1 INTRODUCTION
So far, all methods of predicting future settlement using past observations are based solely on the tem-poral dependence of their quantity. However, the fact that soil properties tend to exhibit a spatial cor-relation structure has been clearly shown by several studies in the past, e.g. Vanmark (1977), DeGroot & Baecher (1993). It is therefore natural to expect that the accuracy of the settlement prediction can be im-proved by taking into account the spatial correlation of ground properties, by which the observed settle-ment data from all of the different observation points can be simultaneously utilised. Furthermore, by in-troducing spatial correlation, it is possible to esti-mate the future settlement of the ground at any arbi-trary point by considering the spatial-temporal structure. This study is actually an attempt to search for such an approach.
2 SPATIAL-TEMPORAL UPDATING AND PREDICTING PROCESS
2.1 Settlement prediction by Asaoka’s Method The basic model used for settlement prediction in this paper is the first-order autoregressive model proposed by Asaoka (1978). The model is applicable for one-dimensional consolidation and is used for predicting the primary settlement based on the pre-viously observed settlement data, as follows:
k k
k y
y =β0+β1 −1+ε (1)
where yk = observed settlement at kth step of
obser-vation, β0 and β1 = constant parameters for the Asaoka model, and εk = observation model error.
[
]
⎩ ⎨ ⎧ ≠ = = m k m k E k m : 0 : 2 ε σ ε ε (2)This implies the temporally independent charac-teristic of εk. In practice, the most essential
informa-tion is usually the final settlement (yf). This can be
simply estimated based on the parameters of the Asaoka model, as follows:
1 0 1 β β − = f y (3)
Settlement Prediction by Spatial-temporal Random Process
P. Rungbanaphan & Y. Honjo
Gifu University, Gifu, Japan
I. Yoshida
Musashi Institute of Technology, Tokyo, Japan
ABSTRACT: A systematic procedure for spatial-temporal prediction of settlement based on Asaoka’s Method is proposed. A probabilistic approach is chosen because it gives a predictive probability distribution of future settlement and allows prior information of the model parameters to be incorporated into the estima-tion. The method is based on Bayesian estimation and Asaoka’s formulation by considering settlement data from all of the observation points for estimating Asaoka’s model parameters. Consecutively observed settle-ment is used for updating the model parameters by taking into account the spatial correlation structure. Auto-correlation distance of the parameters and the observation-model error are also estimated simultaneously based on Bayesian estimation using the observed data. The Kriging method is considered to be a suitable ap-proach for estimating the predicted settlement at any arbitrary location and time based on the estimated pa-rameters. Several case studies are carried out using simulated data as an example. It is concluded that, with relatively strong spatial correlation, the estimation of the model parameters and the final settlement can be significantly improved by taking into account the spatial correlation structure in comparison to the case of ig-noring spatial correlation structure. The sensitivity of this improvement to variations in the auto-correlation distance, observation spacing, and number of observation points is investigated. The accuracy for estimating auto-correlation distance, observation-model error, and the final settlement at an arbitrary location is also dis-cussed.
