Arch. Min. Sci., Vol. 53 (2008), No 2, p

Transkrypt

1 Arch. Mn. Sc., Vol. 53 (2008), No 2, p JANUSZ OSTROWSKI*, ADAM ĆMIEL** THE USE OF A LOGIT MODEL TO PREDICT THE PROBABILITY OF DAMAGE TO BUILDING STRUCTURES IN MINING TERRAINS ZASTOSOWANIE MODELU LOGITOWEGO DO PREDYKCJI PRAWDOPODOBIEŃSTWA USZKODZE- NIA BUDYNKÓW NA TERENACH GÓRNICZYCH The method of assessng the probablty of damage to buldngs n mnng terran accordng to logt models are presented. These models can be appled to plannng underground hard coal explotaton wth fall of roof under buldngs of tradtonal constructon of any age and techncal state and can easly be updated based on current results on surface deformaton and adapted to deal wth partcular mnng and geologcal features whch obtan n land areas each of whch may have ts own characterstcs. It s shown that the probablty of damage to buldngs manly depends on ther techncal state an so called ndex of damage. Key words: deformatons of terran surface, horzontal stran, probablty of damage to buldng structure, logt model Ochrona obektów budowlanych przed szkodlwym wpływam podzemnej eksploatacj złóż kopaln użytecznych jest warunkem prowadzena dzałalnośc górnczej. Możlwość uszkodzena obektów ocena sę na podstawe wynków prognoz zjawsk towarzyszących eksploatacj górnczej oraz oszacowań cech techncznych obektów właścwośc ch podłoża gruntowego. Szczególnym przedmotem takch ocen jest przewdywany skutek oddzaływana deformacj cągłych powerzchn terenu na budynk. Ustalene, które budynk mogą zostać uszkodzone odbywa sę zgodne z formalną procedurą polegającą na porównanu kategor odpornośc danego budynku na wpływy deformacj cągłych podłoża z kategorą terenu górnczego w mejscu lokalzacj budynku. Kategore odpornośc określa sę stosując zazwyczaj tzw. metodę punktową, uwzględnającą najważnejsze cechy technczne budynku. Kategore terenu górnczego wyznacza sę na podstawe prognozy deformacj cągłych, obecne według teor Knothego. Przyjmuje sę, że budynek jest zagrożony uszkodzenem, jeżel oszacowana metodą punktową kategora jego odpornośc jest mnejsza od przewdywanej kategor terenu górnczego w mejscu posadowena budynku. * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING SURVEYING AND ENVIRONMENTAL ENGINEERING, AL. MICKIEWICZA 30, KRAKÓW, POLAND ** AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF APPLIED MATHEMATICS, AL. MICKIEWICZA 30, KRAKÓW, POLAND

2 162 Oszacowane odpornośc budynków na deformacje cągłe podłoża różną sę od wartośc rzeczywstych, a także prognozowane wartośc tych deformacj różną sę od wartośc, które rzeczywśce wystąpą, ze względu na brak możlwośc bezbłędnego opsu tych cech zjawsk, zarówno na podstawe wynków pomarów, jak na drodze modelowana matematycznego. Stwerdza sę systematyczne losowe rozbeżnośc pomędzy wynkam oszacowań prognoz a wynkam obserwacj zachodzących zjawsk co powoduje, że ocena zagrożena budynków wpływam deformacj podłoża od projektowanej eksploatacj górnczej ne może być w pełn warygodna. Szczególne nepewna nejednoznaczna jest taka ocena w przypadku posługwana sę kategoram terenu górnczego kategoram odpornośc budynków. Badana zaprezentowane w publkacj (Ostrowsk, 2006) wykazały, że jednoznaczna ocena zagrożena budynku wpływam deformacj cągłych powerzchn może zostać dokonana poprzez porównane prognozowanej wartośc tzw. neprzekraczalnego odkształcena pozomego z wartoścą odkształcena pozomego wyznaczającego tzw. odporność krytyczną budynku. Wartośc tych odkształceń otrzymuje sę na podstawe prognozy deformacj powerzchn według teor Knothego oraz oceny odpornośc budynku metodą punktową z uwzględnenem systematycznych losowych składowych neadekwatnośc zastosowanego modelu teoretycznego metody oceny do rzeczywstośc. Porównane odpornośc krytycznej budynku z wartoścą neprzekraczalnego odkształcena pozomego w mejscu jego posadowena pozwala na jednoznaczne stwerdzene, czy uszkodzena budynku przekroczą neodczuwalny stopeń ucążlwośc użytkowana (Kwatek, 2002) czy też stopeń ten będze wększy. W perwszym przypadku budynek można uznać za nezagrożony wyłączyć go z procedury dalszej, szczegółowej oceny. Powyższa metoda została nazwana metodą bezpecznego szacowana zagrożena budynku uszkodzenam (Ostrowsk, 2006). Przewdywany stopeń uszkodzena budynku można ocenć na podstawe wartośc lorazu neprzekraczalnych odkształceń pozomych odkształceń wyznaczających odporność krytyczną, nazwanego wskaźnkem uszkodzena. Istotną nformacją charakteryzującą przewdywane uszkodzene budynku jest prawdopodobeństwo wystąpena takego zdarzena. Ze względu na trudnośc w wyznaczanu wartośc odchylena standardowego odkształcena pozomego charakteryzującego odporność budynku na deformacje cągłe podłoża, proponowane dotychczas metody ne są efektywne. Mankament ten jest jednak możlwy do wyelmnowana, jeżel wykorzysta sę wynk obserwacj skutków oddzaływana eksploatacj górnczej w budynkach, które znalazły sę w zasęgu jej wpływów. Na podstawe wynków obserwacj można zbudować logtowy model prawdopodobeństwa wystąpena zdarzena losowego, jakm jest uszkodzene budynku na skutek oddzaływana deformacj podłoża. W modelu logtowym zmenna objaśnana ma zakres wartośc [0,1], co oznacza uszkodzene lub neuszkodzene budynku. Model scharakteryzowany jest przez trzy składowe: składową losową, będącą wektorem utworzonym przez nezależne obserwacje o rozkładach z wykładnczej rodzny rozkładów, składową systematyczną, będącą wektorem nazwanym predyktorem lnowym η o współrzędnych, które są lnowym funkcjam zmennych objaśnających (w budowanym modelu są to cechy technczne budynku oraz wskaźnk charakteryzujące deformacje podłoża), funkcję wążącą składową losową systematyczną. Model buduje sę używając kanoncznej funkcj wążącej, przypsującej lnowemu predyktorow naturalny parametr, jakm jest logt. Parametry modelu logtowego estymuje sę metodą najwększej warogodnośc, a ocenę dobroc dopasowana modelu dokonuje sę na podstawe odchylena, oblczonego jako podwojony logarytm statystyk lorazu warogodnośc. Prawdopodobeństwo p (w zbudowanym modelu jest to prawdopodobeństwo uszkodzena budynku) zwązane jest z wartoścą lnowego predyktora η zależnoścą: e p 1 e Jakość prognozy prawdopodobeństwa uszkodzena budynku ocena sę na podstawe wskaźnków: czułośc prognozy, specyfcznośc prognozy dokładnośc całkowtej. Wskaźnk te wyznacza sę na podstawe prognozowanych lośc budynków uszkodzonych neuszkodzonych w porównanu do lośc budynków faktyczne uszkodzonych neuszkodzonych. Budowa modelu logtowego wymagała opracowana prognozy deformacj powerzchn oraz oszacowana odpornośc budynków na etape oceny zagrożena tych budynków wpływam projektowanej eksploatacj górnczej, a następne zebrana danych dokumentujących zastnałe deformacje oraz uszkodzena budynków po dokonanu eksploatacj. Odpowedne dane zostały uzyskane z czterech rejonów eksploatacj

