Modified Peck Model and Parameter Selection Based on Multi-Source Engineering Evidence

1. Introduction

Tunnel excavation inevitably disturbs the initial stress state of the surrounding ground and induces ground loss, stress redistribution, and associated surface and subsurface deformation [1,2,3,4]. Excessive or nonuniform ground movements may induce differential settlement and angular distortion of overlying buildings [5,6,7,8,9,10] and may also affect foundations [11,12,13], basements [14,15,16], buried utilities [17,18,19], and existing underground structures [20,21,22,23]. Reliable prediction of tunnelling-induced settlement is therefore essential for deformation control and risk assessment [24,25]. Among the available approaches, the Gaussian settlement-trough model proposed by Peck [26] remains one of the most widely used engineering methods because of its simple formulation and clear physical interpretation. Subsequent empirical observations further related the settlement-trough width to tunnel depth and ground conditions [27], while field measurements demonstrated that the characteristics of the settlement trough also vary below the ground surface [28]. Mechanics-based analytical solutions have meanwhile provided alternative descriptions of tunnelling-induced ground movements [29].

Considerable research has subsequently focused on extending settlement prediction from idealized single-tunnel conditions to more complex excavation scenarios. Suwansawat and Einstein [30] described settlement troughs above twin tunnels using a superposition approach, while Chen et al. [31] investigated ground movements induced by parallel EPB tunnelling in silty soils. Fang et al. [32] examined subsurface settlement due to shield tunnelling and provided systematic observations of settlement-trough variation with depth. Xie et al. [33] analyzed the surface settlement induced by a large-diameter shield tunnel in Shanghai. These studies show that settlement magnitude and trough geometry are sensitive to tunnel configuration, observation depth, and project conditions, limiting the direct transfer of a single set of Peck parameters among different projects.

Accordingly, increasing attention has been paid to improved empirical formulations and parameter modification. Lu et al. [34] developed a Gaussian-function-based formulation for predicting tunnelling-induced ground settlement. Wang et al. [35] proposed a new calculation method for tunnelling-caused stratum settlement, in which the settlement influence range and settlement at different depths were considered. Ding et al. [36] investigated shield-tunnelling-induced settlement in soil–rock composite strata, highlighting the influence of heterogeneous ground conditions on settlement prediction. Deng et al. [37] analyzed shield-tunnelling-induced surface settlement along curved sections from the perspective of ground loss. Gao et al. [38] modified the Peck formula for water-rich sandy cobble strata, whereas Li et al. [39] developed a modified Peck formulation for a double-track shield tunnel beneath an expressway subgrade. These studies demonstrate that modification of the maximum settlement and settlement-trough width is often required when the conventional Peck model is applied beyond its original empirical conditions.

Recent studies have further examined the effects of tunnel layout, heterogeneous strata, and construction conditions on settlement-trough characteristics. Kannangara et al. [40] investigated surface settlement induced by twin tunnelling in silty sand, and Hou et al. [41] analyzed four shield tunnels with complex spatial relationships in clay. Guo et al. [42] examined the evolution of relationships between construction parameters and settlement under soil–rock mixed-face conditions, while Li et al. [43] investigated settlement patterns of super-large-diameter shield tunnels in composite strata. Lou et al. [44] considered surface settlement associated with double-line shield tunnelling with a large longitudinal slope, and Wang et al. [45] extended settlement prediction to subsurface observations in sandy cobble strata. More recently, Kong et al. [46] explicitly related Peck-type settlement parameters to tunnel depth, radius, and tunnel gap, further illustrating that maximum settlement and trough width are not invariant empirical quantities.

The influence of construction disturbance and deformation mode has also received increasing attention. Peng et al. [47] investigated the effects of deviations in jacking force and cutterhead torque during shield tunnelling in soft ground. Zhou et al. [48] developed a stochastic-medium-theory-based prediction method for surface settlement induced by bias tunnels, extending settlement prediction to asymmetric tunnel convergence conditions. Zhou et al. [49] further investigated settlement of an existing tunnel subjected to undercrossing excavation through a theoretical prediction framework, demonstrating the importance of evaluating ground deformation at the depth of subsurface structures rather than relying solely on surface settlement. Data-driven approaches have also emerged as a complementary direction; for example, Ye et al. [50] employed machine learning for forecasting shield-tunnelling-induced settlement. Nevertheless, for preliminary design and engineering interpretation, Peck-type empirical models remain attractive because their principal parameters have direct geometric and deformation-related meanings.