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2.2 Bayesian estimation considering spatial correlation structure
In order to improve the estimation and to enable lo-cal estimation, utilisation of Bayesian estimation considering spatial correlation is proposed in this paper. This approach uses prior information of the parameters and the observed settlement data from all observation points to search for the best estimates of the unknown parameters, i.e. model parameters (β1 and β0), auto-correlation distance (η), and the vari-ance of the observation-model error (σε2). The for-mulation consists of two statistical components, namely, the observation model and the prior infor-mation model. These two models will then be com-bined by Bayes’ theorem to obtain the solution. 2.2.1 Observation model
This model relates the observation data to the model parameters. At a specific time step k, let Yk denote
the observed settlement at n observation points, x1, x2,…,xn, where
[
]
T n k k k y x y x Y = ( 1) " ( ) (4)The state vector, θ, is defined as the estimates of model parameters (β ,1* * 0 β ) at the n observation points, as follows:
[
]
T n n x x x x) ( ) ( ) ( ) ( 1 1* 0* 1 0* * 1 β β β β θ = " # " (5)Consequently, the autoregressive model in Eq. (1) can be rewritten in the following matrix form
ε θ + = k k M Y (6) where ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = − − n n n k k k I x y x y M , 1 1 1 ) ( 0 0 ) ( # # # % (7)
[
]
T n k k(x1) ε (x ) ε ε = " (8)In,n denotes an n x n unit matrix. ε is a Gaussian
ob-servation-model error vector with E[ε] = 0 and E[εεT
] = Vε. Vε is a covariance matrix, the compo-nents of which are
n j i j i j i v j i ; , 1,..., : 0 : 2 , = ⎩ ⎨ ⎧ ≠ = = ε ε σ (9) It should be noted that this error is defined as the
combination of the observation error and the model error. These two kinds of errors cannot be separated in practice, and so are assumed to be integrated in the model as shown in Eq. (9).
Given θ, σε2, and Yk-1, the predicted settlement
distribution can be represented by the following multivariate normal distribution
(
)
/2 1/2 1 2, 2 ,σε − = π− ε − θ Y V Y p n k k(
)
(
)
⎥⎦⎤ ⎢⎣ ⎡− − − ⋅ θ − θ ε k k T k k M V Y M Y 1 2 1 exp (10)2.2.2 Prior information model
By assuming two multivariate stochastic Gaussian fields for β0 and β1, the prior information has the
fol-lowing structure: δ θ θ = 0 + (11) where
[
]
T n n x x x x) ( ) ( ) ( ) ( * 0 , 0 1 * 0 , 0 * 0 , 1 1 * 0 , 1 0 β β β β θ = " # " (12)[
]
T n n x x x x) ( ) ( ) ( ) ( 1 1 0 1 0 1 β β β β δ δ δ δ δ = " # " (13) β∗1,0(xi) and β∗0,0(xi) denote the prior mean at
ob-servation point xi of β1 and β0, respectively. δ is the
uncertainty of the prior mean with E[δ] = 0 and
E[δδT] = Vθ,0. Vθ,0 is a prior covariance matrix. By
introducing the spatial correlation structure in the
formulation of Vθ,0, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − = n n n n n n n n n n n n x x x x x x x x x x x x x x x x V ρ ρ ρ ρ σ ρ ρ ρ ρ σ β β θ " # % # " " # % # " 1 1 1 1 2 0 , 0 , , 1 1 1 1 2 0 , 1 0 , 0 0 (14) where 0n,n denotes an n x n zero matrix. σ2β1,0 and
σ2
β0,0 represent the prior variance of β1 and β0,
re-spectively. ρ(|xi - xj|) denotes the auto-correlation
function. The exponential type auto-correlation function is chosen for the current study because it is commonly used in geotechnical applications (e.g. Vanmarcke 1977). The function is given as
(
)
[
η]
ρ xi−xj =exp− xi−xj / (15)
where xi, xj = spatial vector coordinate, and η =
auto-correlation distance.
It should be noted that, for the sake of simplifica-tion, there are two important assumptions about the correlation structure for formulating the above
co-variance matrix. Firstly, β1 and β0 are assumed to be
independent of one another. Secondly, the correla-tion structures of these two parameters are identical, meaning that they share the same auto-correlation distance.