3 163 pokładów węgla kamennego, dokonanych na głębokoścach od 280 m do 570 m, w warstwach złoża o wysokośc od 1,4 m do 3,2 m. Oddzaływanu tych eksploatacj poddane zostały budynk o różnych kategorach odpornośc na wpływy górncze, różnym stane techncznym (od 0,04 do 0,98) różnym weku (od 1 roku do 115 lat). Na skutek oddzaływana eksploatacj w budynkach wystąpły uszkodzena o stopnach zróżncowanych od 0,0 do 0,5. Estymacja prawdopodobeństwa uszkodzena budynków została dokonana na podstawe zmennych objaśnających, spośród których wyróżnono zmenne jakoścowe (kategora terenu górnczego kategora odpornośc budynku) oraz zmenne loścowe (prognozowane według teor Knothego odkształcene pozome ε Kn max, neprzekraczalne odkształcene pozome εnp max, odkształcene pozome granczne ε gr wynkające z oceny odpornośc budynków metodą punktową, stan technczny budynku s t, wek budynku t b, odporność krytyczna ε kr max oraz wskaźnk uszkodzena I nk). Badane zależnośc prawdopodobeństwa uszkodzena budynku od zmennych objaśnających wykonano przyjmując założene, że każdy badany model pownen zawerać zmenne reprezentujące prognozowane deformacje cągłe podłoża budynku oraz cechy technczne budynku zwązane z jego odpornoścą na deformacje. W wynku dokonanych badań stwerdzono, że najbardzej odpowedn do oceny prawdopodobeństwa uszkodzena budynku poddanego oddzaływanu deformacj podłoża, wynkających z wpływów górnczych, jest model w postac: η = +5,190 8,704. s t + 0,170. I nk Wartość predyktora η decydującego o wartośc prawdopodobeństwa uszkodzena budynku zależy od stanu techncznego budynku s t oraz wskaźnka uszkodzena I nk.. Analza rozkładu prawdopodobeństwa uszkodzena budynku według tego modelu wykazuje, że model bardzej zdecydowane reaguje na zmany stanu techncznego s t budynku nż na zmany wskaźnka uszkodzena I nk. Waga stanu techncznego s t jako zmennej objaśnającej uszkodzene budynku jest dużo wyższa nż waga drugej zmennej objaśnającej. Wynka z tego, że budynk w złym stane techncznym są relatywne dużo bardzej zagrożone uszkodzenam nż budynk w dobrym stane, nezależne od wartośc spodzewanych deformacj podłoża. Zaproponowana w publkacj metoda oceny prawdopodobeństwa uszkodzena budynków na terenach górnczych może być stosowana w przypadku projektowana eksploatacj podzemnej z zawałem stropu oraz budynków o konstrukcj tradycyjnej. Słowa kluczowe: deformacje powerzchn terenu, pozome odkształcena, prawdopodobeństwo uszkodzena budynku, model logtowy 1. Introducton One of the most mportant ssues touchng n mnng terrans s the protecton of buldng structures from the harmful mpact of mnes on ther foundatons. A prelmnary assessment of the threat to buldngs wll based on a prognoss of the deformaton of the land surface as well as an estmaton of those techncal features of a buldng, whch affect the resstance of the buldng structure to contnuous deformaton of ts foundatons. The degree of the threat to a buldng structure wll be clarfed by a comparson of both characterstcs. The valdty ths of assessment, and further acton based on t, depend on the relablty of data on both characterstcs. Underestmaton of the threat brngs us too close to the safety lmt for the buldng, or wll lead to ncreased repar costs. An overestmaton of the threat can unnecessarly ncrease the mnng costs and (or) preventve buldng measures, albet guaranteeng a hgh level of safety. In practce, a faultless assessment of the threat to a buldng caused by contnuous deformaton of ts foundatons, s not possble because of the huge degree of complexty

