These authors contributed equally to this work.

The adverse effects of climate warming on the built environment in (sub-)arctic regions are unprecedented and accelerating. The planning and design of climate-resilient northern infrastructure, as well as predicting deterioration of permafrost from climate model simulations, require characterizing permafrost sites accurately and efficiently. Here, we propose a novel algorithm for the analysis of surface waves to quantitatively estimate the physical and mechanical properties of a permafrost site. We show the existence of two types of Rayleigh waves (R1 and R2; R1 travels faster than R2). The R2 wave velocity is highly sensitive to the physical properties (e.g., unfrozen water content, ice content, and porosity) of active and frozen permafrost layers, while it is less sensitive to their mechanical properties (e.g., shear modulus and bulk modulus). The R1 wave velocity, on the other hand, depends strongly on the soil type and mechanical properties of permafrost or soil layers. In situ surface wave measurements revealed the experimental dispersion relations of both types of Rayleigh waves from which relevant properties of a permafrost site can be derived by means of our proposed hybrid inverse and multiphase poromechanical approach. Our study demonstrates the potential of surface wave techniques coupled with our proposed data-processing algorithm to characterize a permafrost site more accurately. Our proposed technique can be used in early detection and warning systems to monitor infrastructure impacted by permafrost-related geohazards and to detect the presence of layers vulnerable to permafrost carbon feedback and emission of greenhouse gases into the atmosphere.

Permafrost is defined as the ground that remains at or below 0

Within the permafrost, the distribution of ice formations is highly variable. Ground ice can be present in distinctive forms including (1) pore ice, (2) segregated ice, and (3) ice wedges

The design and construction of structures on permafrost normally follow one of two broad principles which are based on whether the frozen foundation soil in ice-rich permafrost is thaw stable or thaw unstable. This distinction is determined by the ice content within the permafrost. Ice-rich permafrost contains ice in excess of its water content at saturation and is thaw unstable

Several in situ techniques have been employed to characterize or monitor permafrost conditions. For example, techniques such as remote sensing

Non-destructive seismic testing, including multi-channel analysis of surface waves (MASWs)

Here, we present a hybrid inverse and multiphase poromechanical approach for the in situ characterization of permafrost sites using surface wave techniques. The forward solver is used to numerically calculate the physics-based dispersion curves for both R1 and R2 wave modes given the soil properties. The inverse solver is used to inversely obtain the physical and mechanical properties of soils given the seismic measurements. In our method, we quantify the physical properties such as ice content, unfrozen water content, and porosity, as well as the mechanical properties such as the shear modulus and bulk modulus of permafrost or soil layers.

We also determine the depth of the permafrost table. The role of two different types of Rayleigh waves in characterizing the permafrost is presented based on an MASW seismic investigation at a field site located at SW Spitsbergen, Svalbard. Multiphase poromechanical dispersion relations are developed for the interpretation of the experimental seismic measurements at the surface based on the spectral element method. Our results demonstrate the potential of seismic surface wave testing accompanied by our proposed hybrid inverse and poromechanical dispersion model for assessment and quantitative characterization of permafrost sites.

Figure

We consider the frozen soil specimen to be composed of three phases: solid skeletal frame, pore water, and pore ice. Through the infinitesimal kinematic assumption (Eq.

To obtain the spectral element solution, the Helmholtz decomposition is used to decouple the P waves (P1, P2, and P3) and S waves (S1 and S2).

The displacement vector (

The detailed steps for obtaining the closed-form solutions for P waves and S waves using the eigendecomposition are summarized in Appendix

The dispersion relation of Rayleigh waves is obtained by setting a zero stress condition at the surface (

The aim function is defined as the Euclidean norm between the experimental and numerical results of the dispersion relations. The problem is formulated in Eq. (

In this paper, we used the neighborhood algorithm that benefits from the Voronoi cells to search the high-dimensional parameter space and reduce overall cost function

From a poromechanical point of view, permafrost (frozen soil) is a multiphase porous medium that is composed of a solid skeletal frame and pores filled with water and ice in different proportions. Here, we analyze the seismic wave propagation in permafrost based on the three-phase poroelastodynamic theory. Three types of P wave (P1, P2, and P3) and two types of S wave (S1, S2) coexist in three-phase frozen porous media