Given η, prior means, and prior variances of β1
and β0, the prior distribution of the model
parame-ters is also a multivariate normal distribution of the following form
( )
= − − ⎢⎣⎡−(
−)
−(
−)
⎥⎦⎤ 0 1 0 2 / 1 2 1 exp 2π θ θ θ θ η θ Vθ Vθ p n T (16)It is clear from this formulation that the spatial correlation of soil properties is included in the form
of the spatial correlation of β1 and β0. The
settle-ments themselves are not correlated spatially. The authors believe that this is the most suitable way to introduce the spatial correlation structure to the set-tlement prediction model since it is soil properties that are spatially correlated, not the settlement. 2.2.3 Bayesian estimation
Suppose that the set of observations Yk at the time
step k = 0,1,…,K has already been obtained. By employing Bayes’ theorem, the posterior distribution
of the state vector θ can be formulated as
(
)
( )
∏
(
)
= − ∝ K k k k Y Y p p Y p 1 1 2 2 , , , ,σε η θη θ σε θ (17)where Y denotes the set of all observed data, i.e. Y =
(Y1,Y2,…,YK). By substituting Eq.(10) and (16) into
the above equation, a likelihood function can be
de-fined, with the given values of σε2 and η, as follows:
(
2)
(2 )/2 1/2 /2 2 ; , , n Kn V V K Y L σε η θ = π− + θ − ε −(
)
(
)
⎩ ⎨ ⎧ ⎢ ⎣ ⎡ − − − ⋅ − 0 1 0 2 1 exp θ θ TVθ θ θ(
)
(
)
⎭ ⎬ ⎫ ⎥ ⎦ ⎤ − − +∑
= − K k k k T k k M V Y M Y 1 1 θ θ ε (18)The Bayesian estimator of θ, i.e. θ*, is the one
that maximises the above function. Therefore, it is equivalent to minimising the following objective function
( ) (
)
(
0)
1 0 θ θ θ θ θ = − Vθ− − J T(
)
(
)
∑
= − − − + K k k k T k k M V Y M Y 1 1 θ θ ε (19)By differentiating the above equation with respect to the state vector, we obtain
( )
(
)
(
)
0 2 2 1 1 0 * 1 − − − = = ∂ ∂∑
= − − K k k k T kV Y M M V J θ θ θ θ θ ε θ (20) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + =∑
∑
= − − − = − − K k k T k K k k T kV M V M V Y M V 1 1 0 1 1 1 1 1 * ε θ ε θ θ θ (21)By trial and error, the values of σε2, η, and the
corresponding θ* that give the maximum value of
the likelihood function (L) can be obtained. These values are actually the Bayesian estimators for the current problem.
2.3 Local estimation by the Kriging Method
Based on the calculated statistical inferences of the model parameters at the observation points and the auto-correlation distance, the model parameters at any arbitrary locations can be estimated by the ordi-nary Kriging method (Krige 1966, Matheron 1973, Wackernagel 1998). This method provides an unbi-ased and least error estimator built on the data from a random field. It is also assumed that the random field is second-order stationary. Based on the
esti-mated model parameters (β*1, β*0) at the n
observa-tion points x1, …, xn,the value of β*1 and β*0, at an
arbitrary point x0 can be estimated by the following
equations: ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ n T n n w w x x x x x x # # # 1 * 0 1 * 0 * 1 1 * 1 0 * 0 0 * 1 ) ( ) ( ) ( ) ( ) ( ) ( β β β β β β (22) where
(
)
(
)
(
)
(
)
(
)
(
)
⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 x x x x x x x x x x x x w w n n n n n n ρ ρ ρ ρ ρ ρ µ # " " # # % # " # (23)wi (i = 1, …, n) are the weights attached to the
data at each of the observation points. µ is the
La-grange multiplier used for minimising the Kriging
error, and x0 denotes the spatial vector coordinate at
x0. ρ(|xi - xj|) represents the auto-correlation function
as defined in Eq. (15).
3 SIMULATION EXPERIMENTS
3.1 Random field generation by frequency-domain technique
To investigate the efficiency and practicality of the proposed approach, 2-dimensional random field of the model parameters are generated based on the as-sumed mean, variance, and auto-correlation dis-tance. The observed settlement data is then calcu-lated by Eq. (1), using the generated parameters and the assumed variance of the observation-model
er-ror, σε2. Performing the spatial-temporal updating
procedure previously stated in Section 2.2 based on the generated observed data, the statistical infer-ences of the model parameters at each observation point can be back-calculated. A comparison of these inferences with the simulated ones, namely the true values, reveals the efficiency of the procedure.
