Skip to main content
Log in

Application of grey theory-based model to prediction of land subsidence due to engineering environment in Shanghai

  • Original Article
  • Published:
Environmental Geology

Abstract

Land subsidence is a common geological hazard. The long-term accumulation of land subsidence in Shanghai has caused economic loss to the city. Since the 1990s, the engineering structures have become a new cause of land subsidence. Many factors affect the process of land subsidence. Although such a process cannot be explicitly expressed by a mathematical formula, it is not a “black box” whose internal structure, parameters, and characteristics are unknown. Therefore, the grey theory can be applied to the prediction of land subsidence and provides useful information for the control of land subsidence. In this paper, a grey model (GM) GM (1, 1) with unequal time-intervals was used to predict the subsidence of a high-rise building in the Lujiazui area of Shanghai, and the results were compared with the monitored data. The prediction of subsidence was also corroborated by laboratory tests and the results were compared with measured data and the predicted data by the adaptive neuro-fuzzy inference system (ANFIS). It is found that the GM (1, 1) with unequal time-intervals is accurate and feasible for the prediction of land subsidence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • Bear J (1979) Hydraulics of landwater. McGraw-Hill, New York

    Google Scholar 

  • Cassiani G, Palozzo W, Zoccatelli C (2002) Simulation of subsidence caused by gas production from an offshore field and comparison with field data. J Pet Sci Eng 45:123–134

    Google Scholar 

  • Chai JC, Shen SL, Zhu HH, Zhang XL (2004) Land subsidence due to groundwater drawdown in Shanghai. Geotechnique 54(3):143–148

    Google Scholar 

  • Chai JC, Shen SL, Zhu HH, Zhang XL (2005) 1D analysis of land subsidence in Shanghai. Lowland Technol Int 7(1):33–41

    Google Scholar 

  • Chen J, Zhu GR, Gu AM, Wang CH (2003) Application of Biot consolidation theory to calculation of land subsidence (in Chinese). Hydrogeo Eng Geol 27(2):23–31

    Google Scholar 

  • Cui XD (1998) Application and development of MODFLOW and IDP in digital calculation of land subsidence in Tianjin (in Chinese). Chin J Geol Hazard Control 9(2):122–128

    Google Scholar 

  • Deng JL (1987) Basic methodology of grey system (in Chinese). Publishing House of Huazhong University of Science & Technology, Wuhan

    Google Scholar 

  • Gambolati G, Teatini P, Tomasi L (1999) Stress-strain analysis in productive gas/soil reservoirs. Intl J Num Ana Methods in Geomechanics 23(13):1495–1519

    Article  Google Scholar 

  • Gu XY (1998) Review and prospects of land subsidence computation (in Chinese). Chin J Geol Hazard Control 93(2):81–85

    Google Scholar 

  • Holzer TL, Bluntzer RL (1984) Land subsidence near oil and gas fields, Houston, Texas. Land water 22(4):450–459

    Google Scholar 

  • Lei XW, Bai SW, Meng QS (2000) The application of grey forecasting to analyzing soft foundation subsidence (in Chinese). Rock Soil Mech 21(2):145–147

    Google Scholar 

  • Li QF, Fang Z, Wang HM (2000) A mathematical model and forecast of groundwater workable reserves for Shanghai (in Chinese). Shanghai Geol 23(2):36–43

    Google Scholar 

  • Lin ZC, Lin WS (2001) The application of grey theory to the prediction of measurement points for circularity geometric tolerance. Int J Adv Manuf Technol 17:348–360

    Article  Google Scholar 

  • Lin YH, Wang JS, Pai PF (2004) A grey prediction model with factor analysis technique (in Chinese). J Chin Inst Indust Eng 21(6):535–542

    Google Scholar 

  • Lu C (2005) Study on land subsidence caused by the high-rise building group in shanghai and the model test in the lab (in Chinese). Tongji University, Shanghai

    Google Scholar 

  • Ran QQ, Gu XY (1998) A coupled model for land subsidence computation with consideration of Rheological property (in Chinese). Chin J Geol Hazard Control 9(2):99–103

    Google Scholar 

  • Shen SL, Tohno I, Nishigaki M, Miura N (2004) Land subsidence due to withdrawal of deep-groundwater. Lowland Technol Int 6(1):32–43

    Google Scholar 

  • Sulak RM, Thomas LK, Boade RR (1991) 3D reservoir simulation of Ekofisk compaction drive. J Pet Technol 291:1272–1278