4 164 of the mpact as well as the contrbuton of random factors to the deformaton process. Mathematcal models descrbng these mpacts are always a mere approxmaton to realty and descrbe average states resultng from the nfluence of the determnng factors. The nfluence of random factors s usually defned by standard devaton from the modelled characterstcs of the phenomenon. Methods for assessng the threat to buldngs caused by contnuous deformaton demand a knowledge of the predcted values of the horzontal stran or curvature of the surface of the terran, as well as the lmtng values of the horzontal stran (n short lmtng stran ), whch determne the resstance of the buldng. These values are calculated by the applcaton of determnstc models, as average values. Dependng on the method, standard devatons of ndvdual characterstcs are taken nto account or not. Methods of one-dmensonal analyss (Batkewcz et al., 1977; Sroka et al., 1994) estmate the threat based on the predcted values of the horzontal strans of the surface wth a defnte standard devaton and an error-free estmaton of the values of lmtng horzontal strans. Methods of two-dmensonal analyss addtonally take nto account the standard devatons of ndces characterzng the resstance of the buldng (Sroka et al., 1994; Kwatek, 2004). The method for predctng the costs of repar to buldngs n mnng terrans, wth the applcaton of a logt model s a partcular example of twodmensonal analyss (Kaszowska, 2002). A dsadvantage of the one-dmensonal analyss method s the assumpton that the assessment of buldng resstance s wthout errors, whch s not possble n practce. However two-dmensonal analyss demands a knowledge of the value of the standard devaton of the ndex descrbng the resstance of the buldng structure. Ths value can be acheved only based on premses resultng from the experence of the researcher. Ths artcle presents a proposal for the assessment of the probablty of buldng damage n mnng areas by the applcaton of a logt model. The logt model s sutable n the analyss of such a problem, because t consders the lmtablty of dependent varables and also assumes the applcaton of explanatory varables both quanttatve as well as qualtatve and ther nteracton as predctors. Unlke n competng models whch use quantle functons of other than logstc dstrbuton (e.g. the probt model), the nterpretaton of the estmated parameters s easy and does not depend on the samplng method (retrospectve or prospectve). 2. Formulatng of the problem Assessng the threat of damage to buldngs resultng from deformaton ther foundatons, whch s related to mnng can be done n the followng ways: 1) a comparson of the categores of buldng resstance to contnuous deformaton (table 1) wth the categores of mnng area (table 2) n whch the buldngs are stuated, 2) t can be based on the expert opnons of mnng and constructon specalsts.

5 165 The frst method s appled n the assessment of numerous groups of buldngs of tradtonal constructon and t s compulsory when so-called actvty mnng plans are prepared (Rozporządzene, 2002, part 2, paragraph 2.26, annex 1 and 2). Second method s appled n the case of buldngs of partcular sgnfcance, be t socal, cultural, hstorcal, archtectural, or wth hgh vulnerablty to the mpact of mnng. The applcaton of each of these methods demands a determnaton of the predcted surface deformaton ndexes around the buldngs as well as the nature of ther resstance to contnuous deformaton. Currently, a prognoss of surface deformaton s n lne wth Knothe s theory (Knothe 1984). The assessment of the resstance of a buldng to contnuous deformatons of ts foundatons,.e. the category of buldng resstance, s made based on a score method (Zasady, 1979; Kwatek, 2002). The dea of ths method s to assgn of a defnte number n c of ponts to ndvdual techncal features of the buldng and to defne ts buldng resstance category (KO) based on the sum of ponts (tab. 1). The assessment of the threat to buldngs of mnng-nduced surface deformaton s based on the results of a comparson the predcted (accordng to Knothe s theory) value of horzontal stran ε (as the maxmum value ths ndex at the terran local to the buldng locaton ε Kn max or category of the mnng terran ε (KTG) max ) wth the value of the lmtng horzontal stran expressed by the category of buldng resstance (KO) ε lm. Qualfcaton table of score method Tabela kwalfkacyjna metody punktowej TABLE 1 TABLICA 1 Category of buldng resstance (KO) Lmtng horzontal stran (KO) ε lm [mm/m] Accordng to (Zasady..., 1979) Sum of n c ponts Accordng to (Kwatek, 2002) ε (4) l = 9, ε (3) l = 6, ε (2) l = 3, ε (1) l = 1, ε (0) l = 0, However, the results of assessments prepared wth ths determnstc method mght not be very relable, because they do not take nto account the consderable nfluence of random factors on the behavour of the analysed phenomnon. Partcularly, comparng the category of mnng terran to the category of buldng resstance can lead to unsatsfactory results, for reasons shown by Ostrowsk (2006).

6 166 Categores of mnng terran Kategore terenu górnczego TABLE 2 TABLICA 2 Category of Values of the foreseen deformaton mnng terran Slope Radus of curvature Horzontal stran (KTG) [mm/m] [km] [mm/m] (0) T max = 0,5 (0) R mn = 40 (0) ε max = 0,3 I (I) T max = 2,5 (I) R mn = 20 (II) ε max = 1,5 II (II) T max =5,0 (II) R mn = 12 (III) ε max = 3,0 III (III) T max =10,0 (III) R mn = 6 (IV) ε max = 6,0 IV (IV) T max =15,0 (IV) R mn = 4 (V) ε max = 9,0 V (V) T > 15,0 (V) R < 4 (0) ε > 9,0 It can be deduced from geodetc measurements, that the standard devaton of horzontal strans σ ε s (Popołek & Ostrowsk, 1981; Popołek et al., 1999): avr 0,15 0,25 max (1) avr where ε max s maxmum, average value of horzontal stran. It should be natural, that the categores of mnng land were formulated n tmes, when so-called the basc verson of Knothe s theory was appled to prognoses of deformaton. In that verson the value of coeffcent B bndng vertcal and horzontal deformatons s: B = 0,4r (2) where r s the radus of the dsperson of mpacts. Later research (Popołek & Ostrowsk, 1978; Hejmanowsk et al., 2005) showed that the value of coeffcent B n the explotaton of hard coal deposts and also deposts of copper ores s smaller and ts average value s: B = 0,32r (3) The dfference results from changes n the condtons of depost explotaton over the years, manly due to carryng out the explotaton at deeper and deeper levels. From the above a very mportant concluson can be drawn: the results of prognoses of horzontal strans ε Kn, whch were made accordng to the basc verson of Knothe s theory were overestmated by about 25% compared to values resultng from geodetc observatons.

