A Systematic Review on Inverse GRA Methodologies Developed for the Determination of Dynamic Soil Properties Using Downhole Seismic Array Records

Abstract

Effect of local soil in amplifying bedrock motion during earthquakes (EQs) is an important phenomenon, observed globally. As a result, even at larger distances from the epicentre, significant ground shaking and subsequent damages are witnessed. Understanding local soil effect (LSE) requires information about subsoil type as well as corresponding dynamic soil property curves (DSPCs). Though attempts have been made globally to understand local site effect, majority of such studies used DSPCs developed for other regions. In such cases, whether the outcomes represent actual surface seismic scenario for the region under study, is a matter of debate. It must be mentioned here that while the dynamic soil properties can be determined in laboratory, such findings are affected by sampling disturbance, equipment compliance etc. Downhole seismic array records on the other hand provide actual soil response during various EQs and thus are very useful in determining regional DSPCs through inverse ground response analysis (IGRA). To do IGRA, two main methodologies (based on time domain and frequency domain) exist. This work firstly presents a systematic review on existing IGRA methodologies (both time domain and frequency domain based). Later, in the light of the importance of IGRA to determine regional DSPC, few Research Questions (RQs) are framed. Answers to such RQs are then found in terms of Research Findings (RFs) from the systematic reviews highlighting the limitations of existing methodologies, importance of ground motion characteristics in different methodologies. Based on the present endevour, it is concluded that though frameworks to determine shear modulus (G) based on IGRA exist, no such framework to is available determine damping ratio (β).

Appendices

Appendix A

NLGRA

In NLGRA, the following equation of motion (Eq. 22) is solved numerically at every time step [63]. This time step can be the default time step at which the acceleration values are recorded or some other value set for interpolation.

$$ \left[ M \right]\left\{ {\ddot{u}} \right\} + \left[ C \right]\left\{ {\dot{u}} \right\} + \left[ K \right]\left\{ u \right\} = - \left[ M \right]\left\{ I \right\}\ddot{u}_{g} $$

(A1)

where $\left[M\right]$ is the mass matrix, $\left[C\right]$ is the viscous damping matrix, $\left[K\right]$ is the stiffness matrix, $\left\{\ddot{u}\right\}$ is the nodal relative acceleration vector, $\left\{\dot{u}\right\}$ is the nodal relative velocity vector, $\left\{u\right\}$ is the nodal relative displacement vector. ${\ddot{u}}_{g}$ is the acceleration at the base of the soil column and $\left\{I\right\}$ is the unit vector. Further, the stiffness matrix and damping matrix are updated in each time step to incorporate nonlinear soil behavior. It should be highlighted here that thesoil behavior during an EQ loading is very complex. Depending on the developed γ levels in soil layers, G and β values change throughout the duration of EQ shaking. As γ increases, G decreases, and β increases. The values of G and β at each time step can be obtained by the use of an appropriate constitutive model (stress–strain model), the parameters of which can be adjusted based on DSPC.

ELGRA

The displacement at any point in a soil layer can be expressed as [63];

$$ u_{s} = A_{s} e^{{i\left( {\omega t + k_{s}^{*} z_{s} } \right)}} + B_{s} e^{{i\left( {\omega t - k_{s}^{*} z_{s} } \right)}} $$

(A2)

where A_sis the amplitude of the upward propagating wave and B_sis the amplitude of downward propagating wave. ωis the angular frequency of loading, Z_s is the depth coordinates measured in the downward direction from the top of the soil layer, ${k}_{s}^{*}$ is the complex wavenumber for soil layer, which is defined as;

$$ k_{s}^{*} = \frac{\omega }{{V_{s}^{*} }} = \frac{\omega }{{V_{s} \left( {1 + i\beta } \right)}} $$

(A3)

Further, theγ can be estimated as,

$$ \gamma \left( {z,t} \right) = ik_{s}^{*} \left[ {A_{s} e^{{i\left( {\omega t + k_{s}^{*} z_{s} } \right)}} - B_{s} e^{{i\left( {\omega t - k_{s}^{*} z_{s} } \right)}} } \right] $$

(A4)

In ELGRA, the γ value in each of the soil layers involved are estimated first using Eq. A3. While, calculating the γvalue, initially, low γcorresponding G and β values are assigned to each of the soil layers. Once, γ value is calculated, new values of G and β are selected from G/G_max curve andβ curve, respectively based on theγ value. This way the analysis move forward until the difference between two succeeessive G and β values fall below a certain percentage. Once the G and β values for all the layers are finalized. Response in any layer can be obtained by considering the Eqations mentioned in [63, 70].

Appendix B

List of search terms

• Downhole array

• Modulus degradation

• Damping

• Ground response analysis

• Local site effects

• Inverse analysis