Settlement Prediction by Spatial-temporal Random Process

Transcription

1 Safety, Relablty ad Rs of Structures, Ifrastructures ad Egeerg Systems Furuta, Fragopol & Shozua (eds Taylor & Fracs Group, Lodo, ISBN Settlemet Predcto by Spatal-temporal Radom Process P. Rugbaapha & Y. Hojo Gfu Uversty, Gfu, Japa I. Yoshda Musash Isttute of Techology, Toyo, Japa ABSTRACT: A systematc procedure for spatal-temporal predcto of settlemet based o Asaoa s Method s proposed. A probablstc approach s chose because t gves a predctve probablty dstrbuto of future settlemet ad allows pror formato of the model parameters to be corporated to the estmato. The method s based o Bayesa estmato ad Asaoa s formulato by cosderg settlemet data from all of the observato pots for estmatg Asaoa s model parameters. Cosecutvely observed settlemet s used for updatg the model parameters by tag to accout the spatal correlato structure. Autocorrelato dstace of the parameters ad the observato-model error are also estmated smultaeously based o Bayesa estmato usg the observed data. The rgg method s cosdered to be a sutable approach for estmatg the predcted settlemet at ay arbtrary locato ad tme based o the estmated parameters. Several case studes are carred out usg smulated data as a example. It s cocluded that, wth relatvely strog spatal correlato, the estmato of the model parameters ad the fal settlemet ca be sgfcatly mproved by tag to accout the spatal correlato structure comparso to the case of gorg spatal correlato structure. The sestvty of ths mprovemet to varatos the auto-correlato dstace, observato spacg, ad umber of observato pots s vestgated. The accuracy for estmatg auto-correlato dstace, observato-model error, ad the fal settlemet at a arbtrary locato s also dscussed. INTRODUCTION So far, all methods of predctg future settlemet usg past observatos are based solely o the temporal depedece of ther quatty. However, the fact that sol propertes ted to exhbt a spatal correlato structure has bee clearly show by several studes the past, e.g. Vamar (977, DeGroot & Baecher (99. It s therefore atural to expect that the accuracy of the settlemet predcto ca be mproved by tag to accout the spatal correlato of groud propertes, by whch the observed settlemet data from all of the dfferet observato pots ca be smultaeously utlsed. Furthermore, by troducg spatal correlato, t s possble to estmate the future settlemet of the groud at ay arbtrary pot by cosderg the spatal-temporal structure. Ths study s actually a attempt to search for such a approach. SPATIAL-TEMPORAL UPDATING AND PREDICTING PROCESS. Settlemet predcto by Asaoa s Method The basc model used for settlemet predcto ths paper s the frst-order autoregressve model proposed by Asaoa (978. The model s applcable for oe-dmesoal cosoldato ad s used for predctg the prmary settlemet based o the prevously observed settlemet data, as follows: y β + βy + ( where y observed settlemet at th step of observato, β ad β costat parameters for the Asaoa model, ad observato model error. σ : m E[ m ] ( : m Ths mples the temporally depedet characterstc of. I practce, the most essetal formato s usually the fal settlemet (y f. Ths ca be smply estmated based o the parameters of the Asaoa model, as follows: β y ( f β 7

2 . Bayesa estmato cosderg spatal correlato structure I order to mprove the estmato ad to eable local estmato, utlsato of Bayesa estmato cosderg spatal correlato s proposed ths paper. Ths approach uses pror formato of the parameters ad the observed settlemet data from all observato pots to search for the best estmates of the uow parameters,.e. model parameters (β ad β, auto-correlato dstace (η, ad the varace of the observato-model error (σ. The formulato cossts of two statstcal compoets, amely, the observato model ad the pror formato model. These two models wll the be combed by Bayes theorem to obta the soluto... Observato model Ths model relates the observato data to the model parameters. At a specfc tme step, let Y deote the observed settlemet at observato pots, x, x,,x, where [ y ( x y ( x ] T Y ( The state vector,, s defed as the estmates of model parameters ( β, β at the observato pots, as follows: [ β ( x β ( x β ( x ( ] T ( β x Cosequetly, the autoregressve model Eq. ( ca be rewrtte the followg matrx form Y M + ( where y ( x I (7 y ( x M, [ ( x ( x ] T (8 I, deotes a x ut matrx. s a Gaussa observato-model error vector wth E[] ad E[ T ] V. V s a covarace matrx, the compoets of whch are σ : j ;, j,...,, : j v j (9 It