For a uniform soil or rock mass, Equation (6) can be expressed as:
$$k=\frac{a}{H}+b$$(7)

Where $a=\frac{1000\beta EG}{{(1-v)}_{\gamma}},b=\frac{v}{1-v}+\frac{\beta EG}{{(1-v)}_{\gamma}}$.

In Equation (7), the relationship between the *in situ* stress ratio *k* and depth is inversely proportional. For a certain depth, the value of *k* is mainly dependent on the two parameters *a* and *b*. Moreover, when the buried depth changes slightly in the shallow crust, the thermal expansion coefficient *β*, and thermal gradient *G*, can be treated as constants. Then, the *in situ* stress ratio *k* is taken to consist of a constant term and a hyperbolic term. In addition, the two parameters *a* and *b* are only dependent on rock properties, which reflect the intensity of tectonic movements. However, tectonic stress (including magnitude and direction) often changes significantly in different parts of the study area. As a result, it is not easy to determine the parameters *a* and *b* accurately.

To reflect the influence of local tectonic movements on the *in situ* stress, we introduce the local tectonic correction factors *ξ* and *η*. The following equation provides a more common form of the *in situ* stress ratio *k* for the Earth’s crust:
$$k=\frac{v}{1-v}+\frac{\beta {E}_{rm}G}{(1-v)\gamma}(1+\frac{1000}{H})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\xi +\eta $$(8)

Where *E*_{rm} stands for rock mass elastic modulus. The two parameters *ξ* and *η* are local tectonic correction factors which will be determined by the least-squares approach based on local stress measurements and laboratory tests. For a certain project, the parameters *β*, *G*, *γ*, *ν* and *E*_{rm} in Equation (8) can be easily determined based on laboratory tests and empirical formulas. Then, the *in situ* stress ratio *k* in Equation (8) can be taken as the function of *ξ* and *η*. Furthermore, the least-squares fitting analysis is carried out to determine the initial values of *ξ* and *η* based on local stress measurements. If some stress values deviate considerably, they should be neglected so as to obtain the optimum tectonic correction factors. It is noted that if the study area suffers the same tectonic movements, the parameters *ξ* and *η* can be treated as constants.

When the rock mass elastic modulus is unknown, we can indirectly determine it based on laboratory tests and rock quality information. Guo, *et al*. [22] provided the following relationship between the factors *E*_{rm}/*E*_{i} and *RMDI* through analysing many measurements:
$$\frac{{E}_{rm}}{{E}_{i}}=\phantom{\rule{thinmathspace}{0ex}}0.73\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}RMDI-\phantom{\rule{thinmathspace}{0ex}}0.079$$(9)

Where *E*_{i} is the intact rock elastic modulus, and *RMDI* is a rock mass integrity index, which can be determined by using Equations (10) and (11).

The indicator *RMDI* can be defined as a ratio of intact rock to entire rock mass in certain range. It can reflect (quantitatively) the effect of discontinuities (*e.g*., joints or faults) on rock integrity based on drilling images. Drilling images, from the use of a borehole camera, is often used to obtain the structural characteristics of fractured rock. By using such imaging with acoustic and optical televiewers, continuous and oriented 360^{∘} views of the borehole wall are available, from which the character, inter-relationship, and orientations of lithologic and structural planar features can be easily defined [23, 24]. For a given depth range [*h*_{1}, *h*_{2}], the *RMDI* can be expressed as [25]:
$$RMDI=\frac{{\int}_{{h}_{1}}^{{h}_{2}}f(z)dz}{\mathrm{\Delta}h}$$(10)

Here, *z* is the depth. *f*(*z*) is an integrity index density function of the rock mass which can be determined by Equation (11):
$$f(z)=\{\begin{array}{l}0\text{(Completely\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}fragmented\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}rock)}\\ \delta \text{(Not\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}completely\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}fragmented\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\text{\hspace{0.17em}rock)}\end{array}$$(11)

Where, *δ* is an impact factor for rock fragmentation which depends on filling type and rock mass size. For filling conditions, taking clay filling as an example, *δ* is taken as zero. Otherwise, the value of *δ* is dependent on rock mass size. Based on the published literature [25], as the rock mass changes from intact, massive, bedded, fractured, to non-compact states, the value of *δ* changes from 1.0, 0.8, 0.5, 0.2, to 0.1, respectively.

