.50.60.70.80.91-10110-5(a) (b) (c) Figure /six.pnum: Wave responses, umandu, at the sensor on the top surface induced by a pulse signal of (a) /five.pnum Hz, (b) /one.pnum/zero.pnum Hz, and (c) /two.pnum/zero.pnum Hz in Case /three.pnum. /one.pnum/one.pnum /five.pnum./three.pnum Example /two.pnum - Investigating the inversion performance by decreasing the dominant frequency of excitation for each GA level This example studies the performance of reconstructing the control parameters by using the presented multi-level GA combined with a frequency-continuation scheme that decreases the dominant frequency of the dynamic pulse over the multiple GA levels. The /uniFB01rst, second, and third-level GA use input force signals of central frequencies of /two.pnum/zero.pnum Hz, /one.pnum/zero.pnum Hz, and /five.pnum Hz, respectively. We examined Cases /four.pnum-/six.pnum, which use the models /one.pnum, /two.pnum, and /three.pnum, respectively. The upper and lower limits of estimated control parameters for each GA level are set the same as those in Example /one.pnum. Table /two.pnum shows the values of /uniFB01nally-reconstructed control parameters, and Fig. /seven.pnum shows that the average errors for the best-/uniFB01t individual of all the Cases /four.pnum-/six.pnum become smaller over the generations and the GA levels. The /uniFB01nal values of Ein all the Cases /four.pnum to /six.pnum are smaller than /two.pnum% in Example /two.pnum. Thus, we suggest that the presented multi-level GA-based optimizer combined with a frequency-continuation scheme, which decreases the frequency for each GA level, is able to identify the values of the targeted control parameters as e/uniFB00ectively as the increasing-frequency counterpart. However, this would be the case only in the presented example because the presented example contains a small number of control parameters. In a more complex case (e.g., /two.pnumD or /three.pnumD settings), where there are a large number of control parameters, the decreasing frequency-continuation scheme may not be as e/uniFB00ective as its counterpart of increasing frequency. Table /two.pnum: Example /two.pnum: Reconstructed values of the control parameters for each GA level obtained by decreas- ing the frequency of the dynamic force for each GA level. Cases GA Ls1LfEs1Es2 Average Error level (m) (m) (Pa) (Pa) E Case /four.pnum Target /two.pnum/zero.pnum /two.pnum/five.pnum /two.pnum 108/five.pnum108 /one.pnumst /two.pnum/one.pnum./four.pnum /two.pnum/seven.pnum./three.pnum /two.pnum./two.pnum/nine.pnum 108/five.pnum./five.pnum/five.pnum108/one.pnum/zero.pnum./four.pnum/three.pnum% /two.pnumnd /two.pnum/zero.pnum./one.pnum /two.pnum/four.pnum./nine.pnum /two.pnum./zero.pnum/four.pnum 108/five.pnum./one.pnum/eight.pnum108/one.pnum./six.pnum/three.pnum% /three.pnumrd /two.pnum/zero.pnum./two.pnum /two.pnum/five.pnum./seven.pnum /two.pnum./zero.pnum/two.pnum 108/five.pnum./zero.pnum/four.pnum108/one.pnum./four.pnum/zero.pnum% Individual error /one.pnum./zero.pnum/zero.pnum% /two.pnum./eight.pnum/zero.pnum% /one.pnum./zero.pnum/zero.pnum% /zero.pnum./eight.pnum/zero.pnum% Case /five.pnum Target /four.pnum/zero.pnum /four.pnum/five.pnum /two.pnum 108/five.pnum108 /one.pnumst /four.pnum/one.pnum./zero.pnum /four.pnum/five.pnum./seven.pnum /two.pnum./one.pnum/one.pnum 108/six.pnum./zero.pnum/one.pnum108/seven.pnum./four.pnum/four.pnum% /two.pnumnd /four.pnum/zero.pnum./six.pnum /four.pnum/six.pnum./three.pnum /two.pnum./zero.pnum/five.pnum 108/five.pnum./zero.pnum/nine.pnum108/two.pnum./one.pnum/seven.pnum% /three.pnumrd /four.pnum/zero.pnum./zero.pnum /four.pnum/four.pnum./seven.pnum /two.pnum./zero.pnum/zero.pnum 108/five.pnum./zero.pnum/one.pnum108/zero.pnum./two.pnum/two.pnum% Individual error /zero.pnum./zero.pnum/zero.pnum% /zero.pnum./six.pnum/seven.pnum% /zero.pnum./zero.pnum/zero.pnum% /zero.pnum./two.pnum/zero.pnum% Case /six.pnum Target /five.pnum/zero.pnum /eight.pnum/zero.pnum /two.pnum 108/five.pnum108 /one.pnumst /five.pnum/seven.pnum./five.pnum /eight.pnum/two.pnum./three.pnum /two.pnum./six.pnum/three.pnum 108/four.pnum./five.pnum/one.pnum108/one.pnum/four.pnum./seven.pnum/nine.pnum% /two.pnumnd /four.pnum/nine.pnum./seven.pnum /seven.pnum/seven.pnum./nine.pnum /one.pnum./nine.pnum/eight.pnum 