should be oted that ths error s defed as the combato of the observato error ad the model error. These two ds of errors caot be separated practce, ad so are assumed to be tegrated the model as show Eq. (9. Gve, σ, ad Y -, the predcted settlemet dstrbuto ca be represeted by the followg multvarate ormal dstrbuto p / / ( Y, σ, Y π V T exp ( Y M V ( Y M (.. Pror formato model By assumg two multvarate stochastc Gaussa felds for β ad β, the pror formato has the followg structure: + δ ( where [ β ( x β ( x β ( x ( ] T (,,, β, x [ δ x δ ( x δ ( x δ ( ] T δ ( β( β β β x β,(x ad β,(x deote the pror mea at observato pot x of β ad β, respectvely. δ s the ucertaty of the pror mea wth E[δ] ad E[δδ T ] V,. V, s a pror covarace matrx. By troducg the spatal correlato structure the formulato of V,, we have ρ( x x ρ( x x V, σ β, ρ ( x x ρ( x x, σ β, ρ ρ ( x x ρ( x x ( x x ρ( x x, ( where, deotes a x zero matrx. σ β, ad σ β, represet the pror varace of β ad β, respectvely. ρ( x - x j deotes the auto-correlato fucto. The expoetal type auto-correlato fucto s chose for the curret study because t s commoly used geotechcal applcatos (e.g. Vamarce 977. The fucto s gve as ( x x j exp[ x x j ] ρ /η ( where x, x j spatal vector coordate, ad η auto-correlato dstace. It should be oted that, for the sae of smplfcato, there are two mportat assumptos about the correlato structure for formulatg the above covarace matrx. Frstly, β ad β are assumed to be depedet of oe aother. Secodly, the correlato structures of these two parameters are detcal, meag that they share the same auto-correlato dstace. Gve η, pror meas, ad pror varaces of β ad β, the pror dstrbuto of the model parameters s also a multvarate ormal dstrbuto of the followg form 7

3 / T ( η π V exp ( V ( p ( It s clear from ths formulato that the spatal correlato of sol propertes s cluded the form of the spatal correlato of β ad β. The settlemets themselves are ot correlated spatally. The authors beleve that ths s the most sutable way to troduce the spatal correlato structure to the settlemet predcto model sce t s sol propertes that are spatally correlated, ot the settlemet... Bayesa estmato Suppose that the set of observatos Y at the tme step,,, has already bee obtaed. By employg Bayes theorem, the posteror dstrbuto of the state vector ca be formulated as ( Y, σ, η p( η p( Y, σ, Y p (7 where Y deotes the set of all observed data,.e. Y (Y,Y,,Y. By substtutg Eq.( ad ( to the above equato, a lelhood fucto ca be defed, wth the gve values of σ ad η, as follows: ( + / / (,, ; / σ η π V V L Y exp + T ( V ( T ( Y M V ( Y M (8 The Bayesa estmator of,.e., s the oe that maxmses the above fucto. Therefore, t s equvalet to mmsg the followg objectve fucto J T ( ( V ( + T ( Y M V ( Y M (9 By dfferetatg the above equato wth respect to the state vector, we obta J ( ( T V ( M V Y M ( T T V + M V M V + M V Y ( By tral ad error, the values of σ, η, ad the correspodg that gve the maxmum value of the lelhood fucto (L ca be obtaed. These values are actually the Bayesa estmators for the curret problem.. Local estmato by the rgg Method Based o the calculated statstcal fereces of the model parameters at the observato pots ad the auto-correlato dstace, the model parameters at ay arbtrary locatos ca be estmated by the ordary rgg method (rge 9, Mathero 97, Waceragel 998. Ths method provdes a ubased ad least error estmator bult o the data from a radom feld. It s also assumed that the radom feld s secod-order statoary. Based o the estmated model parameters (β, β at the observato pots x,, x,the value of β ad β, at a arbtrary pot x ca be estmated by the followg equatos: β ( x β ( x where β ( x β ( x β ( x β x ( ( x x ρ( x x T w w ( x x ( w ρ ρ ( ( ( ( w ρ x x ρ x x ρ x x µ w (,, are the weghts attached to the data at each of the observato pots. µ s the Lagrage multpler used for mmsg the rgg error, ad x deotes the spatal vector coordate at x. ρ( x - x j represets the auto-correlato fucto as defed Eq. (. SIMULATION EXPERIMENTS. Radom feld geerato by frequecy-doma techque To vestgate the effcecy ad practcalty of the proposed approach, -dmesoal radom feld of the model parameters are geerated based o the assumed mea, varace, ad auto-correlato dstace. The observed settlemet data s the calculated by Eq. (, usg the geerated parameters ad the assumed varace of the observato-model error, σ. Performg the spatal-temporal updatg procedure prevously stated Secto. based o the geerated observed data, the statstcal fereces of the model parameters at each observato pot ca be bac-calculated. A comparso of these fereces wth the smulated oes, amely the true values, reveals the effcecy of the procedure. Varous techques have bee proposed by several authors for radom feld geerato, e.g. the turg bads method (Mathero 97, frequecy doma techque (Shozua 97, Shozua & Ja 97, ad local average subdvso method (Feto & Vamarce 99. The frequecy doma 77

