Gassy soils have been widely discovered in shallow soil layers in eastern costal area of China, such as Hangzhou Bay area, Zhejiang province. It was generated by the anaerobic decomposition process of organic materials [1] . Since main compositions of biogenic gas is flammable gas (e.g., methane CH₄), the presence of shallow gassy soils may cause significant risk for infrastructure construction, such as fire outbreaks and blasting during underground construction [2] . Therefore, proper characterization of spatial distribution of gassy soils is indispensable prior to the construction of underground projects. Due to the significant cost and human commitments required in the site investigation for gassy soils, only a limited number of gas pressure data can be obtained in engineering practice, which leads to the uncertainty in characterizing spatial distribution of gassy soils. Determining the site investigation scheme (including the number and locations of boreholes) is pivotal to reducing construction risk induced by gassy soils.

Since only a limited number of borehole data can be obtained during geotechnical site investigation, it is prudent to carefully determine the number and locations of boreholes so as to maximize the value of data. However, this is, in general, fundamentally challenging task due to the spatial variability of geotechnical materials and a lack of handy methods and tools for this purpose. Some studies have been performed to investigate the gassy soils in the Hangzhou Bay area. However, most of them focused on the formation and composition of biogenic gas [2] , features and distributions of gas pools [3] , and the exploration methods ( [4] [5] ). Research is rare that focused on how to, efficiently and properly, characterize the spatial distribution of gassy soils with a limited number of borehole data. Determination of the optimal scheme for investigating gassy soils prior to site investigation remains an open question.

This study presents a probabilistic site investigation optimization method to determine the optimal scheme of investigating gassy soils. The proposed method makes uses of the prior knowledge in a quantitative and transparent manner, and explicitly models the spatial variability of gassy soils by the conditional random field. The paper starts with introduction of the proposed method, followed by illustration and verification by a case study in Hangzhou Bay area.

Site investigation of gassy soils are usually carried out to measure and estimate the gas pressure of gassy soils at different locations with a limited number of boreholes. In this research, the number and locations of measuring boreholes are of interest. The main problem solved in this study is to obtain the optimal investigation scheme. Let L and ∆L denote the length of site investigation field concerned and the interval between adjacent boreholes. The investigation schemes (including the number and locations of boreholes) can be represented by a set, $S=\left\{i\Delta L,i=1,2,3,\cdots ,N_B\right\}$, of borehole locations in the horizontal direction, where N_B = INT[L/∆L] denotes the number of locations and INT[·] is a round function that returns the integer part of L/∆L. Note that this study only accounts for one-dimensional spatial variability of gas pressure in the horizontal direction, and the vertical spatial variability is ignored, which can be pursued in future study.

Since no data is available prior to site investigation, the gas pressure of the N_B locations can be arbitrary values. For planning investigation schemes, simulated gas pressure data at some investigation locations are generated using the prior knowledge, which will be introduced in the next subsection. Let $Z_{br}=\left\{P{R}_i,i=1,2,3,\cdots ,N_B\right\}$ be a set of simulated data for a given investigation scheme S, and it consists of gas pressure data PR_i at the N_B borehole locations. Using Z_br, a Kriging-based conditional random field model is applied to modelling the horizontal spatial variability of gas pressure, by which the gas pressure in horizontal direction can be written as [6] [7] :

$Z_c\left(x\right)=Z_k\left(x\right)+\left(Z_{uc}\left(x\right)-Z_{sk}\left(x\right)\right)$(1)

where x is the horizontal coordinate; Z_c(x) is the conditional random field; Z_k(x) is the Kriging interpolation of gas pressure over the domain of interest based on the Z_br [8] ; Z_uc(x) is the unconditional random field of gas pressure; Z_sk(x) is the Kriging interpolation of gas pressure over the domain of interest based on the values of gas pressure at borehole locations simulated via the unconditional random fields. Equation (1) ensures that the realizations of random fields exactly match the Z_br, for which the ordinary Kriging method is employed herein to perform the interpolation over the domain of interest.