108/four.pnum./eight.pnum/eight.pnum108/one.pnum./six.pnum/six.pnum% /three.pnumrd /four.pnum/nine.pnum./seven.pnum /seven.pnum/nine.pnum./nine.pnum /one.pnum./nine.pnum/nine.pnum 108/five.pnum./one.pnum/eight.pnum108/one.pnum./two.pnum/one.pnum% Individual error /zero.pnum./six.pnum/zero.pnum% /zero.pnum./one.pnum/two.pnum% /zero.pnum./five.pnum/zero.pnum% /three.pnum./six.pnum/zero.pnum% 0 20 40 60 80 100 120 140 160051015202530 Figure /seven.pnum:Efor the best-/uniFB01t individual versus the GA iteration in Example /two.pnum. The solid lines at the /five.pnum/one.pnumst and /one.pnum/zero.pnum/two.pnumnd GA iteration indicate the starting of a new GA level and the change in frequency. /one.pnum/two.pnum /five.pnum./four.pnum Example /three.pnum - Utilization of a preliminary inversion using only solid layers for detect- ing the presence of a /uniFB02uid layer, followed by a primary inversion using /uniFB02uid-solid layers In this example, we present a preliminary inversion using only solid layers for identifying if a target model contains a /uniFB02uid-/uniFB01lled cavity. Here, three target models are used: one, which does not contain a /uniFB02uid layer, and the other two, in which a /uniFB02uid-/uniFB01lled cavity is included. For all models, our preliminary GA parameter- estimation method assumes that the target contains only solid layers. Once the preliminary inversion is completed, we continue a primary inversion using a /uniFB02uid-solid model in a manner that is similar to what is described in Example /one.pnum. Such a two-step (preliminary to primary inversion) approach is applied to the following Cases /seven.pnum-/one.pnum/zero.pnum. •Cases /seven.pnum and /eight.pnum consider a target Model /four.pnum, which does not include a /uniFB02uid layer, and contains only three solid layers of /five.pnum m length each. The targeted Young’s modulus of each layer is: Es1= /two.pnum108Pa,Es2= /three.pnum108Pa, andEs3= /five.pnum108Pa, respectively. For Case /seven.pnum, the targeted mass densities of all the layers are all same: s1=s2=s3= /two.pnum/zero.pnum/zero.pnum/zero.pnum kg/m3, whereas, for Case /eight.pnum, they are all di/uniFB00erent from each other:s1= /one.pnum/five.pnum/zero.pnum/zero.pnum kg/m3,s2= /two.pnum/zero.pnum/zero.pnum/zero.pnum kg/m3, ands3= /two.pnum/two.pnum/zero.pnum/zero.pnum kg/m3. •Case /nine.pnum considers Model /five.pnum, which consists of three layers, i.e., two solids and one /uniFB02uid. Model /five.pnum has a total length Ls2of /one.pnum/five.pnum m, and its /uniFB02uid is located between xofLs1= /five.pnum m andxofLf= /one.pnum/zero.pnum m. The Young’s moduli of the solid layers are Es1= /two.pnum108Pa andEs2= /five.pnum108, and their mass densities are s1=s2= /two.pnum/zero.pnum/zero.pnum/zero.pnum kg/m3, respectively. The bulk modulus ( kf) and mass density ( f) of the /uniFB02uid layer are /two.pnum./three.pnum/four.pnum/one.pnum/zero.pnum9N/m2and /one.pnum/zero.pnum/two.pnum/one.pnum kg/m3, respectively. •Case /one.pnum/zero.pnum considers Model /six.pnum, which consists of a /uniFB02uid layer surrounded by solid layers. The /uniFB02uid layer is located between xofLs1= /four.pnum m andxofLf= /nine.pnum./three.pnum m, and its Ls2is /one.pnum/five.pnum m. The properties of the /uniFB02uid layer (kfandf) and the Young’s moduli of the solid layers ( Es1andEs2) are the same as those in Case /nine.pnum except that the mass densities of solid layers are s1= /one.pnum/nine.pnum/zero.pnum/zero.pnum kg/m3ands2= /two.pnum/one.pnum/zero.pnum/zero.pnum kg/m3. In Case /seven.pnum and /eight.pnum, we study the capability of the preliminary inversion to detect the absence of a /uniFB02uid layer by estimating Eandof all solid layers. In Case /nine.pnum and /one.pnum/zero.pnum, by using only solid layers during the preliminary inversion, we investigate what values of /uniFB01nal-converged Eandfrom the preliminary inversion indicate the presence of a /uniFB02uid-/uniFB01lled cavity in the model. /five.pnum./four.pnum./one.pnum Preliminary inversion using all solid-layer model For all Cases /seven.pnum-/one.pnum/zero.pnum, the preliminary inversion method utilizes only one frequency level of GA, where a Ricker pulse signal with a central frequency of /five.pnum/zero.pnum Hz is used as the dynamic force, and the number