7 167 The result of the prognoss of horzontal strans accordng to the basc verson of Knothe s theory vere overestmated by about 25% compared to values resultng from geodetc observatons. Tche results of the prognoss of horzontal strans accordng to the basc verson of Knothe s theory s nfluenced by systematc and random factors wth defnte values. Ths s what causes the dscrepancy between the results of prognoses and the results of observatons. Takng nto consderaton relaton (1) t was possble to estmate that n ths case the probablty of the occurrence of a horzontal stran ε larger than the predcted ε Kn s P ε = 0,21 (Popołek, 1989). To remove ths error, whch results from systematc and random factors, from the result of the prognoss, an not exceedable horzontal stran ε nexc max was defned at work (Ostrowsk, 2006). The value ths ndex, satsfyng the condton P(ε > ε nexc max ) = 0,05, s calculated from the formula: nexc max, 2 Kn max 1 (4) Kn where ε max s the extreme (n pont and n tme) of the horzontal stran calculated accordng to the basc verson of Knothe s theory. The resstance of a buldng to deformaton of foundaton, or strctly to horzontal stran ε, t s dentfed wth lmtng horzontal stran ε lm equal to the bottom lmt of the resstance category (tab. 1), sutable for a gven buldng. The category of resstance s establshed wth the score method. From the accomplshed analyss n the paper (Ostrowsk, 2006) t can be concluded that t s possble to estmate the resstance of buldng as so-called crtcal horzontal strans ε cr res, whch s counted from formula: where: cr ε res cr ε res = 0,65ε lm (5) crtcal horzontal stran (crtcal resstance) markng the lmt, exceedng ths lmt means damage to the buldng, beyond mperceptble degree (Kwatek, 2002) of dffculty ts use, ε lm lmtng horzontal stran marked wth the nterpolaton method from the relaton between the number n c of ponts and horzontal lmtng stran ε (KO) lm. In (Ostrowsk, 2006) a so-called ndex of predctable damage to the buldng I nk was defned. Its value s calculated from the formula: I nk nexc max (6) cr res

8 168 Index I nk determne the predcted degree of damage to the buldng whch can be understood as the extent of necessary repars or preventve measures. These forecasted horzontal deformatons of surface ε (KO) gr, ε Kn, ε nexc max, the resstance of buldngs to contnuous deformatons ε (KO) lm, ε lm, ε cr res, and ndex I nk, were analysed based on the logt model of the probablty of the occurrence of a random epsode, n ths case damage to a buldng. Ths study s amed at fndng the sgnfcance of the respectve characterstcs for the probablty of damage to a buldng and constructng an approprate predcton equaton. 3. Logt model theoretcal bass The nfluence of explanatory varables on the dependent varable can be descrbed by a regresson model (e.g. lnear or non-lnear). However, f the dependent varable s bounded, e.g. [0,1], a classcal regresson model s not adequate, because wthout specal restrctons mposed on the ranges of explanatory values, a dependent varable can take values beyond the establshed range. A proper tool n the modellng of bnary responses (dependent varables) s a logt model whch s one versons of the generalzed lnear model (GLZ). The generalzed lnear model (McCullagh & Nelder, 1989; Agrest, 1990) s characterzed by three components: Y1 1. Random component Y s a vector made by ndependent observatons Y, Y n = 1,, n of dstrbutons from the Nelder-Wedderburn exponental famly of dstrbutons wth densty or mass functon of the form: y b( ) c( y, ) a ( ) f ( y ;, ) e (7) where θ and φ are parameters, and functons a (φ), b(θ ) and c(y, φ) are known as smooth (dfferentable twce) functons. Parameter θ of the random varable dstrbuton Y acts as a locaton parameter and can take dfferent values for respectve observatons Y, = 1,, n. The parameter φ s the same for all the observatons. It can be nterpreted as a scale parameter and t s treated as the nusance parameter A systematc component s a vector η =, the coordnates of whch are lnear n x11 x1m functons of explanatory varables,.e. η = Xβ, where X = s a ma- x n1 xnm

9 169 1 trx of the plan of experment (lke n a classcal lnear model), vector β = m s a vector of unknown parameters. Vector η s called lnear predctor. Its components η take the form: m ; =1,,n, j=1,,m, m<n (8) x j j1 j 3. A functon lnkng a systematc component wth a random varable. If ths s wrtten as µ = E(Y ), the expected value of the varable Y, then expected value µ s connected wth the lnear predctor η by functon g, more precsely where g s a monotonc and dfferentable functon. η = g(µ ), (9) In partcular, let us consder a random component Y made by ndependent bnary random varables Y, = 1,, n of Bernoull dstrbutons wth the mass functons y 1 y (1 p) f ( y, p ) p : y =0,1; p (0,1) (10) Let us take nstead of the parameter p of Bernoull dstrbuton f (y, p) = p y (1 p) 1 y a new parameter θ defned by the equaton: p ( p) log (11) 1 p The new parameter θ called logt, s a functon of parameter p, beng the expected value of a dchotomous response Y.e. p = E(Y). Logt θ (p) s a log odds of a postve response y = 1. Logt θ (p) s a strctly ncreasng smooth functon of probablty p, transformng the nterval (0,1) nto the nterval (, ). Recordng denstes (10) wth new parameters (logts) p log ~ 1 p y log(1 ) (, ) (, ) e f y p f y e (12) t can be seen that, the Bernoull dstrbutons form a 1-parameter exponental famly wth a logt as a natural parameter. Usng a canoncal lnk functon, attrbutng natural parameter θ to a lnear predctor, one can obtan the followng generalzed lnear model (GLZ): p log 1 p m ( p ) x, = 1,,n, j = 1,,m, m < n (13) j j1 j