4 techque s chose for ths study to avod the streag problem whch s foud the turg bads method, ad the dffcultes of mplemetg the local average subdvso method (Feto 99. Ths techque cocetrates o the spectral desty fucto (SDF of the process, whch s defed as the Fourer trasform of the auto-correlato fucto. For a expoetal type auto-correlato fucto, t ca be proved that the SDF S ( ω, ω ( πη + ( ω + ω η whch s a fucto of frequecy doma, ω ad ω Assumg that the power of the employed SDF s eglgble outsde the terval [-ω,,ω, ] ad [- ω,,ω, ], the smulated statoary Gaussa radom feld at ay coordate (x,y ca be expressed as the followg seres of cose fuctos X ( x, y M M j S ( ω j, ω ω ω cos( ω j x + ω y + φ j ( where ω ω, /M, ω ω, /M, ω j - ω, +(j-/ ω, ω -ω, +(-/ ω, ad φ j radom phase agles, uformly ad depedetly dstrbuted the terval (,π. M ad M are the umber of equally dvded tervals of the rage [- ω,,ω, ] ad [-ω,,ω, ], respectvely. Care must be tae whe selectg these rages ad dscretzato tervals to esure that the spectral desty fucto s adequately approxmated.. Improvemet of the estmato by cosderg spatal correlato structure A seres of smulato expermets was performed based o the aforemetoed procedure. It was decded to lmt the umber of smulatos for each expermet to ad ot to let the sample sze exceed. For the purpose of recogzg treds the results, these selectos seem suffcet. For smulato of the model parameters, t s assumed that the mea ad stadard devato of the radom feld of these parameters are β.979, β.9 cm, ad σ β.8, σ β.9 cm, respectvely. These values are chose from the data preseted by Asaoa (978 based o the observatos of obe Port No.. It should be oted that, based o Eq. (, the frst order mea ad varace of the fal settlemet, y f, are cm ad cm, respectvely, assumg statstcal depedecy betwee β ad β. Ths level of ucertaty for fal settlemet estmato s cosdered to be commo egeerg practce. The tal settlemet (y s set as zero for every observato pot. For the curret study, the observato-model error (σ s assumed to be cm. By assgg the desred values of autocorrelato dstace (η, radom values of the model parameters together wth the observed settlemets at each observato pot ca be geerated, as descrbed Secto.. To vestgate the effect of samplg sze, three dfferet layouts of observato plas, wth,, ad observato pots (, are set. All of these are arraged a square grd patter wth eve spacg of s ad total wdth of L, as show Fgure. Based o the geerated observed data, the procedure proposed Secto. s performed by assumg that the pror mea ad stadard devato of the model parameters,.e. β,, β, ad σ β,, σ β, (see Eq. ( ad (, are equal to β, β ad σ β, σ β, respectvely. The auto-correlato dstace ad the observato-model error are also assged the same values as those used for geeratg the smulated data, amely the true values. 8 7 xs L 7xs L xs L 7 8 Fgure. Layout of observato plas. I order to exame the advatages of cosderg spatal correlato structure, the Bayesa estmato, usg the observed settlemet of each pot to estmate the model parameters of that pot tself,.e. gorg spatal correlato structure, s also performed based o the same parameters. The dfferet model parameters are radomly geerated tmes (N sm ad the estmato errors, as a percet of the true values, are calculated. The estmatos based o these two dfferet codtos are compared ad preseted Fgure ad Table. xs L 7xs L xs L 78

5 Mea error of β estmato (% Mea error of β estmato (% Mea error of yf estmato (% Fgure. Estmato error v.s. tme factor for ad s/η. Fgure llustrates the plots of the mea error agast tme factor, T v, of the model parameter ad fal settlemet estmatos for ad s/η.. The tme factor at a specfc tme step s calculated based o the relatoshp betwee β ad the tme terval derved from Asaoa s formulato (Asaoa 978. β s used for ths calculato. The estmato errors are represeted by the term mea error whch s defed as mea error Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure....8 Tme factor, T v N X X est, Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure....8 Tme factor, T v X Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure....8 