of generations and population size are /five.pnum/zero.pnum and /one.pnum/zero.pnum/zero.pnum, respectively. Moreover, the estimated values for Young’s moduli in all solid layers are bounded as /one.pnum 108E50108Pa, and the estimated values of their bounded as /one.pnum/zero.pnum/zero.pnum/zero.pnum/three.pnum/zero.pnum/zero.pnum/zero.pnum kg/m3. Furthermore, the preliminary inversion assumes that the target contains three solid layers of /five.pnum m length each. Fig. /eight.pnum shows the reduction of average errors for the best-/uniFB01t individual for Cases /seven.pnum to /one.pnum/zero.pnum as the number of GA generations increases in the preliminary inversion. Table /three.pnum shows the targeted and reconstructed control parameters from the preliminary inversion for Cases /seven.pnum to /one.pnum/zero.pnum. For Cases /seven.pnum and /eight.pnum, the results show that the preliminary GA-based inversion algorithm can determine that all the layers in the target model are solid layers because the order of magnitude of estimated Young’s moduli are all /two.pnum to /five.pnum 108Pa in all the solid layers, and the order of magnitude of their estimated mass densities are all /one.pnum/five.pnum/zero.pnum/zero.pnum to /two.pnum/two.pnum/zero.pnum/zero.pnum kg/m3. The average error in Cases /seven.pnum and /eight.pnum is /three.pnum./eight.pnum/two.pnum% and /four.pnum./eight.pnum/two.pnum%, respectively. On the contrary, in Cases /nine.pnum and /one.pnum/zero.pnum, the preliminary inversion led to the estimated values of Es2ands2of the second solid layer such that it can be interpreted as a /uniFB02uid layer. Namely, the estimated values of Es2as /two.pnum/two.pnum./nine.pnum/five.pnum108Pa (Case /nine.pnum) and /two.pnum/four.pnum./three.pnum/two.pnum 108 Pa (Case /one.pnum/zero.pnum) are similar to the bulk modulus /two.pnum/three.pnum./four.pnum 108Pa of water, and the estimated values of s2as /one.pnum/zero.pnum/two.pnum/two.pnum kg/m3(Case /nine.pnum) and /one.pnum/zero.pnum/seven.pnum/zero.pnum kg/m3(Case /one.pnum/zero.pnum) are similar to the mass density /one.pnum/zero.pnum/two.pnum/one.pnum kg/m3of water. The average errors in Cases /nine.pnum and /one.pnum/zero.pnum are /two.pnum./one.pnum/six.pnum% and /one.pnum/seven.pnum./one.pnum/three.pnum%, respectively. Thus, after detecting the /uniFB02uid layers in Case /nine.pnum and /one.pnum/zero.pnum, we continue the primary inversion in Cases /nine.pnum and /one.pnum/zero.pnum as follows. /one.pnum/three.pnum Table /three.pnum: Example /three.pnum: Reconstructed values of the control parameters obtained by an preliminary inversion. In Cases /nine.pnum and /one.pnum/zero.pnum, the targeted Es2* (/two.pnum/three.pnum./four.pnum108Pa) adopts the bulk modulus of the /uniFB02uid layer. Cases Es1Es2Es3s1s2s3 Average (Pa) (Pa) (Pa) (kg/m3) (kg/m3) (kg/m3) ErrorE Case /seven.pnumTarget /two.pnum108/three.pnum108/five.pnum108/two.pnum/zero.pnum/zero.pnum/zero.pnum /two.pnum/zero.pnum/zero.pnum/zero.pnum /two.pnum/zero.pnum/zero.pnum/zero.pnum Preliminary values /two.pnum./zero.pnum/one.pnum 108/two.pnum./eight.pnum/eight.pnum108/five.pnum./two.pnum/six.pnum108/one.pnum/nine.pnum/four.pnum/three.pnum /two.pnum/zero.pnum/one.pnum/seven.pnum /one.pnum/eight.pnum/one.pnum/zero.pnum /three.pnum./eight.pnum/two.pnum% Individual error /zero.pnum./five.pnum/zero.pnum% /four.pnum./zero.pnum/zero.pnum% /five.pnum./two.pnum/zero.pnum% /two.pnum./eight.pnum/five.pnum% /zero.pnum./eight.pnum/five.pnum% /nine.pnum./five.pnum/zero.pnum% Case /eight.pnumTarget /two.pnum108/three.pnum108/five.pnum108/one.pnum/five.pnum/zero.pnum/zero.pnum /two.pnum/zero.pnum/zero.pnum/zero.pnum /two.pnum/two.pnum/zero.pnum/zero.pnum Preliminary values /two.pnum./one.pnum/zero.pnum 108/three.pnum./zero.pnum/eight.pnum108/four.pnum./nine.pnum/three.pnum108/one.pnum/five.pnum/six.pnum/zero.pnum /two.pnum/zero.pnum/nine.pnum/two.pnum /two.pnum/four.pnum/four.pnum/eight.pnum /four.pnum./eight.pnum/two.pnum% Individual error /five.pnum./zero.pnum/zero.pnum% /two.pnum./six.pnum/seven.pnum% /one.pnum./four.pnum/zero.pnum% /four.pnum./zero.pnum/zero.pnum% /four.pnum./six.pnum/zero.pnum% /one.pnum/one.pnum./two.pnum/seven.pnum% Case /nine.pnumTarget /two.pnum108/two.pnum/three.pnum./four.pnum/zero.pnum108* /five.pnum108/two.pnum/zero.pnum/zero.pnum/zero.pnum /one.pnum/zero.pnum/two.pnum/one.pnum /two.pnum/zero.pnum/zero.pnum/zero.pnum Preliminary values /two.pnum./zero.pnum/one.pnum 108/two.pnum/two.pnum./nine.pnum/five.pnum108/five.pnum./zero.pnum/seven.pnum108/two.pnum/zero.pnum/two.pnum/four.pnum /one.pnum/zero.pnum/two.pnum/two.pnum /one.pnum/eight.pnum/four.pnum/three.pnum /two.pnum./one.pnum/six.pnum% Individual error /zero.pnum./five.pnum/zero.pnum% /one.pnum./nine.pnum/two.pnum% /one.pnum./four.pnum/zero.pnum% /one.pnum./two.pnum/zero.pnum% /zero.pnum./one.pnum/zero.pnum% /seven.pnum./eight.pnum/five.pnum% Case /one.pnum/zero.pnumTarget /two.pnum108/two.pnum/three.pnum./four.pnum/zero.pnum108* /five.pnum108/one.pnum/nine.pnum/zero.pnum/zero.pnum /one.pnum/zero.pnum/two.pnum/one.pnum /two.pnum/one.pnum/zero.pnum/zero.pnum Preliminary values /two.pnum./five.pnum/two.pnum 108/two.pnum/four.pnum./three.pnum/two.pnum108/six.pnum./five.pnum/three.pnum108/one.pnum/five.pnum/four.pnum/nine.pnum /one.pnum/zero.pnum/seven.pnum/zero.pnum /one.pnum/seven.pnum/zero.pnum/one.pnum /one.pnum/seven.pnum./one.pnum/three.pnum% Individual error /two.pnum/six.pnum./zero.pnum/zero.pnum% /three.pnum./nine.pnum/three.pnum% /three.pnum/zero.pnum./six.pnum/zero.pnum% /one.pnum/eight.pnum./four.pnum/seven.pnum% /four.pnum./eight.pnum/zero.pnum% /one.pnum/nine.pnum./zero.pnum/zero.pnum% 0 10 20 30 40 50 60100101102 Figure /eight.pnum:Efor the best-/uniFB01t individual versus the GA iteration in Example /three.pnum for the preliminary inversion. /five.pnum./four.pnum./two.pnum Primary inversion using a /uniFB02uid-solid model After the preliminary inversion using the solid-only wave solver, the three frequency-level GA-based op- timizer that uses a /uniFB02uid-solid wave model estimates the /uniFB02uid layer’s depth information and the material properties (Eand) of the soil layers. Namely, the preliminary inversion results from Cases /nine.pnum and /one.pnum/zero.pnum are used for determining the bounds of guessed parameters in the /uniFB01rst-frequency set of the primary GA inversion using the /uniFB02uid-solid model. The increasing-frequency continuation approach is utilized with a dynamic pulse of its dominant frequency /five.pnum Hz in the /uniFB01rst frequency-level GA and, then, /one.pnum/zero.pnum and /two.pnum/zero.pnum Hz in the next frequency levels while the values of the estimated parameters are bounded in each frequency set in the following manner. •During the /uniFB01rst frequency-level GA using the /uniFB02uid-solid wave solver, the estimated Ls1andLfvalues are bounded as /two.pnumLs1/seven.pnum m and /eight.pnumLf/one.pnum/three.pnum m, the estimated values of Es1andEs2are bounded as /one.pnum/one.pnum/zero.pnum8E/seven.pnum./five.pnum/one.pnum/zero.pnum8Pa, and the estimated values of s1ands2(i.e., the mass densities of surrounding soil layers) are set as /one.pnum/five.pnum/zero.pnum/zero.pnum /two.pnum/five.pnum/zero.pnum/zero.pnum kg/m3. •In the second frequency-level GA, the values of estimated Ls1andLfare bounded using the same /one.pnum/four.pnum upper and lower limits as the /uniFB01rst frequency-level GA. The values of estimated Es1andEs2are bounded using/five.pnum/zero.pnum% derivations of their /uniFB01nal-reconstructed values obtained during the /uniFB01rst frequency- level GA, while the values of estimated s1ands2are bounded using /two.pnum/zero.pnum% derivations. •Finally, in the last frequency-level GA, the estimated Ls1andLfvalues are bounded by using  /five.pnum% derivations of their /uniFB01nal-reconstructed values obtained during the second frequency-level GA, while/one.pnum/zero.pnum% derivations of the values of estimated Es1andEs2are employed to bound the limits of Young’s moduli, and /one.pnum/zero.pnum% derivations of the /uniFB01nal-reconstructed values of s1ands2are used as theirs bounds in the last frequency-level GA. Table /four.pnum shows the values of /uniFB01nally-reconstructed control parameters in Cases /nine.pnum and /one.pnum/zero.pnum. For Case /nine.pnum, the value of Eat the end of the last-level GA, /one.pnum./seven.pnum/four.pnum%, is smaller than the error obtained in the preliminary result, /two.pnum./one.pnum/six.pnum%, shown in Table /three.pnum. The same can be noted in Case /one.pnum/zero.pnum, where Eat