Despite these advances, most existing Peck modifications have been developed from individual projects or specific geological conditions, and the reported correction coefficients consequently exhibit considerable between-study variability. A cross-project quantitative assessment of whether the correction of maximum settlement and settlement-trough width shows consistent ground-condition dependence remains limited. In addition, many published correction coefficients are reported as ranges rather than point estimates, meaning that the uncertainty embedded in the original engineering evidence should be retained during statistical comparison. Another issue concerns observation depth: parameters calibrated from surface settlement troughs cannot necessarily be transferred directly to buried structures because both trough width and maximum settlement evolve with depth.

To address these issues, this study establishes a multi-source engineering-evidence framework for the modified Peck model, as illustrated in Figure 1. Two complementary databases are employed to address different questions. The first comprises reported correction-coefficient intervals and is used to examine the statistical characteristics and ground-condition dependence of the maximum-settlement coefficient $\alpha$ and trough-width coefficient $\beta$. Descriptive statistics, Mann–Whitney tests, Cliff’s$\delta$, bootstrap resampling, interval Monte Carlo simulation, and construction-method sensitivity analysis are used to evaluate parameter evidence while retaining the uncertainty of the originally reported intervals. The second database, compiled independently from the international shield-tunnelling cases reported by Fang et al. [32], is used to quantify the depth dependence of i(z)/i(0) and S_max(z)/S_max(0). The two evidence streams are not numerically merged; instead, they provide complementary support for parameter selection and physical interpretation. Based on these results, a staged framework is developed for initial parameter selection, project-specific refinement, field back-analysis, dynamic updating, and depth-dependent deformation assessment.

2. Theoretical Framework and Modified Peck Model

2.1. Conventional Peck Model and Settlement-Trough Characteristics

Underground excavation inevitably disturbs the initial stress equilibrium of the surrounding ground and induces ground loss and stress redistribution, resulting in a spatially distributed deformation field around the tunnel. Figure 2 schematically illustrates the propagation of tunnelling-induced ground deformation from the excavation zone to adjacent subsurface structures, the ground surface, and overlying buildings. Such deformation may directly affect existing tunnels, buried pipelines, and other subsurface facilities, while its upward propagation can further generate surface settlement and differential ground movements. In densely built urban areas, these movements may lead to differential settlement, angular distortion, cracking, and other serviceability problems in overlying or adjacent buildings and structures. Therefore, an accurate and practical description of the tunnelling-induced settlement trough is essential for evaluating excavation-induced deformation and the associated risks to both subsurface structures and surface buildings.

Among the available prediction approaches, the empirical model proposed by Peck has been widely used because of its simple formulation and clear engineering interpretation. The transverse surface settlement trough is represented by a Gaussian distribution as follows:

$$S_0(x)=S_{\max ,0}\exp \left(-{\displaystyle \frac{x^2}{2i_0^2}}\right)$$

(1)

where S₀(x) is the predicted settlement at a horizontal distance x from the tunnel centerline, with settlement taken as positive downward; S_max,0 is the maximum settlement at the centerline of the settlement trough; and i₀ is the horizontal distance from the tunnel centerline to the inflection point of the Gaussian settlement curve.

Accordingly, S_max,0 characterizes the magnitude of the settlement trough, whereas i₀ controls its width and spatial extent. The geometric meanings of these characteristic parameters are illustrated in Figure 3.

For the Gaussian settlement trough described by Eq. (1), the settlement volume per unit tunnel length can be obtained by integrating the settlement profile over the transverse direction:

$$V_{s,0}={\displaystyle {\int }_{-\infty }^{+\infty }{S}_0}(x)dx=\sqrt{2\pi }{i}_0{S}_{\max ,0}$$

(2)

where V_s,0 denotes the settlement-trough volume per unit tunnel length.