    Google Scholar 

  • Tang YQ, Xu C (1997) Some environmental geology problem of Shanghai development in 21 century (in Chinese). J Underland Space 17(2):95–98

    Google Scholar 

  • Wang YM (1998) The mechanism and control of the land subsidence in Taiyuan City (in Chinese). J Taiyuan Univ Technol 29(6):599–602

    Google Scholar 

  • Wu HH, Liao YH, Wang PC (2005a) Using grey theory in quality function development to analyse dynamic customer requirements. Int J Adv Manuf Technol 25:1241–1247

    Article  Google Scholar 

  • Wu M, Qiu SJ, Liu JF, Zhao L (2005b) Prediction model based on the grey theory for tacking wax deposition in oil pipelines. J Nat Gas Chem 14:243–247

    Google Scholar 

  • Wu Q, Zhou W, Li S, Wu X (2005c) Application of grey numerical model to landwater resource evaluation. Environ Geol 47:991–999

    Article  Google Scholar 

  • Xue YQ, Zhang Y, Ye SJ (2003) Land subsidence in China and its problems (in Chinese). Quat Sci 23(6):585–592

    Google Scholar 

  • Xu YS, Shen SL, Tang CP, Jiang H (2004) 3D-analysis of land subsidence based on groundwater flow model (in Chinese). Rock Soil Mech 26(suppl):109–112

    Google Scholar 

  • Xue Y, Zhang Y, Ye S et al (2005) Land subsidence in China. Environ Geol 48:713–720

    Article  Google Scholar 

  • Xu YS, Shen SL, Bai Y (2006) State-of-the-art of land subsidence prediction due to groundwater withdrawal in China. ASCE Geotech Spec Publ 148:58–65

    Google Scholar 

  • Yu X (2004) Three methods for prediction of settlement (in Chinese). Building Tech Dev 31(5):32–33

    Google Scholar 

  • Zhang X (2000) Prediction of land subsidence in Suzhou by using gray theory (in Chinese). J Suzhou Inst Urban Constr Environ Protection 13(4):53–57

    Google Scholar 

  • Zhang AG (2002) Continuable development of Shanghai and the prevention and control of land subsidence (in Chinese). In: Wei ZX, Li QF (eds) Proceedings of the national symposium on land subsidence. Shanghai Institute of Geology Survey, Shanghai, pp 17–22

  • Zhou QN, Wang X (2002) The use of the gray theory in the prediction of subsidence (in Chinese). Soil Eng Foundation 16(4):31–33

    Google Scholar 

  • Zhang AG, Wei ZX (2002) Past, present and future research on land subsidence in Shanghai (in Chinese). Hydrogeol Eng Geol 33(5):72–75

    Google Scholar 

  • Zhang YP, Zhang TQ, Gong XN (1999) Grey forecasting to subsidence (in Chinese). Indust Construction 29(4):45–48

    Google Scholar 

  • Zhang ZX, Sun CZ et al (2000) Nero-fuzzy and soft computation (in Chinese). Xi’an Jiaotong Press, Xi’an

    Google Scholar 

Download references

Acknowledgments

This work presented in this paper was supported by the research grant (No. 40372124) from National Natural Science Foundation of China and Shanghai Key Subject (Geotechnical Engineering) Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yi-Qun Tang.

Appendices

Appendix A: Grey prediction model

Grey model with equal time-interval

Equal time-interval model is the most common grey prediction model (Zhou and Wang 2002; Lin et al. 2004; Lei et al. 2000; Zhang et al. 1999). Let the sequence of initial time-interval data be:

$$ x^{{{\left[0 \right]}}} = {\left\{{x^{{{\left(0 \right)}}} {\left({t_{1}} \right)},x^{{{\left(0 \right)}}} {\left({t_{2}} \right)}, \ldots\;,x^{{{\left(0 \right)}}} {\left({t_{n}} \right)}} \right\}} $$
(1)

with time step t i+1t i k = constant, i = 1, ... , n − 1.