the end of the last-level GA, /two.pnum./seven.pnum/nine.pnum%, is much smaller than Efrom the preliminary inversion, /one.pnum/seven.pnum./one.pnum/three.pnum%. Fig. /nine.pnum shows the average error for the best-/uniFB01t individual over GA iterations of the three-level primary GA inversion using /uniFB02uid-solid models for both Cases /nine.pnum and /one.pnum/zero.pnum. During the initial several iterations of each frequency level, the algorithm focuses on exploring widely-varying values of the estimated control parameters within the given limits, and this behavior increases error. Later, the error of the algorithm is converged by /uniFB01nding better-/uniFB01t individuals. Fig. /one.pnum/zero.pnum shows the histograms of the estimated control parameters of the entire population of the individuals during all the GA levels in Case /one.pnum/zero.pnum during the primary inversion. It demonstrates that, at the beginning of each frequency level, GA explores wide ranges of control parameters and eventually converges to the targeted parameters at the end of the last frequency set. Table /four.pnum: Example /three.pnum - the results of the primary inversion in Cases /nine.pnum and /one.pnum/zero.pnum: Estimated values of the control parameters in each GA level obtained by increasing the frequency of the dynamic force for each GA level. Cases GA Ls1LfEs1Es2s1s2 Average Error level (m) (m) (Pa) (Pa) (kg/m3) (kg/m3)E Case /nine.pnumTarget /five.pnum /one.pnum/zero.pnum /two.pnum 108/five.pnum108/two.pnum/zero.pnum/zero.pnum/zero.pnum /two.pnum/zero.pnum/zero.pnum/zero.pnum /one.pnumst /five.pnum./one.pnum /one.pnum/zero.pnum./zero.pnum /two.pnum./zero.pnum/one.pnum 108/five.pnum./four.pnum/one.pnum108/one.pnum/nine.pnum/six.pnum/five.pnum /one.pnum/eight.pnum/four.pnum/three.pnum /three.pnum./three.pnum/eight.pnum% /two.pnumnd /five.pnum./one.pnum /nine.pnum./eight.pnum /two.pnum./zero.pnum/one.pnum 108/five.pnum./four.pnum/five.pnum108/one.pnum/nine.pnum/six.pnum/nine.pnum /one.pnum/eight.pnum/four.pnum/three.pnum /three.pnum./eight.pnum/two.pnum% /three.pnumrd /five.pnum./one.pnum /one.pnum/zero.pnum./one.pnum /one.pnum./nine.pnum/nine.pnum 108/five.pnum./one.pnum/two.pnum108/two.pnum/zero.pnum/one.pnum/one.pnum /one.pnum/nine.pnum/two.pnum/zero.pnum /one.pnum./seven.pnum/four.pnum% Individual error /two.pnum./zero.pnum/zero.pnum% /one.pnum./zero.pnum/zero.pnum% /zero.pnum./five.pnum/zero.pnum% /two.pnum./four.pnum/zero.pnum% /zero.pnum./five.pnum/five.pnum% /four.pnum./zero.pnum/zero.pnum% Case /one.pnum/zero.pnumTarget /four.pnum /nine.pnum./three.pnum /two.pnum 108/five.pnum108/one.pnum/nine.pnum/zero.pnum/zero.pnum /two.pnum/one.pnum/zero.pnum/zero.pnum /one.pnumst /three.pnum./one.pnum /eight.pnum./four.pnum./two.pnum /one.pnum./five.pnum/nine.pnum 108/four.pnum./six.pnum/eight.pnum108/two.pnum/four.pnum/zero.pnum/three.pnum /two.pnum/two.pnum/three.pnum/nine.pnum /one.pnum/five.pnum./three.pnum/six.pnum% /two.pnumnd /four.pnum./one.pnum /nine.pnum./three.pnum /two.pnum./zero.pnum/two.pnum 108/five.pnum./four.pnum/two.pnum108/one.pnum/nine.pnum/two.pnum/two.pnum /one.pnum/nine.pnum/eight.pnum/one.pnum /three.pnum./one.pnum/two.pnum% /three.pnumrd /four.pnum./zero.pnum /nine.pnum./three.pnum /two.pnum./zero.pnum/one.pnum 108/five.pnum./three.pnum/seven.pnum108/one.pnum/nine.pnum/two.pnum/two.pnum /one.pnum/nine.pnum/three.pnum/nine.pnum /two.pnum./seven.pnum/nine.pnum% Individual error /zero.pnum./zero.pnum/zero.pnum% /zero.pnum./zero.pnum/zero.pnum% /zero.pnum./five.pnum/zero.pnum% /seven.pnum./four.pnum/zero.pnum% /one.pnum./one.pnum/six.pnum% /seven.pnum./six.pnum/seven.pnum% /one.pnum/five.pnum 0 20 40 60 80 100 120 140 1600246810121416182022Figure /nine.pnum:Efor the best-/uniFB01t individual versus the GA iteration in Cases /nine.pnum and /one.pnum/zero.pnum during the primary inversion. The vertical solid lines at the /five.pnum/one.pnumst and /one.pnum/zero.pnum/two.pnumnd GA iterations indicate each onset of a new frequency-level GA and the change in the frequency of the excitational Ricker signal. Figure /one.pnum/zero.pnum: The histograms of (a) Ls1, (b)Lf, (c)Es1, (d)Es2, (e)s1, and (f)s2of the entire population of all the individuals at all the generations in Case /one.pnum/zero.pnum during the primary