If the ground-loss ratio V_l is defined as the ratio of the volume of ground loss to the tunnel excavation volume, the ground-loss volume per unit tunnel length can be expressed as AV_l, where A is the tunnel excavation cross-sectional area. Under the commonly adopted volume-compatibility assumption, V_s,0 = AV_l, and the maximum settlement can therefore be written as:

$$S_{\max ,0}={\displaystyle \frac{A{V}_l}{\sqrt{2\pi }{i}_0}}$$

(3)

For a circular tunnel of diameter D, $A=\pi D^2/4$. Equations (1)-(3) show that the conventional Peck model characterizes the transverse settlement trough primarily through the maximum settlement S_max,0 and the trough-width parameter i₀. However, the values predicted using baseline Peck parameters may deviate from field observations under different geological and construction conditions, motivating the introduction of correction coefficients for both the settlement magnitude and trough width.

2.2. Modified Peck Model and Identification of Correction Coefficients

Although the conventional Peck model provides a convenient representation of tunnelling-induced ground settlement, the actual magnitude and width of the settlement trough may deviate systematically from the baseline predictions because of variations in ground conditions, tunnel geometry, construction method, and ground-loss characteristics. To retain the Gaussian form of the conventional Peck model while improving its adaptability to engineering observations, a maximum-settlement correction coefficient $\alpha$ and a settlement-trough-width correction coefficient $\beta$ are introduced. The modified settlement profile is expressed as:

$$S_m(x)=\alpha S_{\max ,0}\exp \left[-{\displaystyle \frac{x^2}{2{(\beta i_0)}^2}}\right]$$

(4)

Accordingly, the characteristic parameters of the modified settlement trough are:

$$S_{\max ,m}=\alpha S_{\max ,0},i_m=\beta i_0$$

(5)

where S_max,m and i_m denote the maximum settlement and settlement trough width corresponding to the modified model, respectively.

Therefore, $\alpha$ quantifies the deviation in settlement magnitude from the baseline Peck prediction, whereas $\beta$ quantifies the corresponding deviation in settlement trough width. When $\alpha =\beta =1$, Eq. (4) red uces to the conventional Peck model given by Eq. (1).

To determine the characteristic parameters of an observed settlement trough, Eq. (4) can be linearized using field monitoring data. For the jth monitoring point, let x_j denote the horizontal distance from the tunnel centerline and S_j the measured settlement magnitude. Taking the natural logarithm of the Gaussian settlement function gives:

$$\ln S_j=\ln S_{\max ,m}+{\displaystyle \frac{1}{i_m^2}}\left(-{\displaystyle \frac{x_j^2}{2}}\right)$$

(6)

Defining:

$$X_j=-{\displaystyle \frac{x_j^2}{2}},Y_j=\ln S_j$$

(7)

Eq. (7) can be written in the linear regression form

$$Y_j=a+b{X}_j+{\epsilon }_j$$

(8)

where a and b are the theoretical intercept and slope, respectively, and ${\epsilon }_j$ represents the residual error.

Comparison between Eqs. (6) and (8) gives

$$a=\ln S_{\max ,m},b={\displaystyle \frac{1}{i_m^2}}$$

(9)

For n monitoring points, the least-squares estimates of the slope and intercept are:

$$\widehat{b}={\displaystyle \frac{{\displaystyle \sum _{j=1}^n(X_j-\overline{X})}(Y_j-\overline{Y})}{{\displaystyle \sum _{j=1}^n{(X_j-\overline{X})}^2}}},\widehat{a}=\overline{Y}-\widehat{b}\overline{X}$$

(10)

The maximum settlement and trough-width parameter fitted from the monitoring data are therefore:

$${\widehat{S}}_{\max ,m}=\exp (\widehat{a}),{\widehat{i}}_m={\displaystyle \frac{1}{\sqrt{\widehat{b}}}}$$

(11)

Relative to the baseline Peck parameters S_max,0 and i₀, the corresponding correction coefficients can then be obtained as:

$$\widehat{\alpha }={\displaystyle \frac{\exp (\widehat{a})}{S_{\max ,0}}},\widehat{\beta }={\displaystyle \frac{1}{i_0\sqrt{\widehat{b}}}}$$

(12)

where a and b denote the theoretical regression coefficients, whereas $\widehat{a}$ and $\widehat{b}$ are their least-squares estimates obtained from the monitoring data.