Various techniques have been proposed by sev-eral authors for random field generation, e.g. the turning bands method (Matheron 1973), frequency domain technique (Shinozuka 1971, Shinozuka & Jan 1972), and local average subdivision method (Fenton & Vanmarcke 1990). The frequency domain
technique is chosen for this study to avoid the streaking problem which is found in the turning bands method, and the difficulties of implementing the local average subdivision method (Fenton 1994). This technique concentrates on the spectral density function (SDF) of the process, which is defined as the Fourier transform of the auto-correlation tion. For an exponential type auto-correlation func-tion, it can be proved that the SDF
(
)
32 2 2 2 1 2 2 1 1 2 1 ) , ( ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + + = ω ω η πη ω ω S (24)which is a function of frequency domain, ω1 and ω2 Assuming that the power of the employed SDF is negligible outside the interval [-ω1,0,ω1,0] and [-ω2,0,ω2,0], the simulated stationary Gaussian random field at any coordinate (x,y) can be expressed as the following series of cosine functions
∑∑
= = ∆ = 2 1 1 1 2 1 2 1 1( , ) 2 ) , ( M k M j k j S y x X ω ω ωω ) cos(ω1jx+ω2ky+φjk ⋅ (25) where ∆ω1 = 2ω1,0/M1, ∆ω2 = 2ω2,0/M2, ω1j = -ω1,0+(j-1/2)∆ω1 , ω2k = -ω2,0+(k-1/2)∆ω2, and φjk =random phase angles, uniformly and independently distributed in the interval (0,2π). M1 and M2 are the number of equally divided intervals of the range [-ω1,0,ω1,0] and [-ω2,0,ω2,0], respectively. Care must be taken when selecting these ranges and discretization intervals to ensure that the spectral density function is adequately approximated.
3.2 Improvement of the estimation by considering spatial correlation structure
A series of simulation experiments was performed based on the aforementioned procedure. It was de-cided to limit the number of simulations for each ex-periment to 100 and not to let the sample size exceed 64. For the purpose of recognizing trends in the re-sults, these selections seem sufficient.
For simulation of the model parameters, it is as-sumed that the mean and standard deviation of the random field of these parameters are β*
1 = 0.9791, β*
0 = 6.94 cm, and σβ1 = 0.0028, σβ0 = 0.59 cm, re-spectively. These values are chosen from the data presented by Asaoka (1978) based on the observa-tions of Kobe Port No. 3. It should be noted that, based on Eq. (3), the first order mean and variance of the final settlement, yf, are 332 cm and 53 cm,
re-spectively, assuming statistical independency be-tween β1 and β0. This level of uncertainty for final settlement estimation is considered to be common in engineering practice. The initial settlement (y0) is set as zero for every observation point. For the current study, the observation-model error (σε) is assumed
to be 1 cm. By assigning the desired values of auto-correlation distance (η), random values of the model parameters together with the observed settlements at each observation point can be generated, as de-scribed in Section 3.1.
To investigate the effect of sampling size, three different layouts of observation plans, with 16, 36, and 64 observation points (n), are set. All of these are arranged in a square grid pattern with even spac-ing of s and total width of L, as shown in Figure 1. Based on the generated observed data, the procedure proposed in Section 2.2 is performed by assuming that the prior mean and standard deviation of the model parameters, i.e. β*
1,0, β*0,0 and σβ1,0, σβ0,0 (see Eq. (12) and (14)), are equal to β*
1, β*0 and σβ1, σβ0, respectively. The auto-correlation distance and the observation-model error are also assigned the same values as those used for generating the simu-lated data, namely the true values.
Figure 1. Layout of observation plans.