According to Section 2.1, the gas pressures (i.e., Z_br) at the borehole locations are needed to simulate the spatial variability using Equation (1). Unfortunately, there is no measuring data prior to site investigation. For planning the investigation scheme, the simulated data can be generated using the prior knowledge (e.g., engineering experience and judgments) on the gas pressure, such as typical ranges of statistics of gas pressures. Consider, for example, that the mean values μ, standard deviations σ and scale of fluctuation λ vary within their respective typical ranges [μ_min, μ_max], [σ_min, σ_max], and [λ_min, λ_max]. Then, the μ, σ and λ can be treated as uniform random variables defined by their respective typical ranges and their random samples can be generated. Let μ_s,_i, σ_s,_i, and ${\lambda }_{s,i}\left(i=1,2,3,\cdots ,N_e\right)$ denote a number, N_e, sets of random samples of μ, σ and λ simulated from the prior knowledge. For each set of μ_s,_i, σ_s,_i and λ_s,_i, a set of the simulated data $Z_{s,i}\left(i=1,2,3,\cdots ,N_e\right)$ is simulated using Karhunen-Loeve (KL) expansion ( [9] [10] ) in this study, which is written as:

$Z_{s,i}\left(x\right)={\mu }_{s,i}+{\displaystyle \underset{j=1}{\overset{\infty }{\sum }}{\sigma }_{s,i}\sqrt{v_j}}{f}_j\left(x\right)\zeta \left(\theta \right)$(2)

where Z_s,_i is the gas pressure simulated using the sample μ_s,_i, σ_s,_i and λ_s,_i; ζ(θ) is independent standard normal random variable; v_j and f_j(x) are the eigenvalues and eigenfunctions of the covariance function, which is taken as a squared exponential correlation function in this study:

$\rho \left(\tau \right)=\exp \left[-\pi {\left(\tau /{\lambda }_{s,i}\right)}^2\right]$(3)

where τ is the separate distance between two locations in the horizontal direction; ρ(τ) is the autocorrelation coefficient between the gas pressures at the two locations. For the sake of conciseness, details of the random field simulation based on KL expansion are not provided here. Interested reader may refer to Huang et al. (2001) [9] and Phoon et al. (2002) [10] for more details.

Using KL expansion, the $Z_{s,i}\left(i=1,2,3,\cdots ,N_e\right)$ can be simulated using each set of samples of μ, σ and λ_, i.e., μ_s,_i, σ_s,_i and λ_s,_i. Then, the gas pressures at borehole locations in an investigation scheme S in Z_s,_i are taken as a simulated data at these locations, denoted by Z_br,i.

For a given set of μ_s,_i, σ_s,_i and λ_s,_i and its corresponding Z_br,i simulated from KL expansion in the preceding subsection, Monte Carlo simulation is performed with Equation (1) to determine the probability distribution of the gas pressure at each location. For this purpose, the Z_br,i is used in Equation (1) to obtain Z_k(x) using Kriging interpolation and μ_s,_i, σ_s,_i and λ_s,_i are used to simulate Z_uc(x) using KL expansion again. By this means, for each set of Z_br,i, a number, N_a, of realizations of conditional random field of the gas pressure can be generated using Equation (1). The N_a realizations, Z_c,i, of the conditional random field consist of the N_a values of the gas pressure at each location. Consider, for example, a threshold value, R, of the gas pressure. As the gas pressure of some location x is greater than R, it is considered risky; otherwise, it is ignorable. These two situations are represented by a pair of complementary events E₀ (Ignorable event) and E_r (Risky event), with respective probabilities of p_r and p₀, which are calculated as

$p_0=\frac{1}{N_a}{\displaystyle \underset{k=1}{\overset{N_a}{\sum }}I\left[Z_{c,i}^k\left(x\right)<R\right]}$(4)

$p_r=1-p_0$(5)

where $Z_{c,i}^k\left(x\right)$ is the gas pressure at position x in the k-th realization of the conditional random field based on Z_br,i; N_a is the total number of realizations of the conditional random field; I[·] is the indicative function, and it is equal to 1 if $Z_{c,i}^k\left(x\right)<R$ or 0 otherwise.