Correspondingly, $\widehat{\alpha }$and $\widehat{\beta }$ are estimated correction coefficients. For notational simplicity, the hats over $\alpha$ and $\beta$ are omitted in the subsequent statistical analysis. The two correction coefficients are not completely independent from the perspective of settlement-trough volume. Based on Eqs. (2) and (5), the modified settlement-trough volume is

$$V_{s,m}=\sqrt{2\pi }(\beta i_0)(\alpha S_{\max ,0})=\alpha \beta V_{s,0}$$

(13)

and hence

$${\displaystyle \frac{V_{s,m}}{V_{s,0}}}=\alpha \beta$$

(14)

Therefore, when a ground-loss ratio is independently available, the consistency between $\alpha \beta$ and the inferred change in settlement-trough volume should be examined. When $\alpha$ and $\beta$ are used as statistical priors rather than deterministic constants, they should also be jointly updated using project-specific monitoring data.

3. Multi-Source Engineering Data and Analysis Methods

3.1. Correction-Coefficient Database

The engineering evidence used in this study consists of two complementary datasets. The first dataset contains reported correction-coefficient intervals and is used to investigate the statistical characteristics and ground-condition dependence of $\alpha$ and $\beta$. The second dataset contains surface and subsurface settlement observations from international shield-tunnelling cases and is used independently to evaluate the depth-dependent evolution of the settlement trough. Because the two datasets describe different quantities and were compiled under different parameter definitions, they were not numerically pooled.

The correction-coefficient database comprises 39 published engineering records. For each record, the ground category, geographical region, construction method, reported lower and upper bounds of $\alpha$ and$\beta$, and source information were retained. The records were classified into four ground categories: soft soil, composite strata, gravelly soil, and sandy soil. Soft soil and composite strata contain 17 and 15 records, respectively, whereas gravelly soil and sandy soil contain only four and three records. Given these sample sizes, inferential comparisons between ground conditions were restricted to the soft-soil and composite-strata groups, while the gravelly-soil and sandy-soil records were retained for descriptive analysis only.

Three of the 39 records involved non-shield construction methods, including shallow-buried tunnelling, mining, and pipe jacking. These records were retained in the complete database to preserve the composition of the collected evidence, but were subsequently excluded in the shield-only sensitivity analysis. In addition, the ground classification of two records could not be fully resolved from the available source descriptions. These records were retained and explicitly flagged rather than being assigned a more precise classification than supported by the original information. The composition of the correction-coefficient database is summarized in Table 1.

3.2. International Dataset for Depth-Effect Assessment

An independent dataset of surface and subsurface settlement observations was compiled from 24 shield-tunnelling cases reported by Fang et al. [32], providing a total of 84 observations at different depths. The cases cover the United Kingdom, the United States, Ireland, Japan, Canada, Mexico, Thailand, mainland China, and Taiwan, and involve a range of shield construction methods, including hand-excavated, mechanical, open-face, slurry, and earth-pressure-balance shields. The extracted variables include the tunnel centerline depth $z_0$, tunnel diameter D, observation depth z, settlement-trough width i(z), and maximum settlement S_max(z). Values not reported in the source were retained as missing and were not imputed.

Because absolute settlement-trough parameters are strongly affected by tunnel burial depth and geometric scale, the overburden thickness above the tunnel crown was defined as:

$$T=z_0-{\displaystyle \frac{D}{2}}$$

(15)

where $z_0$is the depth of the tunnel centerline and D is the tunnel diameter.

The normalized observation depth was then expressed as z/T. The geometric definitions of $z_0$, D, z, and T, together with the corresponding surface and subsurface settlement-trough parameters, are schematically illustrated in Figure 4.