In order to examine the advantages of considering spatial correlation structure, the Bayesian estima-tion, using the observed settlement of each point to estimate the model parameters of that point itself, i.e. ignoring spatial correlation structure, is also per-formed based on the same parameters. The different model parameters are randomly generated 100 times (Nsim = 100) and the estimation errors, as a percent
of the true values, are calculated. The estimations based on these two different conditions are com-pared and presented in Figure 2 and Table 1.
1 2 3 4 1 2 3 4 3xs = L 3xs = L 1 2 3 4 5 6 1 2 3 4 5 6 5xs = L 5xs = L 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 7xs = L 7xs = L n = 64 n = 36 n = 16
Figure 2. Estimation error v.s. time factor for n = 36 and s/η = 0.5
Figure 2 illustrates the plots of the mean error against time factor, Tv, of the model parameter and
final settlement estimations for n = 36 and s/η = 0.5. The time factor at a specific time step is calculated based on the relationship between β1 and the time interval derived from Asaoka’s formulation (Asaoka 1978). β*1 is used for this calculation. The estima-tion errors are represented by the term ‘mean error’ which is defined as mean error X N i truei i true i est N X X X X
∑
= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × − = 1 2 , , , 100 (26) where Xest,i and Xtrue,i denote the estimated value andtrue value, respectively, of the parameter to be esti-mated at each observation point for each simulation. Nx is the total number of estimated values, i.e. Nx = n
× Nsim. The estimated values and true values of final
settlement are calculated based on Eq. (3) using the corresponding values of the model parameters.
Figure 2 clearly shows that the mean errors for the cases of considering the spatial correlation struc-ture are lower that those of ignoring spatial correla-tion structure, regardless of the observacorrela-tion time. This confirms that the estimation can be improved by taking into account the spatial correlation struc-ture. This trend is the same for both model parame-ters and the final settlement estimation.
Table 1. Comparison of error of final settlement estimation be-tween considering and ignoring spatial correlation structure for
estimation at the 50th time step (Tv = 0.424)
* Improvement (%) = [(1) - (2)] × 100 / (1)
To investigate the sensitivity of this improvement for different soil and observation conditions, the same calculations for several sampling sizes (n), and ratios of spacing to auto-correlation distance (s/η) are performed. Then, the mean, bias, and improve-ment values of the errors at the 50th time step, which corresponds to Tv of 0.424, are calculated and
sum-marised in Table 1. The bias value is defined as the mean of (estimated value – true value) as follows:
bias X N i truei i true i est N X X X X
∑
= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × − = 1 , , , 100 (27) The improvement values are the percentreduc-tion of the mean error by considering spatial correla-tion structure, compared with the case of ignoring spatial correlation structure. This value represents the level of improvement of settlement estimation by taking into account the spatial correlation structure.
It can be seen from Table 1 that the improvement values increase with the reduction of s/η ratio. This leads us to conclude that a stronger spatial correla-tion gives a better estimacorrela-tion. Moreover, enlarging the sampling size with constant spatial correlation structure does not greatly improve the accuracy of the estimation, from which we speculate that only neighbouring observations contribute to the im-provement of the estimation. It should be noted that this improvement results from the improvement of model parameter estimations, as shown in Figure 2. The bias values are relatively low for all estimations.
considering spatial corr. Mean (%) (1) Bias (%) Mean (%) (2) Bias (%)
0.5 5.696 -0.344 4.788 -0.570 15.937 0.25 5.631 -0.398 4.082 -0.580 27.498 2 5.695 -0.548 5.662 -0.586 0.576 1 5.612 -0.596 5.320 -0.620 5.200 0.5 5.703 -0.517 4.799 -0.621 15.853 0.25 5.611 -0.314 3.975 -0.584 29.165 0.5 5.680 -0.477 4.678 -0.409 17.641 0.25 5.673 -0.249 3.845 -0.386 32.213
Ignoring spatial corr.