In this paper, a uniform frozen soil layer is used to show the propagation of different types of P and S waves and subsequently the formation of Rayleigh waves (R1 and R2) at the surface. It is assumed that an impulse load with a dominant frequency of 100 Hz is applied at the ground surface. The wave propagation analysis was performed in clayey soils by assuming a porosity (

The phase velocities of R1 and R2 waves are a function of physical properties (e.g., degree of saturation of unfrozen water, degree of saturation of ice, and porosity) and mechanical properties of the solid skeletal frame (e.g., bulk modulus and shear modulus). Figure

Our results also show that an increase in the degree of saturation of ice leads to an increase in the phase velocity of both types of Rayleigh waves. An increase in porosity leads to an increase in the phase velocity of R2. However, an increase in porosity may lead to either a decrease or an increase in the phase velocity of the R1 wave, depending on the level of the degree of saturation of ice. Hence, we use the phase velocity of R2 waves identified by processing the seismic surface wave measurements to characterize the physical properties (e.g., porosity, degree of saturation of ice, or degree of saturation of unfrozen water) of permafrost or soil layers.

The field experiment used in this study was performed by

Surface wave measurement in section 1 (from 0 to 120 m).

In our simulations, the permafrost site is modeled as a three-layered system, consisting of an active layer at the surface followed by a permafrost layer on top of the third layer (permafrost or unfrozen ground, which is to be determined). The ERT results reported by

The mechanical properties of the solid skeletal frame in each layer are then obtained using the R1 wave dispersion relation. The mechanical properties can be then used to determine whether the permafrost site is ice-rich. In fact, the thin ice lenses can not be detected directly when the thickness of ice lenses is smaller than

Figure

Surface wave inversion results for section 1: 0 to 120 m.

Figure

Inversion process for the R2 wave dispersion relation.

We have previously shown the inversion process and results for section 1 from 0 to 120 m. Five additional sections spanning from 120 to 600 m were also studied using a similar approach. The seismic measurements and dispersion relations for each section are given in Appendix

Summary of the inversion results at the offset distance from 0 m to 600 m.

We developed a hybrid inverse and multiphase poromechanical approach to quantitatively estimate the physical and mechanical properties of a permafrost site. The identification of two distinctive types of Rayleigh waves in the surface wave field measurements at permafrost sites is critical for the quantitative characterization of the layers. The identification of the R2 wave allows for the quantitative characterization of physical properties of soil layers independently without making assumptions of the mechanical properties of the layers. This approach simplifies the inversion of the multi-layered three-phase poromechanical model since the dependent optimization variables are largely reduced. The inversion results from the R2 wave dispersion relation can be further used in the characterization of the mechanical properties of soil layers based on the R1 wave dispersion relation. This also increases the stability and convergence rate of the inversion solver and makes the analysis more efficient than the joint inversion analysis.

Additional work on the characterization of permafrost should explore ways to reduce the uncertainty in the proposed hybrid inverse and multiphase poromechanical approach. The uncertainty originates from the non-uniqueness in the inverse analysis (local minima problem) and the limited number of constraints in the inversion analysis. It is recommended to use other geophysical methods to improve the resolution and reduce the uncertainty of the permafrost mapping. With the proposed seismic-wave-based method as the main investigation tool, ERT, GPR, and electromagnetic (EM) tomography can augment the investigation data and supply additional constraints to the inversion analysis.

In this paper, our results demonstrate the potential of seismic surface wave testing accompanied by our proposed hybrid inverse and poromechanical dispersion model for the assessment and quantitative characterization of permafrost sites. Its application for early detection and warning systems to monitor infrastructure impacted by permafrost-related geohazards and to detect the presence of layers vulnerable to permafrost carbon feedback and emission of greenhouse gases into the atmosphere will be the goal of our future studies. Currently, there is no advanced physics-based monitoring system developed for the real-time interpretation of seismic measurements. As such, active and passive seismic measurements can be collected and processed using the proposed hybrid inverse and poromechanical dispersion model for the assessment and quantitative characterization of permafrost sites at various depths in real time. In a future study, we will focus on the development of an early warning system for the long-term tracking of permafrost conditions. The early warning system can be used to collect seismic measurements and predict the physical and mechanical properties of the foundation permafrost. The system then reports periodic variations in physical (mostly ice content) and mechanical properties of the permafrost being monitored. The same method being applied on different dates (e.g., seasonal basis) can be used to record the change in properties of the permafrost site and then warn about the degradation of the permafrost exceeding the threshold. The determination of the value of the threshold (or critical values) will require more in-depth research. The early detection and warning systems can be beneficial in monitoring the condition of the foundation permafrost and preventing excessive thaw settlement and significant loss in strength. Similarly, we can detect the presence of peat (based on the physical and mechanical properties) which is vulnerable to permafrost carbon feedback and emission of greenhouse gases into the atmosphere. It is reported that the soils in the permafrost region hold twice as much carbon as the atmosphere does (almost 1600 billion tonnes)