10 170 Parameters β j are estmated wth a maxmum lkelhood method, whch for GLZ models means a generalzed least squares method; more precsely, an teratvely re-weghted least squares method. Applyng the asymptotc propertes of the maxmum lkelhood estmators one can construct asymptotc confdence ntervals for the estmated parameters and construct asymptotc sgnfcance tests for them. Testng hypothess H 0 : β = β 0 wll be based on the multdmensonal verson of Wald statstcs W = (βˆ β 0 ) T [cov(βˆ)] 1 (βˆ β 0 ) (14) The strong consstence and asymptotc normalty of βˆ mples an asymptotc dstrbuton of the Wald statstcs W, whch s a χ 2 dstrbuton wth degree of freedoms df equal to the rank of covarance matrx cov(βˆ), whch s equal to the number of nonredundant parameters (coordnates) n vector β. The assessment of goodness of ft of the logt model s made by the devance statstc φ m, calculated accordng to the formula: φ m = 2(L s L m ) (15) where: L m logarthm of the maxmum of the lkelhood functon for the consdered model, L s logarthm of the maxmum of the lkelhood functon for the saturated model. Thus devance φ m s a doubled logarthm of the statstcs of the lkelhood rato statstc. The saturated model s a model wth no constrants related to the parameters, the number of whch equals the number of observatons. The saturated model provdes an deal ft to the data. Standardzed devance φ std s calculated from the formula: std (16) n m where: n the sze of the analysed sample, m the number of parameters n the model. The devance allows the constructon the followng statstc: 0 m 0 (17) where φ 0 s the devance of the model n whch the only explanatory varable s constant.

11 171 Ths statstc (17) gve the proporton of devance explaned by the model and makes an analogue of the determnaton coeffcent R 2, whch s known from the lnear regresson analyss. Devance φ m provdes a base for the constructon of the adjustablty measures, whch are partcularly useful n comparng several competng models. One such measure s the Akake nformaton crteron (AIC): AIC = φ m + 2m (18) where the component 2m plays acts a the penalty pad for addng new parameters. Havng several models, we usually choose the one whch has the lowest value of AIC. Other crtera, e.g. the Bayes nformaton crteron by Schwartz BIC, nvolve a dfferent penalty functon. Detaled consderatons referrng to these crtera can be found n (Haste et al., 2001). In the case where the sample sze s very bg and the number of estmated parameters s small (such a stuaton occurs n ths paper) the nfluence of the penalty functon on AIC s very small n relaton to the devance. Thus, t s enough to compare the devances of the consdered models. Probablty p s connected wth the value of a lnear predctor η by the relaton Fg. 1. The dependence of probablty p on lnear predctor η Rys. 1. Zależność prawdopodobeństwa p od lnowego predyktora η e p (19) 1 e The qualty of the prognoss of the probablty of damage to buldngs s assessed based on the estmated logt model, takng the arbtrary crteron of quanttatve compatblty of the prognoss wth realty and plottng a 2 2 table showng four possble outcomes of the prognoss and real buldng damage (table 3).

12 172 In table 3 a means the number of buldngs whch were actually damaged n lne wth the prognoss. Number b s the number of buldngs whch were actually damaged where the prognoss had not ndcate the possblty of damage. Number c s the number of buldngs whch were not damaged where the prognoss had ndcated the possblty of damage. Number d s the number of buldngs whch were not damaged n lne wth the prognoss. Typcal 2 2 table of sample. Entres are number of sample data Typowa tablca próbk z zestawenem danych wejścowych TABLE 3 TABLICA 3 The damaged buldngs (F + ) The ntact buldngs (F ) The predcted damage (P + ) a c The predcted lack of damage (P ) b d The qualty of the assessment of the probablty of damage to buldngs can be judged provded the database s large enough, by an estmaton of: the senstvty of the forecast, defned as condtonal probablty P(P + F + ).e. the probablty that the model predcts the damage to the buldng (P + ) gven the buldng was actually damaged (F + ), s estmated by the rato: a Sens (20) a b the specfcty of the forecast, defned as condtonal probablty P(P F ).e. the probablty that the model does not predct the damage to the buldng, gven the buldng was not actually damaged, s estmated by the rato: d Spec (21) d c total accuracy, defned as probablty P((P + F + ) (P F )) that the prognoss s n lne wth what actually happens, estmated by the rato: a d Ta (22) a b c d A good prognostc system should be characterzed by hgh values of these ndcators.