Tme factor, T v X true, N true, X ( where X est, ad X true, deote the estmated value ad true value, respectvely, of the parameter to be estmated at each observato pot for each smulato. N x s the total umber of estmated values,.e. N x N sm. The estmated values ad true values of fal settlemet are calculated based o Eq. ( usg the correspodg values of the model parameters. Fgure clearly shows that the mea errors for the cases of cosderg the spatal correlato structure are lower that those of gorg spatal correlato structure, regardless of the observato tme. Ths cofrms that the estmato ca be mproved by tag to accout the spatal correlato structure. Ths tred s the same for both model parameters ad the fal settlemet estmato. Table. Comparso of error of fal settlemet estmato betwee cosderg ad gorg spatal correlato structure for estmato at the th tme step (T v. Improvemet (% [( - (] / ( To vestgate the sestvty of ths mprovemet for dfferet sol ad observato codtos, the same calculatos for several samplg szes (, ad ratos of spacg to auto-correlato dstace (s/η are performed. The, the mea, bas, ad mprovemet values of the errors at the th tme step, whch correspods to T v of., are calculated ad summarsed Table. The bas value s defed as the mea of (estmated value true value as follows: bas s /η N X Error of y f estmato Igorg spatal corr. cosderg spatal corr. Mea (% ( Bas (% Mea (% ( Bas (% X est, X X true, N X true, Improvemet (% (7 The mprovemet values are the percet reducto of the mea error by cosderg spatal correlato structure, compared wth the case of gorg spatal correlato structure. Ths value represets the level of mprovemet of settlemet estmato by tag to accout the spatal correlato structure. It ca be see from Table that the mprovemet values crease wth the reducto of s/η rato. Ths leads us to coclude that a stroger spatal correlato gves a better estmato. Moreover, elargg the samplg sze wth costat spatal correlato structure does ot greatly mprove the accuracy of the estmato, from whch we speculate that oly eghbourg observatos cotrbute to the mprovemet of the estmato. It should be oted that ths mprovemet results from the mprovemet of model parameter estmatos, as show Fgure. The bas values are relatvely low for all estmatos. 79

6 . Estmato of auto-correlato dstace ad observato-model errors I the prevous secto, the true values of autocorrelato dstace (η ad observato-model error (σ are assumed to be ow ad are used the estmato procedure. I practce, however, these parameters are uow ad eed to be estmated based o the observed data. It was proposed Secto. that these parameters ca be estmated by a optmzato procedure based o Bayesa estmato. By performg umercal expermets for several cases, the error of ths estmato ad the sestvty of the model parameter estmato to ths error ca be vestgated as show below. Table summarses the statstcal fereces of the error of auto-correlato dstace ad observato-model error estmato for samplg sze, (see Fg. at the th tme step (T v.. η est /η deotes the rato of the estmated auto-correlato dstace (η est to ts true value (η. L s the total wdth of the group of observato pots as show Fgure. The umber of smulatos for each tral s. The other parameters, such as β, β, σ β, σ β, y, are assged the same values as Secto.. Table. Error of auto-correlato dstace ad observatomodel error estmato for at the th tme step (T v. σ (cm.. L /η s /η η est /η Error of σ estmato Mea SD COV Mea (% SD (% COV NP NP NP NP NP NP Note: NP mples the estmato s ot possble uder the assged codto. Table clearly shows that the error of η estmato s much hgher tha that of σ estmato. Wth the mea of error below %, t s cocluded that σ ca be accurately estmated by the proposed approach. However, the huge error of η estmato ca be reduced by creasg the L/η rato ad reducg the value of σ. It should be oted that the estmato may ot be able to perform at relatvely low values of L/η rato together wth hgh values of σ. I addto, t should be ept md that Table shows oly the estmato errors at the th tme step. Ay estmato at a earler stage ca gve the hgher level of error due to the lac of observed data. Wth the large error of η estmato, the sestvty of the model parameter estmato to ths error s of terest. To vestgate ths sestvty, several values of auto-correlato dstace are assumed ad used for estmatg the model parameters based o the proposed method. Fgure shows the error of model parameter ad fal settlemet estmatos for dfferet ratos of the assumed value to the true value of the