By accumulative addition, generally once, a smooth sequence is obtained:

$$ x^{{{\left[1 \right]}}} = {\left\{{x^{{{\left(1 \right)}}} {\left(1 \right)},x^{{{\left(1 \right)}}} {\left(2 \right)}, \ldots\;,x^{{{\left(1 \right)}}} {\left(n \right)}} \right\}} $$
(2)

where \(x^{{{\left(1 \right)}}} {\left(k \right)} = {\sum\nolimits_{i = 1}^k {x^{{{\left(1 \right)}}} {\left({t_{i}} \right)}}}.\)

Based on the above expression, we can obtain the differential equation:

$$ \frac{{dx^{{{\left(1 \right)}}}}}{{dt}} + ax^{{{\left(1 \right)}}} = c $$
(3)

where \(c = {\sum\nolimits_{i = 1}^{n - 1} {b_{i} x^{{{\left(1 \right)}}} {\left({i + 1} \right)}}}\) and the denoting coefficient vector \({\mathop a\limits^ \wedge} = {\left[{a,c} \right]}^{\rm T}.\)

The solution is:

$$ \hat{x}^{{{\left(1 \right)}}} {\left({k + 1} \right)} = {\left[{x^{{{\left(0 \right)}}} {\left({t_{1}} \right)} - \frac{c}{a}} \right]}\exp {\left({- ak} \right)} + \frac{c}{a} $$
(4)

where coefficients a and c are the parameters to be identified and can be obtained by the least square method.

Grey model with unequal time-interval

When the time step t i+1t i ≠ constant, the initial data sequence becomes unequal time-interval data sequence. Generally, the initial data sequence obtained in the practical work is of this type. The unequal time-interval data sequence can be transformed to the equal time-interval by means of equalization. Then, the GM (1, 1) with unequal time-intervals can be established by applying the method of establishing the GM (1, 1) with equal time-interval.

A-1 Modeling procedure

Let the unequal time intervals be k i = t i+1t i , i = 1 ... n − 1

  1. (a)

    The average time interval, m is calculated by (Deng 1987):

    $$ m = \frac{1}{{n - 1}}{\sum\limits_{i = 1}^{n - 1} {k_{i}}} = \frac{1}{{n - 1}}{\left({t_{n} - t_{1}} \right)} $$
    (5)
  2. (b)

    The difference between the average time point and observed time point is calculated:

    $$ p_{i} = (i - 1)m - {\left({t_{i} - t_{1}} \right)}, \quad i = 1\;\ldots\;n $$
    (6)
  3. (c)

    The new time-interval sequence x (0) 1 is formed with time step m:

    $$ x^{{{\left(0 \right)}}}_{1} = {\left\{{x^{{{\left(0 \right)}}}_{1} {\left(1 \right)},x^{{{\left(0 \right)}}}_{1} {\left(2 \right)}, \ldots\;,x^{{{\left(0 \right)}}}_{1} {\left(n \right)}} \right\}} $$
    (7)

    where \(x^{{{\left(0 \right)}}}_{1} {\left(i \right)} = \frac{{p_{i}}}{{k_{i}}}x^{{{\left(0 \right)}}} {\left({t_{{i + 1}}} \right)} + \frac{{k_{i} - p_{i}}}{{k_{i}}}x^{{{\left(0 \right)}}} {\left({t_{i}} \right)}, \quad i = 1\;\ldots\;n.\)

  4. (d)

    The GM (1, 1) with equal time-interval is established on x (0) 1.

After once accumulative addition, a smooth sequence is obtained:

$$ x^{{{\left(1 \right)}}}_{1} = {\left\{{x^{{{\left(1 \right)}}}_{1} {\left(1 \right)},x^{{{\left(1 \right)}}}_{1} {\left(2 \right)}, \ldots\;,x^{{{\left(1 \right)}}}_{1} {\left(n \right)}} \right\}} $$
(8)

where \(x^{{{\left(1 \right)}}}_{1} {\left(k \right)} = {\sum\nolimits_{i = 1}^k {x^{{{\left(1 \right)}}}_{1} {\left({t_{i}} \right)}}}.\)

Based on the above equation, the differential equation is established:

$$ \frac{{dx^{{{\left(1 \right)}}}_{1}}}{{dt}} + ax^{{{\left(1 \right)}}}_{1} = c $$
(9)

where \(c = {\sum\nolimits_{i = 1}^{n - 1} {b_{i} x^{{{\left(1 \right)}}} {\left({i + 1} \right)}}}\) and the denoting coefficient vector \({\mathop a\limits^ \wedge} = {\left[{a,c} \right]}^{\rm T}.\)

Then, the solution is:

$$ \hat{x}^{{{\left(1 \right)}}}_{1} {\left({m + 1} \right)} = {\left[{x^{{{\left(0 \right)}}}_{1} {\left(m \right)} - \frac{c}{a}} \right]}\exp {\left({- am} \right)} + \frac{c}{a} $$
(10)

where coefficients a and c can be obtained by the least square method.