13 Research materal The analyzed research materal conssts of geologcal-mnng documentaton of the accomplshed explotaton of coal beds, the results of geodetc measurements of surface deformaton as well as assessments of buldng structures and records of damage to buldngs whch can be lnks to ths mnng actvty. The data were collected from four regons: Berun (regon No. 1), Byczyna (regon No. 2), Kostuchna (regon No. 3) and Lazska (regon No. 4). Mnng explotaton underneath these regons was carred out by a wall system wth fall of roof, to a depth H = m, n depost layers of thcknesses g = 1,4 3,2 m. The speed of the progress of explotaton was v = 8 28 m/ week. So, t was at small or average speed. Thus, t was possble to gnore the nfluence of speed n the growth of surface deformaton on damage to buldngs. TABLE 4 Basc statstcal data of the characterstcs of surface deformaton of the terran as well as buldngs n the analyzed regons TABLICA 4 Podstawowe dane statystyczne charakterystyk deformacj terenu budynków w analzowanych rejonach FEATURE REGIONS Database Database Database Database Database ALL BIE BYC KOS LAZ Number KO = KO = KO = KO = KTG = KTG = I KTG = II KTG = III KTG = IV KTG = V s u (0 0,1) s u (0,1 0,2) s u (0,2 0,5) Age of buldng t b [years] 44 (20-106) 45 (1-115) 40 (2-96) 42 (16-104) 44 (1-115) 0,53 0,62 0,63 0,59 0,60 s t (0,07 0,85) (0,04 0,96) (0,15 0,98) (0,19 0,95) (0,05 0,98) H [m] g [m] 3,1-3,2 2,1-2,8 1,7-2,2 1,4 1,4-3,2 KO category of buldng resstance, KTG category of mnng terran, s u degree of damage, s t techncal state

14 174 Surface deformatons the terran local to every buldng was defned by theoretcal modelng wth the applcaton of Knothe theory, and assgnng representatve values to process characterstcs and the parameters of the model, whch are based on the results of measurements n ndvdual regons. Ths procedure facltated a determnaton of the most relable values of deformaton ndexes, partcularly the extreme nstantaneous values of horzontal strans ε Kn max, at the locaton of every analyzed buldng. The calculatons were made accordng to the basc verson of the theory, whch made t possble to determne the category of mnng terran by a wdely appled nterpretaton of these values. Buldngs of tradtonal constructon, such as those n small towns, suburban areas and countrysde, were nfluenced by explotaton wth fall of roof. The resstance of buldngs to the nfluence of contnuous deformaton was estmated by the score method (Zasady..., 1979 encl. 18), by a team of hgh class buldng experts from the AGH Unversty of Scence and Technology n Cracow (Projekt badawczy, 1997). Every buldng was classfed nto a category of resstance to the nfluences of contnuous surface deformatons. Damage to buldngs, such as scratches on walls, was classfed by a degree of damage s u n a scale from 0.0 to 1.0 accordng to the classfcaton n the lterature (Wodyńsk & Kocot 1996). Also, techncal state s t, the degree of techncal wear s z as well as degree of natural wear s n of every buldng were assessed. Basc statstcal nformaton, characterzng mnng explotaton and deformaton of the foundatons of analyzed buldngs as well as buldng assessments n ndvdual regons are lsted n table 4 (5 databases were made). These arse from the set of results of calculatons and estmatons put n table 4, supplemented by detaled values of surface deformaton ndexes. They contaned nformaton characterzng ndvdual regons (base BIE, database BYC, database KOS, base LAZ) as well as all the regons together (database ALL). Prelmnary analyss of the sets contaned n the created databases (BIE, BYC, KOS and LAZ) showed that there were features of homogenety between them. Thus basc modelng was made on the total set ALL, and the obtaned results were checked on the component databases. 5. Analyss of estmated models The estmaton of buldng damage probablty was done based on the data sets presented n table 4 by use of the Generalzed lnear and nonlnear models a logt model manager module from STATISTICA TM software. Qualtatve varables (category of mnng terran and category of buldng resstance) as well as quanttatve varables (all other ndcators) were dstngushed among the explanatory varables. Qualtatve and quanttatve varables are the components of lnear predctor η determned n formula (8). Statstcal analyss of the logt model allows one to determne values of parameters β j, descrbng how much the partcular explanatory varables affect the probablty of buld-

15 175 ng damage. Probablty P(U = 1) of damage of -th buldng may be calculated based on a determned value of predctor η n lne wth formula (19). The examnaton of the dependence of buldng damage probablty on explanatory varables was performed by assumng that each tested model should encompass some varables representng contnung deformaton of buldng foundatons as well as those techncal features whch play a part n ts resstance to such deformaton. Damage or lack of damage depends on buldngs reacton to the nfluence of deformatons of foundatons. Therefore, an explanaton statng that possble damage can be caused solely by deformaton of foundatons or only by specfc features of the buldng may not be correct, as wll be suggested below. Categores of mnng area KTG as well as horzontal strans ε were taken nto consderaton to descrbe surface deformaton characterstcs. On the other hand, categores of buldng resstance to contnuous deformaton KO and lmtng horzontal strans ε lm resultng from an evaluaton of buldng resstance by usng a score method, were analyzed as explanatory varables pertanng to buldng characterstcs. Buldng age t b as well as techncal state s t were dstngushed as ndependent explanatory varables too. Even though these features affect buldng resstance to contnuous deformaton only to a certan degree, the accuracy of ts value determnaton s much hgher than those of lmtng horzontal strans ε lm and categores of buldng resstance KO. Accordng to the classfcaton (Wodyńsk & Kocot 1996) used for buldng damage evaluaton n the examned areas, those buldngs wth a degree of damage s u 0.05 were consdered as ntact. In order to obtan the most relable model for buldng damage probablty t was necessary to conduct a large number of analyses whose results were not always satsfactory. In the case of the model where the category of mnng terran KTG as well as the category of buldng resstance KO were used as quanttatve explanatory varables, the sgnfcance levels obtaned was wthn α = 0, (for KTG) and α = 0,13 0,32 (for KO), whch ruled ths model out. More satsfactory models for assessng the sgnfcance to the probablty of buldng damage of explanatory varables are dscussed below. Models based on surface deformaton ndexes modeled accordng to the Knothe theory as well as on evaluaton of buldng resstance usng a score method are presented frst. MODEL No. 1 Explanatory varables: extreme nstantaneous horzontal stran accordng to the basc verson of Knothe theory ε Kn max and category of buldng resstance to contnuous deformaton (4 categores: KO1, KO2, KO3, KO4 shown n table 1) were consdered. Lnear predctor η: 0,065 0,170 Kn max 0,833 I [ KO] 3 1,375 I 4 KO 1,861 I [ KO ] 1 0,347 I 2 KO [ ] [ ] (23)