auto-correlato dstace, amely η a /η. These calculatos are performed based o the codtos, s/η., ad T v., ad the umber of smulatos for each tral s. The other parameters are assged the same values as stated Secto.. As mght be expected, for a extremely low value of η a /η,.e.., the case of cosderg spatal correlato structure gves smlar results as that of gorg spatal correlato structure. However, for a relatvely wde rage of η a /η,.e. from. to, a hgh level of mprovemet ca be smlarly obtaed by tag to accout the spatal correlato structure. The, at a extremely hgh value of η a /η,.e. 8, the dfferece betwee these two approaches becomes smaller aga. From these results, we coclude that the sestvty of the model parameter ad fal settlemet estmatos to the error of η estmato s farly low. I other words, the estmates of these parameters wth a smlar level of accuracy be obtaed, eve though the error of η estmato s relatvely hgh. Mea error of β estmato (% Mea error of β estmato (% Mea error of yf estmato (% Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure 8 η a /η Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure 8 η a /η Igorg Spatal Correlato Structure Cosderg Spatal Correlato Structure 8 η a /η Fgure. Estmato error v.s. η a /η rato for, s/η. at the th tme step (T v. 8

7 . Estmato of fal settlemet at a arbtrary locato As metoed prevously, oe of the advatages of the proposed method s ts ablty to estmate the settlemet at ay arbtrary locato. By applyg the rgg method (see Secto., the model parameter at a specfc pot ca be approxmated based o the estmated parameters at the observato pots. The, the fal settlemet at that pot ca be predcted usg Eq. (. To vestgate the level of error for ths predcto, a seres of umercal examples was performed, the results of whch are show ad dscussed ths secto. L xs.l.l L xs.s.s.s.s Fgure. Observato pla ad locatos of the pots to be estmated Fgure shows the pla of the observato pots ad the locato of the pots to be cosdered for settlemet predcto. Several calculatos are performed based o the codtos, s/η., ad T v., the results of whch are summarsed Table. The umber of smulatos for each tral s. The values of other parameters are also the same as those assged Secto.. Table summarses the mea ad bas of the estmato error for fal settlemet, y f. These values are calculated accordg to Eq. ( ad (7, wth the true values determed by the smulated model parameters of the same radom feld at that pot. Cocerg the large error for η estmato as dscussed the prevous secto, Table also shows the comparso betwee the estmato usg the true value versus that usg the estmated value of both η ad σ. Accordg to Table, ote that the mea of η est /η rato ad the error of σ estmato for the curret codto s.98 ad. (%, respectvely. It ca be see that the dfferece betwee the mea error of the estmato usg the true value (Case A ad that usg the estmated value of both η ad σ (Case B s relatvely low despte the huge error for η estmato. Ths proofs that the proposed method s practcal for estmatg fal settlemet at a arbtrary locato usg observed data. s Table. Error of fal settlemet estmato at several locatos (see Fg. for, s/η., σ. (cm at the th tme step (T v. Error of y f estmato Pot Case A Case B Dfferece Mea (% ( Bas (% Mea (% ( Bas (% (% Case A refers to the case whch the calculato s performed based o the true values of both η ad σ Case B refers to the case whch the calculato s performed based o the estmated values of both η ad σ Dfferece (% [( - (] / ( To further vestgate the practcablty of ths method, the extesos of the above table for dfferet s/η rato ad observato perod are performed ad summarsed Table. Due to the fact that the auto-correlato dstace caot be estmated by the proposed method some codtos,.e. low L/η rato or short observato perod, the calculatos usg the estmated value of both η ad σ (Case B Table are replaced by those usg estmated value of σ ad assumed value of η (Case C, whch s assumed to be tmes of true value. Cocerg the level of error for η estmato show Table, ths assumpto seems reasoable. The advatages of cludg the spatal correlato structure to the settlemet estmato ca clearly be see from Table. For the ste that the spatal correlato of the sol parameters s relatvely strog,.e. s/η., the fal settlemet ca be predcted wth a smlar level of accuracy at the pot located wth the group of observato pots or wth the legth of auto-correlato dstace aroud the group,.e. at pots,, ad. Ths level of accuracy wll be reduced wth the crease of the dstace from