In addition, studies [26, 27] show that horizontal stress mainly consists of two parts: one is the horizontal component of gravitational stress, and the other is the horizontal tectonic stress. When the stress field is mainly caused by gravity, horizontal stresses can be estimated by Equation (2). However, as the horizontal tectonic movement predominates, the following equations provide the basic relationship for elastic rock behaviour, with the constraints that one of the principal stresses is vertical and varies linearly with depth [3]:
$$\begin{array}{r}{\sigma}_{V}=\gamma H\\ {\sigma}_{H}=\lambda {\sigma}_{V}+{\sigma}_{T}\\ {\sigma}_{h}=\lambda ({\sigma}_{V}+{\sigma}_{T})\end{array}\}$$(12)

Where σ_{H} and σ_{h} are the major and minor horizontal stresses, respectively, σ_{T} is the horizontal tectonic stress, and *λ* is the lateral pressure coefficient due to gravitational loading with *λ* = ν/(1−ν).

As mentioned before, the ratio *k* = (σ_{H} + σ_{h})/(2σ_{V}). Then, Equation (8) can be rewritten as:
$$\frac{{\sigma}_{H}+{\sigma}_{h}}{2{\sigma}_{V}}=\frac{v}{1-v}+\frac{\beta {E}_{rm}G}{{(1-v)}_{\gamma}}(1+\frac{1000}{H})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\xi +\eta $$(13)

In Equation (12), both σ_{H} and σ_{h} have the common part in σ_{T}. Then, we can get the relationship between σ_{H} and σ_{h} by removing σ_{T}.

$$\begin{array}{r}{\sigma}_{H}=\frac{{\sigma}_{h}}{\lambda}-(1-\lambda )\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}{\sigma}_{V}\\ {\sigma}_{V}=\gamma H\end{array}\}$$(14)

Based on Equations (13) and (14), we can obtain:
$$\begin{array}{c}{\sigma}_{H}=\lambda {\sigma}_{V}+2\beta {E}_{rm}G\xi (H+1000)+2\eta (1-v)\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{V}=\gamma H\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{h}=\lambda {\sigma}_{V}+2\lambda \beta {E}_{rm}G\xi (H+1000)+2\eta v\end{array}\}$$(15)

According to Equation (15), the *in situ* stress at an arbitrary depth can be estimated. Figure 4 illustrates the flowchart of the improved method.

Figure 4 Flow chart describing the improved method for estimating in situ stress proposed in this study.

To verify the reliability of this method in estimation of *in situ* stress, it is necessary to compare the calculated results with field measurements.

In stage 5, we define
$$a=\frac{\underset{i-1}{\sum ^{n}}\phantom{\rule{thinmathspace}{0ex}}|{{\sigma}_{ci}}^{2}-{{\sigma}_{mi}}^{2}|}{\underset{i=1}{\sum ^{n}}\phantom{\rule{thinmathspace}{0ex}}{{\sigma}_{ci}}^{2}},\phantom{\rule{thinmathspace}{0ex}}i=1,2,3,...,n$$(16)

Where σ_{ci} and σ_{mi} are the calculated and measured stresses at point *i*, respectively. According to Equation (16), if α ≤ 0.2, the process is terminated and we will have the equations for estimating *in situ* stresses at an arbitrary point; otherwise, it is used to recalculate the parameters *ξ* and η by Equation (8).

However, if the precision is not satisfied after 10 times, the user should decrease the accuracy of the estimation (*i.e*., increase the value of α). Also, another solution is to increase the number of *in situ* stress measurements.

It is noted that if at a location the *in situ* stresses are measured in competent parts of the same formation, then the corresponding stresses in an adjacent location can be estimated by the relationship:
$$\begin{array}{c}\frac{{\sigma}_{H1}}{{\sigma}_{H2}}=\frac{{\lambda}_{1}{\sigma}_{V1}+2{\beta}_{1}{E}_{rm1}G\xi ({H}_{1}+1000)+2\eta (1-{v}_{1})}{{\lambda}_{2}{\sigma}_{V2}+2{\beta}_{2}{E}_{rm2}G\xi ({H}_{2}+1000)+2\eta (1-{v}_{2})}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{V}=\gamma {H}_{2}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{{\sigma}_{h1}}{{\sigma}_{h2}}=\frac{{\lambda}_{1}{\sigma}_{V1}+2{\lambda}_{1}{\beta}_{1}{E}_{rm1}G\xi ({H}_{1}+1000)+2\eta {v}_{1}}{{\lambda}_{2}{\sigma}_{V2}+2{\lambda}_{2}{\beta}_{2}{E}_{rm2}G\xi ({H}_{2}+1000)+2\eta {v}_{2}}\end{array}\}$$(17)

Where the suffix 1 denotes the hard (or soft) rock in which stress measurements have been carried out: suffix 2 refers to the hard (or soft) rock at an adjacent location where the *in situ* stresses are to be estimated.

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