The velocities of the three types of P waves are determined by a third-degree characteristic equation

The inversion results for the sections ranging from 120 to 600 m are summarized in Fig.

Surface wave inversion results for section 2: 120 to 240 m.

Surface wave inversion results for section 3: 240 to 360 m.

Surface wave inversion results for section 4 (from 360 to 480 m).

Surface wave inversion results for section 5 (from 480 to 600 m).

Summary of dispersion measurements for sections 1 to 5.

The Green–Lagrange strain tensor (

The strain tensor of pore water

The constitutive models defined as the relation between the stress and strain tensors for solid skeleton, pore water, and pore ice are given in Eq. (

The momentum conservation considers the acceleration of each component and the existing relative motion of the pore ice and pore water phases with respect to the solid skeleton. The momentum conservation for the three phases is given by Eq. (

Through the infinitesimal kinematic assumptions, the stress–strain constitutive model, and conversation of momentum, the field equation can be written in the matrix form, as shown in Eq. (

By performing divergence operation (

Equation (

By setting

Now, the P wave potentials can be written as follows:

The solutions for the S wave potentials can be solved in a similar manner. Eq. (

Since

Finally, the solution of the S wave potentials can be written as follows:

By including both incident wave and reflected wave, the potentials
for a layer with finite thickness can be written as in Eq. (

The matrix of effective stress, pore water pressure, and pore ice pressure in the frequency domain is shown in Eq. (

According to the Cauchy stress principle, the traction force (

By assuming that no wave reflects back to a semi-infinite element, a one-node element with infinite thickness is applied. The matrix for the displacement components in the one-node layer is written as Eq. (

The stiffness assembling method is shown in Fig.

Construction of the global stiffness matrix in which

The matrices

An example is given to further explain and validate the decomposition of the global stiffness matrix. It is assumed that the porosity is 0.5 for all three layers; the degree of saturation of unfrozen water is 0.1, 0.3, and 0.6; the shear modulus of soil skeleton is 6.85, 10, and 10 GPa; the bulk modulus of soil skeleton is 15, 15, and 21 GPa. Figure

Decomposition of the global stiffness matrix.

The components of the

The components of the

The data and code that support the findings of this study can be found in

Conceptualization was done by HL and PM; methodology was developed by HL and PM; investigation and data visualization were preformed by HL and PM; HL drafted the original manuscript; and PM and AS reviewed and edited the manuscript.

Pooneh Maghoul has the patent Systems and Methods for In-situ Characterization of Permafrost Sites pending, together with Hongwei Liu, Guillaume Mantelet, and Ahmed Shalaby, University of Manitoba.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to acknowledge the National Science Centre, Poland (NCN) UMO-2016/21/B/ST10/02509, for the support of the MASW permafrost measurements. The authors are grateful to Mariusz Majdański, Artur Marciniak, and Bartosz Owoc for sharing the data. The authors also acknowledge the financial support of the New Frontiers in Research Fund – Exploration Grant (NFRF-2018-00966), the Natural Sciences and Engineering Research Council of Canada (NSERC) – Discovery Grant program (RGPIN-2016-06019), the Mathematics of Information Technology and Complex Systems (Mitacs) Accelerate program, and the University of Manitoba Graduate Enhancement of Tri-Council Stipends (GETS) program.

This research has been supported by the New Frontiers in Research Fund – Exploration Grant (grant no. NFRF-2018-00966), the Natural Sciences and Engineering Research Council of Canada (NSERC) – Discovery Grant program (grant no. RGPIN-2016-06019), the Mathematics of Information Technology and Complex Systems (Mitacs) Accelerate program, and the University of Manitoba Graduate Enhancement of Tri-Council Stipends (GETS) program.

This paper was edited by Evgeny A. Podolskiy and reviewed by Ludovic Moreau, Rowan Romeyn, and one anonymous referee.