$$ {\mathop a\limits^ \wedge} = {\left[{a,c} \right]}^{\rm T} = {\left({B^{T} B} \right)}^{{- 1}} B^{\rm T} y_{n} $$
(11)

where

$$ B = {\left[{\begin{array}{*{20}c}{{- \frac{1}{2}{\left[{x^{{{\left(1 \right)}}}_{1} {\left(1 \right)} + x^{{{\left(1 \right)}}}_{1} {\left(2 \right)}} \right]}}}& {1} \\ {{- \frac{1}{2}{\left[{x^{{{\left(1 \right)}}}_{1} {\left(2 \right)} + x^{{{\left(1 \right)}}}_{1} {\left(3 \right)}} \right]}}}& {1} \\{\cdots}& {\cdots} \\{{- \frac{1}{2}{\left[{x^{{{\left(1 \right)}}}_{1} {\left({n - 1} \right)} + x^{{{\left(1 \right)}}}_{1} {\left(n \right)}} \right]}}}& {1} \\\end{array}} \right]}, \quad y_{n} = {\left\{{x^{{{\left(0 \right)}}}_{1} {\left(2 \right)},x^{{{\left(0 \right)}}}_{1} {\left(3 \right)}, \cdots, x^{{{\left(0 \right)}}}_{1} {\left(n \right)}} \right\}}^{\rm T} .$$

The generating number \(\hat{x}^{{{\left(0 \right)}}}_{1} {\left(i \right)}\) is obtained by accumulative subtraction of the predicted value \(\hat{x}^{{{\left(1 \right)}}}_{1}.\)

  1. (e)

    \(\hat{x}^{{{\left(0 \right)}}}_{1} {\left(i \right)}\) is reduced to the unequal time-interval sequence:

    $$ \hat{x}^{{{\left(0 \right)}}} {\left({t_{i}} \right)} = \frac{{m - p_{i}}}{m}\hat{x}^{{{\left(0 \right)}}}_{1} {\left({i + 1} \right)} + \frac{{p_{i}}}{m}x^{{{\left(0 \right)}}}_{1} {\left(i \right)}, \quad i = 1\;\ldots\;n $$
    (12)

A-2 Analysis and checking of prediction precision

Whether the predicted value based on the GM (1, 1) model is reliable or not, it must be verified by some means of checking and criterion of evaluation. In order to increase the prediction precision, the remaining difference must be processed further. In other words, the remaining difference of the initial data sequence must be identified. Only when the remaining difference meets the requirement, can the prediction succeed. The reliability of the prediction precision is guaranteed by the analysis of correlation extent and the checking of the post-difference.

  1. (1)

    Analysis of correlation extent

Correlation extent is distinguished by the similarity quantity among the curves and the correlation coefficient of a single sequence (Deng 1987), x (0) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{(0)}\) at point i is

$$ \xi (i) = \frac{{\min {\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{0} (i) - x^{0} (i)} \right|} + \beta {\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{0} (i) - x^{0} (i)} \right|}}}{{{\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{0} (i) - x^{0} (i)} \right|} + \beta \max {\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{0} (i) - x^{0} (i)} \right|}}} $$
(13)

where β is discrimination coefficient, 0 <  β <  1 and usually it takes 0.5.

Then the correlation extent between x (0) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ^{{(0)}}\) is

$$ d = \frac{1}{n}{\sum\nolimits_{i = 1}^n {\xi (i)}} .$$
  1. (2)

    Checking of post-difference

In the model presented in this paper, the post-difference is judged by the ratio between the mean-square deviation S 2 of the remaining difference and the mean-square deviation S 1 of the initial data. Let the initial data sequence be {x (0) (i)} and the predicted data sequence be {x (1) (i)}. The remaining difference is:

$$ \varepsilon {\left(i \right)} = x^{{{\left(0 \right)}}} {\left({t_{i}} \right)} - \hat{x}^{{{\left(0 \right)}}} {\left({t_{i}} \right),} \quad i = 1,2, \ldots\;,n $$
(14)

The ratio of post-difference is:

$$ C = \frac{{S_{1}}}{{S_{2}}} $$
(15)

where

$$ S_{1} = {\sqrt {\frac{1}{{n - 1}}{\sum\nolimits_{i = 1}^n {{\left[{x^{{{\left(0 \right)}}} {\left({t_{i}} \right)} - \bar{x}^{{{\left(0 \right)}}}} \right]}^{2}}}}}, S_{2} = {\sqrt {\frac{1}{{n - 1}}{\sum\nolimits_{i = 1}^n {{\left[{\varepsilon {\left(i \right)} - \bar{\varepsilon}} \right]}^{2}}}}} .$$

The probability for the small error is

$$ P = {\left\{{{\left| {\varepsilon {\left(i \right)} - \bar{\varepsilon}} \right|} < 0.6745S_{1}} \right\}} $$
(16)

The prediction precision indexes of the model are shown in Table 6.