For each tunnelling case, the surface observation was adopted as the reference condition. The normalized settlement-trough width and maximum settlement were therefore defined as i(z)/i(0) and S_max(z)/S_max(0) respectively. These normalized ratios describe the relative evolution of settlement-trough geometry with observation depth. As the observation position approaches the tunnel, changes in i(z)/i(0) characterize the variation in trough width, whereas changes in S_max(z)/S_max(0) reflect the corresponding evolution of settlement magnitude.

It should be emphasized that the depth-effect dataset and the correction-coefficient database describe different quantities. The former provides i(z) and S_max(z) at different observation depths, whereas the latter provides the correction coefficients $\alpha$ and $\beta$ relative to a baseline Peck model. Because these quantities were not defined under a unified parameter-identification procedure, the two datasets were not numerically pooled. Instead, the correction-coefficient database was used to investigate the statistical distributions and ground-condition dependence of $\alpha$ and $\beta$, while the international dataset was used as an independent source of evidence for evaluating the depth dependence of settlement-trough characteristics.

3.3. Statistical and Uncertainty Analysis

The statistical analysis comprised descriptive comparison, between-group inference, interval uncertainty propagation, construction-method sensitivity analysis, and depth-effect assessment, as summarized in Figure 5. First, the midpoint of each reported $\alpha$ and $\beta$ interval was used for descriptive analysis. The mean, standard deviation, median, interquartile range (IQR), and 5th–95th percentile range were calculated for each ground category. Because the gravelly-soil and sandy-soil groups contained only four and three records, respectively, formal between-group inference was restricted to soft soil and composite strata.

Differences between the soft-soil and composite-strata groups were evaluated using a two-sided Mann-Whitney U test, with Cliff’s $\delta$ adopted as a nonparametric effect-size measure. A positive $\delta$ indicates a tendency for higher parameter values in soft soil, whereas a negative value indicates the opposite. Uncertainty in the difference between group medians was further quantified using 20,000 nonparametric bootstrap resamples. To retain the uncertainty contained in the originally reported parameter intervals, interval Monte Carlo sampling was also performed. For each realization, one value was independently sampled from a uniform distribution bounded by the reported lower and upper limits of each record, after which the median difference between the soft-soil and composite-strata groups was calculated. This procedure was repeated 20,000 times for both $\alpha$ and $\beta$, yielding the median between-group difference, its 95% interval, and the probability that the median parameter value in soft soil exceeds that in composite strata. The uniform distribution was used only to represent uncertainty within the reported intervals in the absence of additional probabilistic information.

To assess the influence of construction method, a shield-only sensitivity analysis was conducted by excluding the three non-shield records and recalculating the Mann–Whitney U test and Cliff’s $\delta$. The full-database and shield-only results were then compared to evaluate the robustness of the inferred ground-condition effects.

The depth-effect dataset was analyzed independently. Spearman rank correlations were calculated between the normalized observation depth z/T and the normalized responses i(z)/i(0) and S_max(z)/S_max(0), with linear trends fitted for visualization. Because multiple observations could originate from the same tunnelling case, a slope was additionally estimated for each case with observations at multiple depths, and the consistency of the expected slope direction was evaluated using an exact binomial test. This case-level analysis avoids treating repeated observations within the same project as fully independent samples. The significance level was set at 0.05, while interpretation was based jointly on statistical significance, effect size, bootstrap uncertainty, interval Monte Carlo results, and case-level directional consistency.

4. Results

4.1. Distributions of the Correction Coefficients

The reported intervals of the maximum-settlement correction coefficient $\alpha$ and the settlement-trough-width correction coefficient $\beta$ exhibit considerable variability among the collected engineering records. Figure 6 and Figure 7 present the reported $\alpha$ and $\beta$ intervals for soft-soil conditions, respectively, whereas Figure 8 and Figure 9 show the corresponding intervals for composite strata. The soft-soil and composite-strata groups contain substantially more records than the gravelly-soil and sandy-soil groups and therefore provide the primary basis for subsequent between-group inference.