16 176 where the category of resstance ndcator s defned by [ ] 1, when KO k I k KO (24) 0, when KO k Naturally, a gven buldng shows only one category of resstance, therefore only one of the four predctor components pertanng to category of resstance s actve (non-zero) n predctor (23). Wald tests confrm the sgnfcance of all No. 1 model parameters. All p-values were 0,001, except for the parameter correspondng to the second resstance category whch shows a p-value equal to 0,04. The dagnostc ndcators of the model are: φ std Sens Spec Ta 1,217 63,6% 71,6% 67,4% MODEL No. 2 Explanatory varables: lmtng horzontal stran ε lm, resultng from an evaluaton of buldng resstance usng the score method as well as the category of mnng terran KTG (6 categores were consdered and marked wth symbols 0, I, II, III, IV, V n accordance wth table 2). Lnear predctor η: [ ] 1,44 0,38 [ ] lm 0,40I 0 KTG 0,15I I KTG 0,07 I II KTG 0,50I [ KTG] [ KTG ] [ KTG] III 0,10I IV 0,13I V [ ] All p-values for varous categores of mnng terran (0,076; 0,498; 0,705; 0,098; 0,724, respectvely) exceeded the assumed sgnfcance level α = 0,05, and only the nfluence of horzontal lmtng stran ε l (p 0,001) was statstcally sgnfcant. Model No. 2, therefore, descrbed n formula (25) s not relable. MODEL No. 3 Explanatory varables: lmtng horzontal stran ε lm, resultng from an evaluaton of buldng resstance usng the score method as well as extreme nstantaneous horzontal stran accordng to the basc verson of the Knothe theory ε Kn max. Lnear predctor η: lm 0, 143 Kn max (25) 1,090 0,368 (26) The nfluence of both varables on buldng damage probablty remans sgnfcant (p-values for both parameters are 0.001). The model dagnostc ndcators are:

17 177 φ std Sens Spec Ta 1,265 69,0% 60,0% 64,6% MODEL No. 4 Explanatory varables: techncal state s t of buldng as well as extreme nstantaneous horzontal stran accordng to the basc verson of the Knothe theory ε Kn max. Lnear predctor η: t Kn max 5,410 9,317 s 0,172 (27) The nfluence of both varables on buldng damage probablty remans sgnfcant (p-values for both parameters are 0.001). Model dagnostc ndcators φ std Sens Spec Ta 0,976 63,6% 79,0% 77,2% MODEL No. 5 Explanatory varables: buldng age t b as well as extreme nstantaneous horzontal stran accordng to the basc verson of the Knothe theory ε Kn max. Lnear predctor η: b Kn max 1,948 0,037t 0,164 (28) The nfluence of both varables on buldng damage probablty remans sgnfcant (p-values for both parameters are 0,001). The model dagnostc ndcators are φ std Sens Spec Ta 1,235 65,6% 68,4% 66,9% Apart from these models, some other models based on not exceedable horzontal nexc stran ε max (see formula 4) as well as on horzontal stran determnng crtcal buldng cr resstance ε res (see formula 5) were also tested. Models for pars of explanatory varables, smlar to those n models No. 3, No. 4 and No. 5, were created. MODEL No. 6 Explanatory varables: not exceedable horzontal stran ε nexc max stran determnng crtcal buldng resstance ε res Lnear predctor η: cr. as well as horzontal cr res nexc max 1,090 0,566 0,119 (29)

18 178 The values of p-values of explanatory varables, standardsed devaton φ std and statstcs Sens, Spec and Ta are equal to the values of those parameters defned n model No. 3. MODEL No. 7 nexc Explanatory varables: not exceedable horzontal stran ε max state s t of a buldng. Lnear predctor η: as well as techncal t nexc max 5,410 9,317s 0,143 (30) The values of p-values of explanatory varables, standardsed devaton φ std as well as statstcs Sens, Spec and Ta are equal to the values of those parameters defned n model No. 4. MODEL No. 8 nexc Explanatory varables: not exceedable horzontal stran ε max Lnear predctor η: and buldng age t b. b nexc max 1,948 0,037t 0,137 (31) The values of p-values of explanatory varables, standardsed devaton φ std and statstcs Sens, Spec and Ta are equal to the values of those parameters defned n model No. 5. Comparson of models where horzontal stran s modeled accordng to the basc verson of the Knothe theory and the comparson of resstance estmates usng a score method wth the models featurng not exceedable horzontal stran and crtcal resstance, proves that there are no sgnfcant dfferences. These models vary only n the values of some parameters β, but ths has no effect on standardsed devaton φ std and statstcs Sens, Spec and Ta. Moreover two models of buldng damage probablty were examned, takng nto consderaton damage ndex I nk (see formula 6) whch was used to determne probable degree of buldng damage. MODEL No. 9 Explanatory varables: buldng age t b and damage ndex I nk. Lnear predctor η: 1, 640 0, 031t 0, 234 (32) b I nk

