. The e ect of \velocity reversals" (soft soil layers con ned between sti er ones) over 1D Site Response calculations has been studied for many years now, but a complete characterization of their in uence is still pending. In a recent article titled Never Fear Velocity Reversals , Yagoda-Biran, Kerpel and Kamai analyzed a set of three-layer systems with di erent properties. Their study was focused on numerical calculations leading to insights concerning the in uence of the reversal over the fundamental resonant mode. The same systems are analyzed in this note, yet from a theoretical standpoint in this case, working directly with the analytical transfer functions. A new interpretation of their results is provided, along with a reasoning that evinces diculties when it comes to extrapolating insights to more complex layerings. 1.Introduction The analysis of wave propagation in layered media is, naturally, one of the recur- rent themes in Geotechnical Earthquake Engineering and Seismology. In those cases where the intensity of the wavefront is small enough to not to trigger irreversible deformation in the medium, Linear Elastodynamics has proved itself an adequate tool to model the seismic wave propagation phenomenon. Those occasions when it fails entail large deformation of the soil, carrying irreversible changes, are usually associated to propagation in the shallow softer layers of the earth. Assuming the earthquake source is located deep enough, the front impinges over the ground almost-vertically, as the decreasing sti ness bends the rays towards the normal of the free surface [1]. Under the assumption of one-dimensional propaga- tion perpendicular to both the free surface and to all other internal interfaces, the di erent components decouple and SH, SV and P waves can be studied separately and then summed back, given that there is no mode conversion. An approach to assess the evolution of the wave eld across the layers is the \Propagator Matrix Method" [2]. The classic Seismology textbooks [1, 3] present a general treatment for inclined SH waves and inclined P-SV waves. No inclination is considered here, thus the expressions that will appear represent a particular case. An approach for the 1D case, often used in geotechnical engineering, is presented in [4]. There is no de nitive understanding over the e ect of each layer over the nal global transfer function [5]. In particular, the e ect of so-called velocity reversals is still inquired nowadays. The term \velocity reversal" refers to intermediate layers that break the monotonicity of the sti ness distribution along depth: in usual circumstances, one encounters that the wave velocity increases the deeper the layer Date : June 2021. Civil Engineering Institute Ecole Politechnique F ed erale de Lausanne (EPFL), Lausanne, Switzerland . 1 is (this is not only due to di erent geological conditions, but also to the cumulative e ect of overburden pressure), a velocity reversal is a layer that does not abide by this trend, as it is softer than both the layer immediately above and the one immediately below. One could think of it as a soft inclusion in a sti medium. Yagoda-Biran, Kerpel and Kamai [6] carried out a thorough parametric inves- tigation on the e ect of the velocity reversals over the displacement-to-amplitude transfer function of a two-layer system over a half-space. Chie y, they found that: (1) The amplitude of the tranfer function at the rst resonance peak scales proportionally to the impedance ratio between reversal and half-space in a 3-layer system (bottom layer being the half-space). (2) A criterion to ensure minor in uence of reversals can be furnished for such systems. This work tackles the same problem but from a purely analytical basis. Utiliz- ing the exact expression of the transfer function (which can be obtained from the propagator matrices), an argument based on relative magnitude of di erent terms during the ( rst) resonance peak will be leveled. All the results are made available to the research community: the Mathematica notebook utilized to carry out the computations can be found on GitHub (see Supplementary Material). 2.Scaling of fundamental mode amplitude The transfer function for a system made up of 2 layers overlying a half-space are given next. See the appendix for derivation of these results. The width, shear modulus, density and shear-wave velocity of the top (bottom) layer are h1,1,1, Vs1(h2,2,2,Vs2). The circular frequency of the wave is denoted by !. The expression studied by Madera [7] gives the displacement-displacement trans- fer function: TF(!)1=^utop ^uhalf= cos(r1) cos(r2)Z1sin(r1) cos(r2); (1) wherer1=h1!=Vs1,r2=h2!