To provide a representative basis for descriptive comparison, the midpoint of each reported interval was calculated. Figure 10 compares the midpoint distributions of $\alpha$ and $\beta$ between soft soil and composite strata, and the corresponding descriptive statistics for all four ground categories are summarized in Table 2.

For $\alpha$, the soft-soil group has a median of 0.650 and an interquartile range of 0.530–0.700, compared with a median of 0.550 and an interquartile range of 0.350–0.700 for composite strata. The midpoint distribution of $\alpha$ therefore shows a tendency toward higher values in soft soil. In contrast, the median values of $\beta$ are relatively close, at 0.795 for soft soil and 0.750 for composite strata, with substantial overlap between their interquartile ranges. Owing to the limited numbers of gravelly-soil and sandy-soil records, their statistics are reported for descriptive purposes only and are not used for population-level inference.

4.2. Ground-Condition Dependence and Interval Uncertainty

The statistical comparison between soft soil and composite strata is summarized in Table 3 and Figure 11. For $\alpha$, the median midpoint value is 0.650 in soft soil and 0.550 in composite strata. The Mann–Whitney U test gives p=0.150, while Cliff’s $\delta =0.302$, indicating a moderate directional tendency toward higher $\alpha$ values in soft soil. The bootstrap estimate of the median difference is 0.100, with a 95% confidence interval of [-0.107, 0.350].

When the originally reported parameter intervals are retained through interval Monte Carlo sampling, the median between-group difference in $\alpha$ is 0.104, with a 95% interval of $[-0.050,0.262]$. The probability that the median $\alpha$ value in soft soil exceeds that in composite strata is 0.907. Although both the bootstrap and Monte Carlo intervals include zero, the direction of the effect remains predominantly positive. The available evidence therefore indicates a ground-condition-related tendency in $\alpha$, but does not establish a statistically robust deterministic difference between the two groups.

For $\beta$, the two groups show substantially greater overlap. Cliff’s $\delta =-0.059$ and the Mann–Whitney U test gives$p=0.791$. The bootstrap median difference is 0.000 with a 95% confidence interval of$[-0.200,0.175]$, while the interval Monte Carlo analysis gives a median difference of -0.012 with a 95% interval of $[-0.163,0.136]$. The probability that the median $\beta$ value in soft soil exceeds that in composite strata is 0.442. These results provide insufficient evidence for a systematic ground-condition dependence of $\beta$ within the present database.

4.3. Construction-Method Sensitivity

The complete correction-coefficient database contains three records obtained using non-shield construction methods. To evaluate whether these records materially influence the inferred ground-condition differences, the analysis was repeated using shield-tunnelling records only. After exclusion of the three non-shield records, all 17 soft-soil records were retained, whereas the number of composite-strata records decreased from 15 to 13.

For $\alpha$, the median values in soft soil and composite strata are 0.650 and 0.500, respectively. Cliff’s $\delta$ increases from 0.302 for the complete database to 0.394 for the shield-only dataset, while the corresponding p-value decreases from 0.150 to 0.071. Thus, restricting the analysis to shield-tunnelling records strengthens the directional difference in $\alpha$, although the result remains above the conventional significance threshold of 0.05.

In contrast, $\beta$ remains insensitive to the ground-condition classification after restriction to shield-tunnelling records. The median values are 0.795 for soft soil and 0.700 for composite strata, with Cliff’s $\delta =0.032$ and $p=0.900$. The construction-method sensitivity analysis therefore does not alter the principal statistical pattern observed in the complete database: $\alpha$ exhibits a directional ground-condition signal, whereas $\beta$ shows little evidence of a consistent between-group difference.

Table 4. Sensitivity analysis based on shield-tunnelling records only.

Parameter	n(Soft soil/Composite strata)	Median (Soft soil)	Median (Composite strata)	Cliff’s$\delta$	p
$\alpha$	17/13	0.650	0.500	0.394	0.071
$\beta$	17/13	0.795	0.700	0.032	0.900

Table 4. Sensitivity analysis based on shield-tunnelling records only.