the group of observato pots to the pot to be cosdered. O the other hads, for the ste that the sol parameters ted to be depedet,.e. s/η, the mea errors of the settlemet estmato by the proposed method are smlar, regardless of the locatos. The dfferece betwee the errors for the estmatos at the earler stage,.e. at T v.8 (the th tme step, ad at the later stage,.e. at T v. (the th tme step, s otceable oly for the case that the spatal correlato s strog ad, especally, at the pots wth the rage of spatal correlato dstace. These results emphasze the mert of cosderg spatal correlato structure for the local estmato, especally, whe the sol parameters are strogly correlated space, whch s qute usual. Furthermore, the estmato errors for Case A ad Case C are smlar ay codtos. Ths also cofrms the sesblty of the proposed 8

8 approach wth the value of auto-correlato dstace, whch maes the approach practcal eve though the true value of the auto-correlato dstace s dffcult to be obtaed. Table. Error of fal settlemet estmato at several locatos (see Fg. wth dfferet s/η rato ad observato perod for, σ. (cm s/ η L /η Pot.. Case A refers to the case whch the calculatos are performed based o the true values of both η ad σ Case C refers to the case whch the calculatos are performed based o the estmated values of σ ad the assumed values of η ( η CONCLUSION Mea error of y f estmato (% at T v.8 at T v. Case A Case C Case A Case C A methodology was preseted for observato based settlemet predcto wth cosderato of spatal correlato structure. The spatal correlato s troduced amog the sol propertes ad the settlemets at varous pots are spatally correlated through these parameters, whch aturally descrbe the pheomeo. The proposed spatal-temporal formulato s cosdered to have the followg two ma advatages: ( The settlemet predcto ca be mproved by cosderg the spatal correlato structure. It s cocluded that a stroger spatal correlato structure gves a better estmato for both model parameters ad fal settlemet. ( The settlemet predcto at ay arbtrary pot becomes possble through the terpolato of the sol parameters usg the rgg Method. I addto, t was foud that, whle the observato model error ca be estmated accurately, the error of auto-correlato dstace estmato s cosderably hgh. The error wll be eve larger whe the rato of total wdth of the observatos to the autocorrelato dstace reduces or the observatomodel error creases. However, t was proved that the accuracy of settlemet predcto s cosderably sestve to the relatvely wde rage of autocorrelato dstace estmated ad the accuracy of the fal settlemet predcto at a arbtrary locato usg the true value of the auto-correlato dstace s also close to that usg the estmated value. Therefore, t ca be cocluded that the proposed method s practcal for the settlemet predcto. Exteso of ths study to the actual observato data of groud settlemet a large area s of terest for future research. REFERENCES Asaoa, A Observatoal procedure of settlemet predcto. Sol ad Foudatos 8(: 87. DeGroot, D. J. & Baecher, G. B. 99. Estmatg autocovarace of -stu sol propertes. Joural of Geotechcal Egeerg 9(: 7. Feto, G. A. 99. Error evaluato of three radom-feld geerators. Joural of Egeerg Mechacs, ASCE (: Feto, G. A. & Vamarce, E. H. 99. Smulato of radom felds va local average subdvso. Joural of Egeerg Mechacs, ASCE (8: Hoshya, M. & Yoshda, I. 99. Idetfcato of codtoal stochastc gaussa feld. Joural of Egeerg Mechacs, ASCE (: -8. Hoshya, M. & Yoshda, I Process ose ad optmum observato codtoal stochastc felds. Joural of Egeerg Mechacs, ASCE (: -. Jourel, A. G. & Hujbregts, Ch. J Mg Geostatstcs. New Yor, N.Y: Academc Press, Ic. rge, D. G. 9. Two dmesoal weghted movg averagg tred surfaces for ore evaluato. Proc., of Symp. o Math., Statstcs ad Comp. Appl. for Ore Evaluato. Mathero, G. 97. The trsc radom fuctos ad ther applcatos. Adv. Appl. Probab.. Shozua, M. 97. Smulato of multvarate ad multdmesoal radom processes. Joural of the Acoustcal Socety of Amerca 9(, part : 7 7. Shozua, M. & Ja, C. -M. 97. Dgtal smulato of radom processes ad ts applcatos. Joural of Soud ad Vbrato (: 8. Srdhara, A., Murthy, N. S. & Praash, Rectagular hyperbolar method of cosoldato aalyss. Géotechque 7(: 8. Ta, S. A. 99. Hyperbolc method for settlemets clays wth vertcal dras. Caada Geotechcal Joural :. Vamarce, E. H Probablstc modelg of sol profles. Joural of the Geotechcal Egeerg Dvso, ASCE (GT: 7. Waceragel, H Multvarate Geostatstcs: A Itroducto wth Applcatos. d ed. Germay: Sprger- Verlag Berl Hedelberg. 8