=Vs2, ^utopis the spectral amplitude of the displace- ment at the ground level while ^ uhalfis at the interface between the bottom layer and the half-space, Z1= (11=22)1=2is the impedance contrast between rst and second layer. Likewise, the displacement-amplitude transfer function is: 2TF(!)1=^utop S= cos(r1) cos(r2)Z1sin(r1) cos(r2) + iZ1Z2sin(r1) cos(r2) + iZ2sin(r2) cos(r1); (2) where Srefers to the impinging S-wave amplitude coming from the half-space, while Z2= (22=halfhalf)1=2is the impedance contrast between the second layer and the half-space. Let us state the obvious relation between the two transfer functions: 2 TF=1 TF+ iZ1Z2sin(r1) cos(r2) + iZ2sin(r2) cos(r1): (3) The consequential e ect of the presence of the velocity reversal (second layer) is thatZ1>1, whereas Z2<1. See how it follows that Z2=Z1< Z 2and hence we may assume Z2=Z1<< 1 for as long as there is a velocity reversal and a 2 bedrock halfspace sti er than the upper layers. The possibility of exploiting Z2=Z1 as if it was a small parameter in the expression allows to pursue the following argumentation. Let us focus on 2TF1=Z1, which, from eq. (2), is 2TF(!)1 Z1=cos(r1) cos(r2) Z1sin(r1) cos(r2) + iZ2 Z1(cos(r1) sin(r2) +Z1cos(r2) sin(r1)); (4a) let us call"=Z2=Z1and assume " 1, so =cos(r1) cos(r2) Z1sin(r1) cos(r2) +"i (sin(r1) cos(r2) +Z1sin(r2) cos(r1)); (4b) and, see eq. (3), recognize the rst two addens as TF1=Z1, =TF1=Z1+O("): (4c) This last result means that the displacement-to-amplitude transfer function, TF, can be thought to be the displacement-to-displacement transfer function plus a small correction term order ". In particular, see that, in absence of material damp- ing dissipation,jTF1j0 during resonance (resonance frequency !=!1), so TF1=Z1j!=!1=O("), the residual amplitude (which is small but remains nite even in the case of no material damping) corresponds to the amplitude of the fun- damental resonance mode, and its boundedness (unlike in the case of the other transfer function) can be interpreted as a consequence of the radiational damping, which is acounted for by the imaginary addends in eq. (2), absent from eq. (1). Another remarkable consequence of eq. (3) is that, since the resonance peak for both TF and TFcoincides, the resonance frequency can be thought as independent of the properties of the half-space; in other words, the presence of the reversal decouples the resonance of the top strata from the features of the half-space. Let us unpack the amplitude at resonance next. One can estimate both upper and lower bounds of the amplitude during resonance. First, the lower bound: jTF(!=!1)j2 jZ1Z2cos(r1) sin(r2) +Z2cos(r2) sin(r1)j!=!1(5a) since both since and cosine are bounded by 1, 2 Z1Z2+Z2=2 Z1Z2(1 + 1=Z1|{z} <1)2 Z1Z2= 2rhalfhalf 11: (5b) See that this lower bound depends on the properties of the upper layer yet not on the reversal's, even though the result is derived exploiting its presence. Additionally, an upper bound for the amplitude at resonance can also be argued: jTF(!=!1)j2 jZ1Z2cos(r1) sin(r2) +Z2cos(r2) sin(r1)j!=!1(6a) =2=Z2 jZ1cos(r1) sin(r2) + cos(r2) sin(r1)j!=!1; (6b) 3 the rst addend in the numerator can be greater than one, unlike the second addend; if we want to minimize the denominator this term must be deactivated and such thing would happen when r2 1, thus sin( r2)0, while cos( r2)1, so 2=Z2 jsin(r1)j; (6c) moreover, assuming that the denominator is close to 1 in this situation, nally yields 2 Z2= 2rhalfhalf 22: (6d) The properties of the upper layer do not appear in the formula of the upper bound. 2.1.Veri cation. In order to verify the results we resort to computing numerical transfer functions and comparing them to eq. (5b) and eq. (6d). The set of pro les to be analyzed is the one presented in [6]: there are a total of 140 pro les, the upper layer shear-wave velocity ranging from 400 to 800 m =s, in increments of 100 m =s, while the lower layer shear-wave velocity ranges between 300 and 800 in increments of 50 m =s; only the combinations that yield Vs1Vs2 are considered. The density in each layer is assumed to be a linear function of the wave velocity in the layer de ned by assigning density 1800 kg =m3when the velocity is 200m =s and 2100 kg =m3when the velocity is 900 m =s. The thickness of the two layers sum up to 50 m in all cases, the breakdown being de ned by h1=h2= 10=40;20=30;30=20;40=10. The damping factor used in [6] was 10% across all layers; the prior derivation of the bounds did not include the damping for the sake of clarity, but it can be added on the derivation easily, just consider that jTF(!=!1)1j=4d[8] (instead of zero). Thus, the expression of the lower bound, originally eq. (5b) that includes material damping is jTF(!=!1)j2  4d+q11 halfhalf(7a) while the upper bound, after updating eq. (6d) in similar fashion, jTF(!