Parameter	n(Soft soil/Composite strata)	Median (Soft soil)	Median (Composite strata)	Cliff’s$\delta$	p
$\alpha$	17/13	0.650	0.500	0.394	0.071
$\beta$	17/13	0.795	0.700	0.032	0.900

4.4. Depth Dependence of Settlement-Trough Characteristics

The international shield-tunnelling dataset reveals a pronounced depth dependence of settlement-trough geometry. As shown in Figure 12 and Table 5, the normalized settlement-trough width i(z)/i(0) decreases markedly with increasing normalized observation depth z/T. Based on 42 subsurface observations from 17 cases, the pointwise Spearman rank correlation coefficient is $\rho =-0.898$ with$p=6.97\times {10}^{-16}$. Moreover, all 17 cases containing observations at multiple depths exhibit negative within-case slopes. The corresponding exact binomial test gives $p=7.63\times {10}^{-6}$, demonstrating strong case-level consistency in the direction of change.

The normalized maximum settlement S_max(z)/S_max(0) shows a different pattern. Based on 56 subsurface observations from 22 cases, the Spearman rank correlation coefficient is $\rho =0.287$, with$p=0.0319$. Although the pointwise relationship is substantially more scattered than that of the settlement-trough width, 20 of the 22 cases exhibit positive within-case slopes. The exact binomial test gives $p=6.06\times {10}^{-5}$, indicating a highly consistent positive trend at the case level.

These results show that, as the observation position approaches the tunnel, the settlement trough generally becomes narrower while the maximum settlement tends to increase. The depth dependence is particularly strong and consistent for the trough-width parameter, whereas the maximum-settlement response exhibits greater between-case scatter. Some individual cases show local non-monotonic variations, which may reflect differences in ground conditions, monitoring configurations, and construction-induced disturbances; therefore, the identified trends should be interpreted as general cross-case patterns rather than as strictly monotonic behavior in every project.

5. Discussion

5.1. Interpretation and Dependence of the Correction Coefficients

The two correction coefficients represent different geometric characteristics of the modified Peck settlement trough. The coefficient $\alpha$ scales the maximum settlement, whereas $\beta$ modifies the trough width. In the present database, $\alpha$ exhibits a directional dependence on ground condition: the median value is 0.650 for soft soil and 0.550 for composite strata, with Cliff’s $\delta =0.302$. Although the between-group difference is not statistically significant ($p=0.150$), interval Monte Carlo analysis gives a probability of 0.907 that the median $\alpha$ in soft soil exceeds that in composite strata. The signal becomes stronger after restricting the analysis to shield-tunnelling records ($\delta =0.394$,$p=0.071$). These results support the use of $\alpha$ as a ground-conditioned prior rather than as a deterministic stratum-specific constant.

The observed tendency is consistent with the greater compressibility and lower stiffness of soft soils, together with construction disturbance and post-shield consolidation. In composite strata, relatively stiff layers may restrain overall deformation, whereas the position and inclination of the soil–rock interface and local construction conditions can introduce substantial scatter. Ground type therefore provides only a first-order explanation for the variability in $\alpha$.

By contrast, $\beta$ shows no robust ground-condition dependence. Its distributions for soft soil and composite strata overlap substantially, and this conclusion is unchanged in the shield-only analysis. The inferred $\beta$ is influenced not only by ground properties but also by the baseline formulation of i, tunnel-depth definition, monitoring-section extent, and tunnel configuration. Therefore, the current evidence does not support assigning $\beta$ solely according to ground type. In addition, $\alpha$ and $\beta$ should not be interpreted independently because $V_{s,m}/V_{s,0}=\alpha \beta$. Their joint selection should therefore remain consistent with the implied settlement-trough volume and, where available, the independently estimated ground-loss ratio.

5.2. Engineering Implications and Parameter Updating

The international dataset demonstrates a systematic depth dependence of the settlement trough. The normalized trough width i(z)/i(0) decreases markedly with increasing z/T ($\rho =-0.898$), and all 17 cases with multiple observation depths exhibit negative within-case slopes. In contrast, S_max(z)/S_max(0) generally increases with depth; despite greater scatter, positive slopes are observed in 20 of 22 cases. These results indicate that the settlement field becomes progressively narrower and more concentrated as the observation position approaches the tunnel.