inversion. /one.pnum/six.pnum This Example /three.pnum shows a proof of concept, potentially promising that ( i) we can detect /uniFB02uid-/uniFB01lled cavities in a /two.pnumD or /three.pnumD domain by using the all-solid-element-based preliminary inversion of E,G, andand /uniFB01nding particular solid elements (of a high value of E, a low value of G, and the value of , being similar to /one.pnum/zero.pnum/two.pnum/one.pnum kg/m3); then, (ii) we can use a /uniFB02uid-solid-element-based wave solver and identify the geometry of a /uniFB02uid- solid interface and E,G, andof solid elements, surrounding /uniFB02uid, in the /two.pnumD or /three.pnumD domain. The details of such /two.pnumD//three.pnumD extensions are shown below. /six.pnum Discussion We can extend this work into /two.pnumD and /three.pnumD settings in a manner such that one can identify the geometry of /uniFB02uid-/uniFB01lled cavities and the material properties of surrounding soils by using an FWI algorithm (instead of GA), which can be optionally combined with an arti/uniFB01cial neural network (ANN). Although the GA can be a useful method to identify a small number of control parameters (e.g., up to /one.pnum/zero.pnum) under a feasibility study like this presented research, the GA is considered to be a much less viable solution method than the FWI that is powered by the PDE-constrained optimization method when the number of control parameters is large in a case—such as one, where the material properties E,G, andof all the solid elements in a /two.pnumD or /three.pnumD setting are to be estimated. Namely, under the framework of FWI, we can employ the following two-step approach (i.e., preliminary to primary inversion) to ( i) detect the existence of the /uniFB02uid-/uniFB01lled voids and ( ii) accurately identify the geometry of the /uniFB02uid-solid interface and the material proprieties of soils in /two.pnumD//three.pnumD settings. •Preliminary inversion of FWI: First, one can approximately detect the existence of /uniFB02uid-/uniFB01lled cavities in a /two.pnumD or /three.pnumD domain by using the all-solid-element-based inversion of E,G, andof solids. That is, in each iteration of this preliminary FWI, the forward solution is obtained for a guessed pro/uniFB01les of E, G, andin all the elements, which are solids, in a computational domain. Per each iteration, one can compute the value of a gradient of a mis/uniFB01t function—between measurement signal data at sensors and their computed counterparts due to a guessed pro/uniFB01le—with respect to control parameters by using the solution of an adjoint problem, which utilizes the mis/uniFB01t at each sensor location as a driving force in an all-solid model. By using the gradient vector, the guessed pro/uniFB01le can be updated, and this process is repeated until a termination condition is satis/uniFB01ed. Once one /uniFB01nds a particular area (i.e., an area of interest) of solid elements—with a high value of elasticity but a low value of shear modulus and a mass density that is similar to that of water, one can consider such an area as a /uniFB02uid-/uniFB01lled cavity. •Primary inversion of FWI: Subsequently, once this preliminary FWI is completed, we would switch the wave solver (for both forward and adjoint problems) such that /uniFB02uid elements occupy the area of interest that was detected as a potential /uniFB02uid-/uniFB01lled cavity from the preliminary inversion. In this primary FWI, one will update the following control parameters— E,G, andof the solid elements surrounding the /uniFB02uid area, as well as the geometry parameters of the interface between /uniFB02uid and solid elements. In that, we may have several possible methods of parameterizing the shape of the interface. Among them is using the Cartesian coordinates of the discretized nodes on the interface as control parameters. Our initial guessed pro/uniFB01le of the control parameters of the interface will be based on the result from the preliminary all-solid FWI. For instance, Fig. /one.pnum/one.pnum depicts a possible scenario of how one may approximate the initially-guessed geometry of the /uniFB02uid-solid interface based on the result from the preliminary all-solid FWI. At each iteration of this primary FWI, one /uniFB01rst solves the forward problem