19 The nfluence of both varables on buldng damage probablty remans sgnfcant (p-values for both parameters are 0.001). The model dagnostc ndcators are: φ std Sens Spec Ta 1,215 62,0% 72,3% 66,9% MODEL No. 10 Explanatory varables: techncal state of buldng s t as well as damage ndex I nk. Lnear predctor η: t I nk 179 5,190 8,704s 0, 170 (33) The nfluence of both varables on buldng damage probablty remans sgnfcant (p-values for both parameters are 0.001). The model dagnostc ndcators are: φ std Sens Spec Ta 0,978 75,5% 79,4% 77,3% 6. Test result nterpretaton Table 5 contans the values of the most sgnfcant statstcs whch feature n the above models,.e. standardzed devatons φ std and total accuracy Ta of the prognoss. Lst of statstcal values for the evaluaton of logt models TABLE 5 TABLICA 5 Zestawene wartośc statystyk do oceny lnowych predyktorów η model logtowych Model φ std Ta [%] nr 1 ( ε Kn max, KO) 1,217 67,4 nr 2 (ε lm, KTG) nr 3 (ε lm, ε Kn nr 6 (ε cr res, ε max max ) nexc ) nr 4 (s t, ε Kn max ) nr 7 (s t, ε nexc max ) 1,265 64,6 0,976 77,2 nr 5 (t b, ε Kn max ) nr 8 (t b, ε max ) 1,235 66,9 nr 9 (t b, I nk ) 1,215 66,9 nr 10 (s t, I nk ) 0,978 77,3

20 180 It appears from the data contaned n table 5 that models No. 4 (see formula 27), No. 7 (see formula 30) and model No. 10 (see formula 33) have the hghest matchng qualty and the best total accuracy of the prognoss. The estmaton of buldng damage probablty accordng to models No. 4 and No. 7 renders dentcal results, whch may be explaned by the rather small nfluence of horzontal strans ε Kn nexc max or ε max as compared to nfluence of buldng techncal state s t. Snce values of standardzed devance φ std as well as total accuracy Ta for models No. 4, No. 7 and No. 10 are equal, these models may be consdered to be equally relable. The lack of sgnfcant dfferences between these models motvates the use one of them based on premses other than those whch result from purely evaluaton. Snce model No. 10 contans ndcator I nk, used to defne probable degree of buldng damage s u ts applcaton to buldng damage probablty prognoss s recommended. Table 6 contans a lst of buldng damage probablty values calculated accordng to formula 33 for determned ranges of techncal state s t and expected damage ndex I nk. Probablty of buldng damage accordng to model No. 10 (formula 33) TABLE 6 TABLICA 6 Rozkład prawdopodobeństwa uszkodzena budynku według modelu nr 10 (wzór 33) Techncal Damage ndex I nk state s t 1,0 2,0 3,0 4,0 5,0 6,0 7,0 0,1 0,99 0,99 1,00 1,00 1,00 1,00 1,00 0,2 0,97 0,98 0,98 0,99 0,99 1,00 1,00 0,3 0,94 0,95 0,96 0,96 0,97 0,97 0,98 0,4 0,87 0,89 0,91 0,92 0,93 0,94 0,95 0,5 0,74 0,77 0,80 0,82 0,85 0,87 0,89 0,6 0,54 0,58 0,62 0,66 0,70 0,73 0,76 0,7 0,33 0,37 0,40 0,45 0,49 0,53 0,58 0,8 0,17 0,20 0,22 0,25 0,29 0,32 0,36 0,9 0,08 0,09 0,11 0,12 0,14 0,17 0,19 1,0 0,03 0,04 0,05 0,06 0,07 0,08 0,09 When analyzng buldng damage probablty dstrbuton accordng to table 6, t may be noted that ths model reacts sgnfcantly better to buldng techncal state s t changes than to changes n damage ndex I nk. The weght of techncal state s t taken as an explanatory varable n buldng damage s much hgher than the weght of the second explanatory varable. It should be remembered, and ths s sgnfcant n the evaluaton of probablty values shown n table 6, that a buldng s consdered to be damaged f damage degree s u > 0,05 n accordance wth the (Wodyńsk & Kocot, 1996) scale. It should be stated that only one techncal feature of buldng nfluencng ts resstance to contnuous deformaton of foundatons s represented n model No. 10. No other

21 181 features are taken nto account when usng the score method for buldng resstance evaluaton. However, those models where other features are ncluded n the explanatory varables,.e. buldng resstance category KO (model No. 1) and lmtng horzontal stran ε lm (model No. 3), appeared to be less relable due to the concomtant values of standardzed devatons φ std and total accuracy Ta. Based on the results obtaned a buldng resstance evaluaton usng categores s not very accurate. A buldng s reacton to horzontal strans of ts foundatons s largely dependent upon ts techncal state. All other techncal features whch nfluence a buldng s resstance to contnuous surface deformaton are reflected ndrectly n the damage ndex I nk. However, the relatvely low nfluence of ths explanatory varable on buldng damage probablty leads to the concluson that other techncal features of buldng, taken ndvdually, may be less sgnfcant to the damage of that buldng than s t, ts overall techncal state. The buldng damage probablty model accordng to formula 33 was verfed usng sets contaned n the component bases BIER BYC, KOS and LAZ. Qualty measures of the prognoss, such as Sens, Spec and Ta were determned for each set. The results are shown below. Regon Sens Spec Ta Berun 61,7% 64,3% 62,3% Byczyna 70,6% 73,5% 72,2% Kostuchna 68,8% 64,9% 67,0% Lazska 63,3% 61,5% 62,8% The results of ths verfcaton may be consdered satsfactory snce they confrm the relablty of model No. 10 (see formula 33) when appled to an estmaton of the buldng damage probablty n those unts whch are exposed to the nfluence of horzontal strans on foundatons. 7. Concluson The method of assessng the probablty of damage to buldngs n mnng land accordng to logt model No. 10 (formula 33) can be appled to plannng underground hard coal explotaton wth fall of roof (n 1.5 m to 3.5 m thck layers of coal beds at the depth range of 250 m to 600 m) under buldngs of tradtonal constructon of any age and techncal state. The model can easly be updated based on current results on surface deformaton and adapted to deal wth partcular mnng and geologcal features whch obtan n land areas each of whch may have ts own characterstcs. Ths research work fnanced by statutory nvestgatons KOTGGGG AGH nr

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