This depth dependence has direct implications for deformation assessment of the built environment. Surface-calibrated Peck parameters may be suitable for evaluating ground movements affecting surface buildings, but direct extrapolation to buried pipelines, existing tunnels, basements, or other subsurface structures may misrepresent both the magnitude and spatial concentration of deformation. Depth-dependent values of i(z) and S_max(z) should therefore be incorporated when subsurface structural responses are assessed.

For engineering application, the statistical results should be treated as initial parameter priors rather than fixed design values. A staged parameter-selection and updating strategy is therefore proposed, as summarized in Figure 13. During preliminary design, the median may be adopted as the central estimate, the interquartile range as the preferred search interval, and the 5th–95th percentile range as a broader uncertainty envelope. Before construction, these ranges should be refined using comparable projects, site investigation, tunnel geometry, groundwater conditions, and numerical analysis. During tunnelling, $\alpha$ and $\beta$ should be jointly updated using field monitoring, while the consistency of $\alpha \beta$ with the inferred ground loss or settlement-trough volume should be checked. For subsurface structures, the updated parameters should be combined with the depth-dependent corrections identified above. The corresponding data-derived initial priors are summarized in Table 6.

5.3. Limitations and Future Research

The present study is subject to four main limitations. First, the 39 records are study-level parameter intervals rather than raw settlement profiles fitted under a unified procedure, so heterogeneity in parameter definition and fitting method cannot be fully eliminated. Second, the sample sizes remain limited, particularly outside the soft-soil and composite-strata groups, and ground condition may be partially confounded with project-specific factors. Third, the interval Monte Carlo analysis assumes uniform uncertainty within each reported interval; this assumption represents a lack of information within the interval rather than a physical probability distribution. Fourth, the international depth-effect data were obtained from a single compilation source and therefore provide complementary physical evidence rather than direct external validation of the correction-coefficient database.

Future work should prioritize the collection of raw transverse settlement profiles together with tunnel geometry, overburden conditions, groundwater information, construction parameters, and monitoring time. Re-estimating $\alpha$ and $\beta$ under a unified objective function would allow geological and methodological effects to be separated more clearly. With a larger standardized database, hierarchical statistical models could then be used to quantify the contributions of ground and construction variables, while independently validated depth-dependent relationships could be developed for subsurface deformation prediction.

6. Conclusions

This study developed a multi-source engineering-evidence framework for evaluating and selecting the correction parameters of the modified Peck model. The main conclusions are as follows.

(1) The maximum-settlement correction coefficient $\alpha$ exhibits a clear directional dependence on ground condition. Soft soil shows a higher median $\alpha$ than composite strata, and the effect remains directionally consistent after uncertainty propagation and shield-only sensitivity analysis. However, the between-group difference does not reach conventional statistical significance. Therefore, $\alpha$ is better treated as a ground-conditioned prior than as a fixed stratum-specific constant.

(2) The settlement-trough-width correction coefficient $\beta$ shows substantial overlap between soft soil and composite strata, with little evidence of a systematic ground-condition effect. Accordingly, $\beta$ should not be prescribed solely according to ground type and should instead be calibrated using project-specific geological, geometric, and construction information.

(3) Settlement-trough characteristics vary systematically with observation depth. The normalized trough width i(z)/i(0) decreases markedly with increasing z/T, whereas S_max(z)/S_max(0) generally increases. Thus, the deformation field becomes narrower and more concentrated toward the tunnel, indicating that surface-calibrated Peck parameters should not be transferred directly to subsurface deformation assessment.

(4) The correction coefficients are most appropriately used within a staged parameter-updating framework. Statistical distributions can provide initial priors, which should subsequently be refined using project-specific information and field monitoring. Joint updating of $\alpha$ and $\beta$ should remain consistent with settlement-trough volume or independently estimated ground loss, while depth-dependent corrections should be incorporated when evaluating subsurface structures. This provides a practical basis for deformation assessment of both surface buildings and underground infrastructure.