by using the guessed pro/uniFB01le, then, solves the adjoint problem, and computes the gradient of a mis/uniFB01t function with respect to control parameters. When one computes the gradient with respect to the geometric control parameters, one can use the concept of a moving boundary linked with the total derivative (Jeong et al., /two.pnum/zero.pnum/zero.pnum/nine.pnum) that allows for the analytical evaluation of the gradient with respect to the shape parameters. By using the gradient, one updates the mesh such that it re/uniFB02ects the updated geometry parameters. Thus, this primary FWI should be linked, by virtue of the application programming interface (API), to a high-performance mesh generator (e.g., CUBIT) such that a new mesh is generated per each inversion iteration. Then, one repeats to calculate the forward and adjoint solutions and the gradient of a mis/uniFB01t for the new mesh and, then, updates the guessed pro/uniFB01le of control parameters as well as the mesh until a termination condition is satis/uniFB01ed. The multiple frequency-continuation scheme can be employed under this primary FWI as well. As a forward and adjoint solver, one could potentially use a spectral element method (SEM) wave solver (Komatitsch and Tromp, /one.pnum/nine.pnum/nine.pnum/nine.pnum; Komatitsch et al., /two.pnum/zero.pnum/zero.pnum/zero.pnum, /two.pnum/zero.pnum/zero.pnum/two.pnum; Komatitsch and Tromp, /two.pnum/zero.pnum/zero.pnum/two.pnum; Tromp /one.pnum/seven.pnum (a)(b) (c)(d)Figure /one.pnum/one.pnum: An exemplary sketch of our proposed method in /two.pnumD//three.pnumD settings: (a) an initial guess of a prelimi- nary inversion, using a structured mesh, (b) the /uniFB01nal result of the preliminary inversion, (c) an initial guess of a primary inversion, using an unstructured mesh, and (d) the /uniFB01nal result of the primary inversion. et al., /two.pnum/zero.pnum/zero.pnum/eight.pnum). It has already addressed the large computational cost in /two.pnumD//three.pnumD wave propagation anal- yses by virtue of a naturally-arisen diagonal mass matrix and explicit time integration, and the /uniFB02uid- solid interface conditions and the absorbing boundary conditions are already implemented within the SEM solver (Lloyd et al., /two.pnum/zero.pnum/one.pnum/eight.pnum). However, the moving boundary concept and the total derivative— associated with updating the geometries of a /uniFB02uid-solid interface—are not implemented in an SEM solver yet. Thus, the implementation of such new modeling in an SEM solver will enable the pro- posed primary FWI in /three.pnumD to be used for generating very accurate geophysical characterization data in Karst areas. Alternatively, the /two.pnumD//three.pnumD extension can be investigated by using the combination of an arti/uniFB01cial neural net- work (ANN) and FWI (henceforth, ANN/two.pnumFWI) in the following two-step approach. •Preliminary inversion of ANN/two.pnumFWI: ANN can be used as an inversion solver in the preliminary in- version of ANN/two.pnumFWI. We note that the considered inverse problem will be solved by using ANN as the combination of classi/uniFB01cation and regression problems not as a typical regression problem. The input-layer feature of the training data would be the measurement signals at sensor locations while the output-layer feature would be the classi/uniFB01cation of the possible type of elements (i.e., solid ver- sus /uniFB02uid) in a computational domain and the material properties ( E,G, and) of solid elements (if a given element is classi/uniFB01ed as a solid element). By using the data, an ANN will be trained such that it can classify whether each element is a solid or /uniFB02uid, and, then, once it is classi/uniFB01ed as a solid, the ANN will estimate its material properties. The details on the training of ANN are omitted in this paper because it is quite extensive and beyond the scope of this paper. •Primary inversion of ANN/two.pnumFWI: Once the preliminary inversion is completed, one can /uniFB01ne-tune its result by using the aforementioned primary FWI. Our proposed method described above can be used as both initial screening (i.e., a preliminary inversion using all solid elements) and in-depth cavity characterization (i.e., a primary inversion using /uniFB02uid-solid