Error of yf estimation 16 n Improvement (%) * 36 64 s /η 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 0.2 0.4 0.6 0.8 1 Time factor, Tv M e an er ro r o f β 1 e sti m a ti o n ( % )
Ignoring Spatial Correlation Structure Considering Spatial Correlation Structure
0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Time factor, Tv M e a n e rro r o f β 0 e sti m a tio n ( % )
Ignoring Spatial Correlation Struct ure Considering Spatial Correlation Structure
0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 Time factor, Tv M e a n e rro r o f yf e stim a tio n ( % )
Ignoring Spatial Correlat ion Structure Considering Spatial Correlation St ruct ure
3.3 Estimation of auto-correlation distance and observation-model errors
In the previous section, the true values of auto-correlation distance (η) and observation-model error (σε) are assumed to be known and are used in the es-timation procedure. In practice, however, these pa-rameters are unknown and need to be estimated based on the observed data. It was proposed in Sec-tion 2.2 that these parameters can be estimated by an optimization procedure based on Bayesian estima-tion. By performing numerical experiments for sev-eral cases, the error of this estimation and the sensi-tivity of the model parameter estimation to this error can be investigated as shown below.
Table 2 summarises the statistical inferences of the error of auto-correlation distance and observa-tion-model error estimation for sampling size, n = 36 (see Fig. 1) at the 50th time step (Tv = 0.424). ηest/η
denotes the ratio of the estimated auto-correlation distance (ηest) to its true value (η). L is the total
width of the group of observation points as shown in Figure 1. The number of simulations for each trial is 50. The other parameters, such as β*
1, β*0, σβ1, σβ0, y0, are assigned the same values as in Section 3.2.
Table 2. Error of auto-correlation distance and
observation-model error estimation for n = 36 at the 50th time step (Tv =
0.424)
Note: NP implies the estimation is not possible under the as-signed condition.
Table 2 clearly shows that the error of η estima-tion is much higher than that of σε estimation. With the mean of error below 3%, it is concluded that σε
can be accurately estimated by the proposed ap-proach. However, the huge error of η estimation can be reduced by increasing the L/η ratio and reducing the value of σε. It should be noted that the estimation may not be able to perform at relatively low values of L/η ratio together with high values of σε. In addi-tion, it should be kept in mind that Table 2 shows only the estimation errors at the 50th time step. Any estimation at an earlier stage can give the higher level of error due to the lack of observed data.
With the large error of η estimation, the sensitiv-ity of the model parameter estimation to this error is of interest. To investigate this sensitivity, several values of auto-correlation distance are assumed and used for estimating the model parameters based on
the proposed method. Figure 3 shows the error of model parameter and final settlement estimations for different ratios of the assumed value to the true value of the auto-correlation distance, namely ηa/η.
These calculations are performed based on the con-ditions n = 36, s/η = 0.5, and Tv = 0.424, and the
number of simulations for each trial is 100. The other parameters are assigned the same values as stated in Section 3.2. As might be expected, for an extremely low value of ηa/η, i.e. 0.1, the case of
considering spatial correlation structure gives simi-lar results as that of ignoring spatial correlation structure. However, for a relatively wide range of ηa/η, i.e. from 0.5 to 4, a high level of improvement
can be similarly obtained by taking into account the spatial correlation structure. Then, at an extremely high value of ηa/η, i.e. 8, the difference between
these two approaches becomes smaller again. From these results, we conclude that the sensitivity of the model parameter and final settlement estimations to the error of η estimation is fairly low. In other words, the estimates of these parameters with a simi-lar level of accuracy be obtained, even though the error of η estimation is relatively high.