=!1)j2  4d+q22 halfhalf: (7b) One acknowledges that the numerical results lie below eq. (7b), see Figure 1, and over eq. (7a), Figure 2. Since the results lie between an upper bound, that depends only on the properties of the half-space and the rst layer's, and a lower bound that depends only on the properties of the half-space and the second layer's, it is logical to think that the actual amplitudes must depend on some function of the three (upper, lower and half- space). In [6], the authors put forward the idea of homogenizing the two layers into an equivalent one with velocity given by the \time-averaged velocity" (harmonic mean), then compared its resonance amplitude to the one of the strati ed model, obtaining remarkable agreement (even though the harmonic mean is known not to be optimal to homogeneize a continuum that undergoes fundamental resonance [9]). 4 0.3 0.4 0.5 0.6 0.7 0.8 0.90123456 (ρ2μ2/ρhalfμhalf)12AmplitudeVsL1=400m/s VsL1=500m/s VsL1=600m/s VsL1=700m/s VsL1=800m/sFigure 1. Veri cation upper bound: amplitude of rst resonance peaks as a function of impedance contrast bottom stratum (re- versal) v. half-space, for di erent combinations of parameters. Markers relate to thickness ratios: circles, 10/40; triangles, 20/30; squares, 30/20; and diamonds, 40/10. 3.Reversals in more complex lithography: no easy assessment The prospect for a de nition of small parameter and for dividing the terms in clear-cut groups of di erent magnitude vanishes when moving to more complex scenarios. Recall that the small parameter was the key to the split of the transfer function into two parts, one controlling the position of the fundamental mode, and the second one controlling its amplitude. This feature being absent leaves no room for a reasoning analogous to the one carried out for the 2-layer one. Let us illustrate this situation regarding at a 3-layer site overlying a half-space. In such a scenario the reversal can be located either between the rst and the third layer ( Z1>1), or between the second and the half-space ( Z2>1). The 3-layer-over-halfspace transfer function is 2TF(!)1= cos(r1) cos(r2) cos(r3)Z1sin(r1) sin(r2) cos(r3) Z1Z2sin(r1) cos(r2) sin(r3) Z2cos(r1) sin(r2) sin(r3) iZ1Z2Z3sin(r1) cos(r2) cos(r3) iZ2Z3cos(r1) sin(r2) cos(r3) iZ3cos(r1) cos(r2) sin(r3) + iZ1Z3sin(r1) sin(r2) sin(r3); (8) See how the split on the grounds of the assumption of one velocity reversal in either position is not straightforward, as one can not trace a division in terms of 5 0.4 0.5 0.6 0.7 0.8 0.90123456 (ρ1μ1/ρhalfμhalf)12Amplitude VsL1=400m/s VsL1=500m/s VsL1=600m/s VsL1=700m/s VsL1=800m/sFigure 2. Veri cation lower bound: amplitude of rst resonance peaks as a function of impedance contrast top stratum v. half- space, for di erent combinations of parameters. Markers relate to thickness ratios: circles, 10/40; triangles, 20/30; squares, 30/20; and diamonds, 40/10. magnitude the way it was done in the previous section, which may have enabled pursuing a similar argumentation. 4.Final Remarks An analytical interpretation of the results presented in [6] has been laid out in the text. The key step in the derivation was to leverage the presence of a small parameter in order to split the displacement-amplitude transfer function into a por- tion with larger magnitude that still vanishes during resonance (identi ed with the displacement-displacement transfer function) and small-magnitude term represent- ing the radiational damping which controls the amplitude during resonance. The validity of the assumption, upon which the rest of the results rest, has been accredited with a derivation of an upper and a lower bound of the amplitude, which naturally follows from the simpli ed expressions. These bounds have also been veri ed later via comparison to numerical results obtained through the conventional approach (evaluating propagator matrices). In summary: In the 2-layer case, the presence of the reversal renders the fundamental resonance frequency independent of the material properties of the half- space. The e ect of velocity reversals over the dynamic response can be described precisely in the 2-layer case, but doing so is not straightforward when it comes to sites with more than two layers overlying a rigid half-space. 6 This work also proves the possibility of extracting useful general insights by working directly in analytical terms. Let us mention, just like it was done in the original paper [6], that the results are circumscribed to small-intensity seismic-wave propagation owing to the linearity assumption. A further study that could account for the e ect of large magnitude earthquakes on the soil must take into account non-linear material response neces- sarily [10]. Acknowledgements The author is grateful to Alexey Popkov ( github.com/AlexeyPopkov ), since code developed by him has been used to generate the plots. Supplementary Material AMathematica notebook titled Reversals containing the evaluation of the re- sults in this manuscript can be found in the author's Github page: github.com/jgarciasuarez .