ele- ments). We also note that initial screening may be done by using other geophysical methods, such as GPR and electrical resistivity. Furthermore, engineers or geologists can somehow know the existence of water- /uniFB01lled underground cavities or internal soil erosion activities because of surface signs (e.g., depression or subsidence) or geological/hydrogeological phenomena (e.g., sinkhole raveling activities) in most of practi- cal geoengineering application sites. Thus, engineers or geologists may have a combination of preliminary data from several di/uniFB00erent methods, including our proposed preliminary inversion, that will greatly bene/uniFB01t our proposed primary inversion using the /uniFB02uid-solid elements in most cases. It is critical, in the geo-engineering analysis and design, to know the dimension and location of an un- /one.pnum/eight.pnum derground cavity. Thus, obtaining the accurate locations and shapes of /uniFB02uid-/uniFB01lled voids and the material properties of surrounding soils from our proposed method will be instrumental to geotechnical engineers who perform further stability analysis by using the result from our proposed method. /seven.pnum Summary We present a new multi-level GA-based inversion method combined with a frequency-continuation scheme. Three levels of GA are used, and, in each level, a pulse signal with a di/uniFB00erent dominant frequency is utilized to tackle the solution multiplicity of the presented inverse problem. Numerical results show the following /uniFB01ndings about the proposed inversion method. First, it is feasible to estimate the depths of solid-/uniFB02uid in- terfaces and Young’s moduli of the surrounding solid layers by employing an acoustic-elastodynamic wave model and a measured dynamic response at a sensor on the top surface. Second, combining the multi- level GA-based optimizer with a frequency-continuation scheme improves the performance of identifying targeted control parameters. Third, the frequency-continuation scheme where the frequency is progres- sively decreased for each GA level is as e/uniFB00ective as the conventional frequency-continuation scheme (i.e., increasing the frequency for each level) in the presented /one.pnumD setting. Lastly, a preliminary inversion using a solid-only model can address the uncertain presence of a /uniFB02uid layer in a domain by uncovering that one of the layers has a very large value of Young’s modulus (with a similar value to that of the bulk modulus of water) and a mass density being similar to that of water. After the preliminary inversion, a primary inversion—based on multiple frequency-level GA using a /uniFB02uid-solid model—can adjust the /uniFB02uid layer’s depth information and the material properties of the soil layers. We also discussed how to extend the presented method in realistic /two.pnumD//three.pnumD settings. Such an extension may e/uniFB00ectively detect water-/uniFB01lled un- derground cavities and also estimate the risk of potential urban cave-in and Karst sinkholes. Please also note that, in typical engineering projects, we do not rely on a single geophysical method no matter how advanced its theories or computation capacities are. Thus, the presented method and its extension can be an important supplement to existing testing methods rather than a stand-alone solution in practice. Data Availability Some or all data, models, or code generated or used during the study are available from the corresponding author by request. •MATLAB code (.m format) of the presented inverse modeling that contains the optimization solver and the forward and adjoint wave solvers. •MATLAB datasets (.mat format) of the presented numerical results. Acknowledgment This material is based upon work supported by the National Science Foundation under Award CMMI- /one.pnum/eight.pnum/five.pnum/five.pnum/four.pnum/zero.pnum/six.pnum and CMMI-/two.pnum/zero.pnum/four.pnum/four.pnum/eight.pnum/eight.pnum/seven.pnum. Any opinions, /uniFB01ndings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily re/uniFB02ect the views of the National Science Foun- dation. The authors are also grateful for the support by the Faculty Research and Creative Endeavors (FRCE) Research Grant at Central Michigan University. /one.pnum/nine.pnum