Figure 3. Estimation error v.s. ηa/η ratio for n = 36, s/η = 0.5
at the 50th time step (Tv = 0.424)
Mean SD COV Mean (%) SD (%) COV 10 2 1.789 0.960 0.537 2.100 2.493 1.187 5 1 2.220 0.809 0.364 1.900 2.452 1.290 2.5 0.5 3.706 1.125 0.304 1.500 2.315 1.543 10 2 3.736 2.358 0.631 1.500 2.315 1.543 5 1 4.928 1.993 0.404 1.100 2.092 1.902 2.5 0.5 NP NP NP NP NP NP 0.5 1.0 s /η L /η σε (cm) Error of σε estimation ηest/η 0.00 0.04 0.08 0.12 0.16 0.20 0 2 4 6 8 10 ηa/η M ean er ro r o f β 1 es ti m at io n ( % )
Ignoring Spat ial Correlation Structure Considering Spat ial Correlation Struct ure
0.0 0.8 1.6 2.4 3.2 4.0 0 2 4 6 8 10 ηa/η M e a n e rro r o f β 0 e stim a tio n (% )
Ignoring Spatial Correlation Structure Considering Spatial Correlation Structure
0 2 4 6 8 0 2 4 6 8 10 ηa/η M e a n e rro r o f yf e stim a tio n ( % )
Ignoring Spat ial Correlation St ructure Considering Spat ial Correlation St ructure
3.4 Estimation of final settlement at an arbitrary location
As mentioned previously, one of the advantages of the proposed method is its ability to estimate the set-tlement at any arbitrary location. By applying the Kriging method (see Section 2.3), the model pa-rameter at a specific point can be approximated based on the estimated parameters at the observation points. Then, the final settlement at that point can be predicted using Eq. (3). To investigate the level of error for this prediction, a series of numerical exam-ples was performed, the results of which are shown and discussed in this section.
Figure 4. Observation plan and locations of the points to be es-timated
Figure 4 shows the plan of the observation points and the location of the points to be considered for settlement prediction. Several calculations are per-formed based on the conditions n = 36, s/η = 1.0, and Tv = 0.424, the results of which are summarised
in Table 3. The number of simulations for each trial is 100. The values of other parameters are also the same as those assigned in Section 3.2.
Table 3 summarises the mean and bias of the es-timation error for final settlement, yf. These values
are calculated according to Eq. (26) and (27), with the true values determined by the simulated model parameters of the same random field at that point. Concerning the large error for η estimation as dis-cussed in the previous section, Table 3 also shows the comparison between the estimation using the true value versus that using the estimated value of both η and σε. According to Table 2, note that the mean of ηest/η ratio and the error of σε estimation for
the current condition is 4.928 and 1.1 (%), respec-tively. It can be seen that the difference between the mean error of the estimation using the true value (Case A) and that using the estimated value of both η and σε (Case B) is relatively low despite the huge error for η estimation. This proofs that the proposed method is practical for estimating final settlement at an arbitrary location using observed data.
Table 3. Error of final settlement estimation at several
loca-tions (see Fig. 4) for n = 36, s/η = 1.0, σε = 1.0 (cm) at the 50th
time step (Tv = 0.424)
* Case A refers to the case which the calculation is performed
based on the true values of both η and σε
** Case B refers to the case which the calculation is performed
based on the estimated values of both η and σε
*** Difference (%) = [(2) - (1)] × 100 / (1)
To further investigate the practicability of this method, the extensions of the above table for differ-ent s/η ratio and observation period are performed and summarised in Table 4. Due to the fact that the auto-correlation distance cannot be estimated by the proposed method in some conditions, i.e. low L/η ra-tio or short observara-tion period, the calculara-tions us-ing the estimated value of both η and σε (Case B) in Table 3 are replaced by those using estimated value of σε and assumed value of η (Case C), which is
as-sumed to be 5 times of true value. Concerning the level of error for η estimation shown in Table 2, this assumption seems reasonable.