Table 5 Predicted precision of test data
Table 6 Predicted precision indexes of model

A-3 Modeling remaining difference sequence

If the values of P, C both are within the allowable ranges listed in the above table, the prediction value can be computed. Otherwise, the remaining difference should be revised. By the modeling of the remaining sequence, the model precision increases and the reliability of the prediction precision can be guaranteed. Let the remaining difference sequence be:

$$ \begin{aligned} q^{{{\left(0 \right)}}} {\left(k \right)} &= x^{{{\left(1 \right)}}} {\left(k \right)} - \hat{x}^{{{\left(1 \right)}}} {\left(k \right)}, \quad k = 2, \ldots\;,n\\ q^{{{\left(0 \right)}}} &= {\left\{{q^{{{\left(0 \right)}}} {\left(i \right)}, \ldots\;,q^{{{\left(0 \right)}}} {\left(n \right)}} \right\}}\\ q^{{{\left(0 \right)}}} &= {\left\{{q^{{{\left(0 \right)}}} {\left(1 \right)}, \ldots\;,q^{{{\left(0 \right)}}} {\left({n^{'}} \right)}} \right\}}, \quad {n}^{\prime} = n - i\\ \end{aligned} $$

Then the remaining difference sequence model is:

$$ \frac{{dq^{{{\left(1 \right)}}}}}{{dt}} + {a}^{\prime}x^{{{\left(1 \right)}}} = {c}^{\prime} $$
(17)

We can obtain the function with time of the remaining difference sequence:

$$ {\mathop q\limits^ \wedge}^{1} {\left(t \right)} = {\left[{q^{{{\left(0 \right)}}} {\left(1 \right)} - \frac{{{c}^{\prime}}}{{{a}^{\prime}}}} \right]}\exp {\left({- {a}^{\prime}t} \right)} + \frac{{{c}^{\prime}}}{{{a}^{\prime}}} $$
(18)

Then the computing formula for the reduction value of the revised remaining difference model is:

$$ {\mathop {x_{1}}\limits^ \wedge}{\left({m + 1} \right)} = {\left[{x^{{{\left(0 \right)}}}_{1} {\left(1 \right)} - \frac{c}{a}} \right]}\exp {\left({- am} \right)} + \frac{c}{a} - \delta {\left({m - i} \right)}{\left({- {a}^{\prime}} \right)}{\left[{q^{{{\left(0 \right)}}} {\left(1 \right)} - \frac{{{c}^{\prime}}}{{{a}^{\prime}}}} \right]}\exp {\left({- {a}^{\prime}t} \right)} $$
(19)

where \(\delta {\left({k - i} \right)} = \left\{{\begin{array}{*{20}c}{1}& {{k\,\geqslant\,i}} \\{0}& {{k\,<\,i}} \\\end{array}} \right., i = n - {n}^{\prime}; {n}^{\prime}\) is the number of the remaining difference sequence {q 0 i }.

If the reduction value from Eq. (19) does not meet the requirement, the remaining difference sequence should be modeled again until the requirement is met. Then the prediction can be done.

Appendix B Principle and structure of ANFIS

Combining artificial neural networks with the fuzzy logic inference, ANFIS is obtained. It belongs to Sugeno fuzzy system and it consists of the inputs (initial values) and the target (predicted values) (Zhang et al. 2000). The typical fuzzy rule is as follows.

If x is A and y is B, then Z = f(x,y), where A and B are the fuzzy set of the inputs and z = f(x,y) is the accurate function of the target. Generally, f(x,y) is polynomial about input variables x, y. If f(x,y) is one-order polynomial, the obtained fuzzy inference system is one-order Sugeno fuzzy model.

Figure 12 shows the inference process of one-order Sugeno fuzzy model which has two inputs x, y and one output z, so it contains of two fuzzy if-than rules:

  • If x is A 1 and y is B 1, then f 1 = p 1 xq 1 yr 1

  • If x is A 2 and y is B 2, then f 2 = p 2 xq 2 yr 2

where A i and B i are fuzzy sets corresponding with input variables.