In order to quantify the uncertainty in the presence of gassy soils, Monte Carlo simulation is adopted to repeatedly simulate gas pressure from the prior knowledge. Based on each set of simulated data $Z_{br,i}\left(i=1,2,\cdots ,N_e\right)$, the probabilities of E₀ and E_r are calculated as p₀,_i and $p_{r,i}\left(i=1,2,\cdots ,N_e\right)$ using Equations (4)-(5). After that, the mean values (i.e., p₀,_e and p_r,_e) of p₀ and p_r corresponding to the N_e sets of simulated data $Z_{br,i}\left(i=1,2,\cdots ,N_e\right)$ are obtained:

$p_{\text{0,}e}=\frac{\text{1}}{N_e}{\displaystyle \underset{i=1}{\overset{N_e}{\sum }}{p}_{\text{0},i}}$(6)

$p_{r,e}=1-p_{0,e}$(7)

where N_e is the total number of simulated data $Z_{br,i}\left(i=1,2,\cdots ,N_e\right)$. Then, the larger value between p₀,_a and p_r,_a is used to determine whether gassy soils present at some location or not, i.e.,

$p_a=\max \left\{p_{0,e},p_{r,e}\right\}$(8)

If p_a = p₀,_e and is greater than a certain value (e.g., 0.9) selected for decision-making, no gassy soils present at the location; otherwise, if p_a = p_r,_e and is greater than the selected threshold of probability value, gassy soils occur at the location. The investigation scheme shall guarantee that the p_a values in the whole domain of interest are greater than the prescribed threshold probability value.

Gassy soils were found to be distributed in a crossing section of Hangzhou Metro Line 1. The proposed approach is applied to determining the optimal scheme for investigating gassy soils in the cross section of the Hangzhou Metro Line 1. Although the gassy soils may be distributed within depths ranging from 25m to 32m, only the horizontal spatial variability of gas pressure is taken into account in this study. The horizontal length of cross section concerned in this study is 3000m along the Hangzhou Metro Line 1. The influence of gassy soils on the construction of underground projects depends on the gas pressure. Generally speaking, gassy soils are considered to be dangerous if the gas pressure is greater than or equal to 50 kPa; otherwise, the risk induced by gassy soils is ignorable. Therefore, the R is set as 50 kPa in this study. In other words, E₀ (Ignorable event) and E_r (Risky event) are defined as the event with gas pressure less than 50 kPa and its complementary event (see Table 1 ), respectively.

For a given investigation scheme, the proposed approach gives the locations where the gas pressures are very likely to be risky (i.e., greater than 50 kPa in this example), which are shown in Figure 3 by open squares. To verify the results obtained from the proposed approach, gas pressure data were obtained from 59 locations along the cross section, as shown in Figure 4 by open circles. All the gas pressures are greater than 50 kPa, which indicates the occurrence of the risky event at the 59 locations. The results are compared with those shown in Figure 3 . For a given investigation scheme, the ratio of correct predictions is calculated as R_cp = N_r/59, where N_r is the number of locations with p_a = p_r,_e (i.e., E_r has a larger probability

than E₀) among the 59 locations. Figure 5 shows the variation of R_cp as the borehole interval ∆L decreases from 60 to 10 m, corresponding to S1 - S6 shown in Table 3 . The R_cp increases considerably as ∆L decreases. For S5 with ∆L = 20 m, the R_cp is greater than 0.9. As ∆L decreases to 10 m for S6, the R_cp increases slightly. These observations verify the proposed approach and support that S5 with ∆L = 20 m is the optimal investigation scheme among the six candidate ones with a trade-off between investigation efforts and predication accuracy.

This study developed a site investigation optimization method to characterize the spatial distribution of shallow gassy soils along the horizontal direction based on prior knowledge. The horizontal spatial variability of gas pressures is modeled using Kriging-based conditional random field. Then, Monte Carlo simulation is performed to simulate the horizontal spatial distribution of gassy soils with considering the uncertainty in random field parameters given the prior knowledge. Based on the simulated results, the presence of gassy soils was determined based on a prescribed probability threshold value, i.e., 0.9 (“very likely”). Proper investigation schemes shall provide sufficient information to predict the presence of gassy soils and to judge whether the gas pressures are ignorable or risky at different locations of the site concerned. The optimal scheme is then determined as a trade-off between investigation efforts and prediction accuracy. For illustration and validation, the proposed approach was applied to characterize gassy soils along a cross section of Hangzhou Metro Line 1. Results were compared with gas pressure data measured at site. It was shown that the optimal investigation scheme obtained from the proposed method allows characterizing the horizontal spatial distribution of gassy soils reasonably. Last but not the least, it is worthy pointing out that future developments of the proposed approach shall be devoted to 2D/3D spatial variability modeling, efficient optimization algorithms, and decision-making tools and criterion to facilitate site investigation optimization of gassy soils.