The advantages of including the spatial correla-tion structure into the settlement estimacorrela-tion can clearly be seen from Table 4. For the site that the spatial correlation of the soil parameters is relatively strong, i.e. s/η = 0.25, the final settlement can be predicted with a similar level of accuracy at the point located within the group of observation points or within the length of auto-correlation distance around the group, i.e. at points 1, 2, and 3. This level of accuracy will be reduced with the increase of the distance from the group of observation points to the point to be considered. On the other hands, for the site that the soil parameters tend to be independent, i.e. s/η = 10, the mean errors of the settlement esti-mation by the proposed method are similar, regard-less of the locations. The difference between the er-rors for the estimations at the earlier stage, i.e. at Tv
= 0.208 (the 25th time step), and at the later stage, i.e. at Tv = 0.424 (the 50th time step), is noticeable
only for the case that the spatial correlation is strong and, especially, at the points within the range of spa-tial correlation distance. These results emphasize the merit of considering spatial correlation structure for the local estimation, especially, when the soil pa-rameters are strongly correlated in space, which is quite usual. Furthermore, the estimation errors for Case A and Case C are similar in any conditions. This also confirms the insensibility of the proposed
Mean (%) (1) Bias (%) Mean (%) (2) Bias (%)
1 9.374 -0.077 9.637 0.409 2.808 2 10.341 -0.423 10.474 -0.047 1.284 3 10.232 -0.160 10.463 0.513 2.255 4 13.392 3.980 13.797 5.051 3.024 5 15.828 2.758 16.548 3.348 4.554 Point Error of yf estimation
Case A * Case B ** Difference
(%) *** 1 2 3 4 5 6 1 2 3 4 5 6 1 2 4 2.5s 10s 5 L = 5xs 3 0.5s 0.5s 0.5s L = 5xs 0.5L 0.5L
approach with the value of auto-correlation distance, which makes the approach practical even though the true value of the auto-correlation distance is difficult to be obtained.
Table 4. Error of final settlement estimation at several loca-tions (see Fig. 4) with different s/η ratio and observation
pe-riod for n = 36, σε = 1.0 (cm)
* Case A refers to the case which the calculations are
per-formed based on the true values of both η and σε
** Case C refers to the case which the calculations are
per-formed based on the estimated values of σε and the assumed
values of η (= 5η)
4 CONCLUSION
A methodology was presented for observation based settlement prediction with consideration of spatial correlation structure. The spatial correlation is intro-duced among the soil properties and the settlements at various points are spatially correlated through these parameters, which naturally describe the phe-nomenon.
The proposed spatial-temporal formulation is considered to have the following two main advan-tages:
(1) The settlement prediction can be improved by considering the spatial correlation structure. It is concluded that a stronger spatial correlation struc-ture gives a better estimation for both model pa-rameters and final settlement.
(2) The settlement prediction at any arbitrary point becomes possible through the interpolation of the soil parameters using the Kriging Method.
In addition, it was found that, while the observa-tion model error can be estimated accurately, the er-ror of auto-correlation distance estimation is consid-erably high. The error will be even larger when the ratio of total width of the observations to the
auto-correlation distance reduces or the observation-model error increases. However, it was proved that the accuracy of settlement prediction is considerably insensitive to the relatively wide range of auto-correlation distance estimated and the accuracy of the final settlement prediction at an arbitrary loca-tion using the true value of the auto-correlaloca-tion dis-tance is also close to that using the estimated value. Therefore, it can be concluded that the proposed method is practical for the settlement prediction.
Extension of this study to the actual observation data of ground settlement in a large area is of inter-est for future research.
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Case A * Case C ** Case A * Case C ** 1 5.397 5.684 2.807 3.121 2 6.545 6.926 3.065 3.269 3 7.146 7.793 4.219 4.839 4 11.055 11.397 10.808 11.050 5 16.624 17.668 17.073 18.126 1 11.418 11.720 9.374 9.570 2 11.697 12.222 10.341 10.439 3 11.000 11.571 10.232 10.515 4 13.419 14.217 13.392 14.078 5 15.430 16.437 15.828 16.990 1 16.258 16.525 16.266 17.134 2 15.403 15.801 15.416 16.023 3 12.763 12.996 12.937 13.173 4 15.293 15.151 15.248 15.118 5 15.495 15.546 15.783 15.841 Point
Mean error of yf estimation (%)
at Tv = 0.208 at Tv = 0.424 s/η L /η
0.25 1.25
1 5