Fig. 12
figure 12

The inference process of one-order Sugeno fuzzy system

Supposing that the S type membership functions of input variables x and y are:

$$ S_{{A_{i}}} (x,a_{i}, b_{i}) = \frac{1}{{1 + e^{{- a_{i} (x - b_{i})}}}} $$
(20)
$$ S_{{B_{i}}} (y,c_{i}, d_{i}) = \frac{1}{{1 + e^{{- c_{i} (x - d_{i})}}}} $$
(21)

where i = 1, 2, {a i , b i } and {c i , d i } being two group characteristic parameters of S type membership functions.

S type membership functions are changed with the change of the values of characteristic parameters, that is, the membership functions of A i and B i are changed. The inference process can be equivalent to the ANFIS structure, shown in Fig. 13. The ANFIS structure consist of five layers and their functions are as follows.

Fig. 13
figure 13

The ANFIS structure equivalent to one-order Sugeno

The function of the first layer is to compute the fuzzy membership of inputs. Every node of this layer is the adaptive node and has node function. Namely,

$$ O_{{1,i}} = S_{{A_{i}}} (x,a_{i}, b_{i}), \quad i=1, 2 $$
(22)
$$ O_{{1,j}} = S_{{B_{{j - 2}}}} (y,c_{{j - 2}}, d_{{j - 2}}), \quad j=3, 4 $$
(23)

where O 1,i is the No. i output of the first layer and is the membership of corresponding output variable of fuzzy sets A i and B i .

The function of the second layer is to compute the fitness of every rule. Every node of this layer is a fixed node labeled \(\prod.\) Its output is the product of all input signals, representing the inspiriting strength of one rule, shown as:

$$ O_{{2,1}} = O_{{1,1}} \cdot O_{{1,3}} = S_{{A_{1}}} (x,a_{1}, b_{1}) \cdot S_{{B_{1}}} (y,c_{1}, d_{1}), \,\hbox{marking}\, W_{1} $$
(24)
$$ O_{{2,2}} = O_{{1,2}} \cdot O_{{1,4}} = S_{{A_{2}}} (x,a_{2}, b_{2}) \cdot S_{{B_{2}}} (y,c_{2}, d_{2}), \,\hbox{marking}\,W_{2} $$
(25)

The function of the third layer is to compute the unitary value of fitness. Every node of this layer is a fixed node labeled N. The ratio of inspiring strength of one rule to the sum of all the inspiring strengths is obtained:

$$ O_{{3,1}} = \bar{W}_{1} = \frac{{W_{1}}}{{W_{1} + W_{2}}} $$
(26)
$$ O_{{3,2}} = \bar{W}_{2} = \frac{{W_{2}}}{{W_{1} + W_{2}}} $$
(27)

The function of the fourth layer is to compute the output of every rule. Every node of this layer is an adaptive node which has node function, being:

$$ O_{{4,1}} = \bar{W}_{1} z_{1} = \bar{W}_{1} (p_{1} x + q_{1} y + r_{1}) $$
(28)
$$ O_{{4,2}} = \bar{W}_{2} z_{2} = \bar{W}_{2} (p_{2} x + q_{2} y + r_{2}) $$
(29)

where {p i , q i , r i } (i = 1, 2) is the set of parameters of corresponding nodes, namely conclusion parameters.

The function of the fifth layer is to compute the output of fuzzy system. The single node of this layer is a fixed node labeled ∑ which compute the sum of all the incoming signals and by way of the total output:

$$ O_{5} = z = {\sum {\bar{W}_{i} z_{i} =}}\bar{W}_{1} z_{1} + \bar{W}_{2} z_{2} $$
(30)

This network consists of undetermined characteristic parameters (a i , b i , c i , d i (i = 1, 2) of membership functions and p i , q i , r i (i = 1, 2) of conclusion parameters). ANFIS dynamically adjusts these parameters in the trained process to achieve the adaptive study. So the mapping relationship of inputs and outputs can be more accurately described by this trained network.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, YQ., Cui, ZD., Wang, JX. et al. Application of grey theory-based model to prediction of land subsidence due to engineering environment in Shanghai. Environ Geol 55, 583–593 (2008). https://doi.org/10.1007/s00254-007-1009-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00254-007-1009-y

Keywords

Navigation