Arch. Min. Sci., Vol. 53 (2008), No 2, p. 161–182

Arch. Min. Sci., Vol. 53 (2008), No 2, p. 161–182
161
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 USZKODZENIA BUDYNKÓW NA TERENACH GÓRNICZYCH
The method of assessing the probability of damage to buildings in mining terrain according to logit
models are presented. These models can be applied to planning underground hard coal exploitation with
fall of roof under buildings of traditional construction of any age and technical state and can easily be
updated based on current results on surface deformation and adapted to deal with particular mining
and geological features which obtain in land areas each of which may have its own characteristics. It is
shown that the probability of damage to buildings mainly depends on their technical state an so called
index of damage.
Key words: deformations of terrain surface, horizontal strain, probability of damage to building structure,
logit model
Ochrona obiektów budowlanych przed szkodliwymi wpływami podziemnej eksploatacji złóż kopalin
użytecznych jest warunkiem prowadzenia działalności górniczej. Możliwość uszkodzenia obiektów ocenia
się na podstawie wyników prognoz zjawisk towarzyszących eksploatacji górniczej oraz oszacowań cech
technicznych obiektów i właściwości ich podłoża gruntowego. Szczególnym przedmiotem takich ocen
jest przewidywany skutek oddziaływania deformacji ciągłych powierzchni terenu na budynki.
Ustalenie, które budynki mogą zostać uszkodzone odbywa się zgodnie z formalną procedurą polegającą
na porównaniu kategorii odporności danego budynku na wpływy deformacji ciągłych podłoża z kategorią
terenu górniczego w miejscu lokalizacji budynku. Kategorie odporności określa się stosując zazwyczaj tzw.
metodę punktową, uwzględniającą najważniejsze cechy techniczne budynku. Kategorie terenu górniczego
wyznacza się na podstawie prognozy deformacji ciągłych, obecnie według teorii Knothego. Przyjmuje się,
że budynek jest zagrożony uszkodzeniem, jeżeli oszacowana metodą punktową kategoria jego odporności
jest mniejsza od przewidywanej kategorii terenu górniczego w miejscu posadowienia budynku.
*
**
AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING SURVEYING AND ENVIRONMENTAL
ENGINEERING, AL. MICKIEWICZA 30, 30-059 KRAKÓW, POLAND
AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF APPLIED MATHEMATICS, AL. MICKIEWICZA 30,
30-059 KRAKÓW, POLAND
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Oszacowane odporności budynków na deformacje ciągłe podłoża różnią się od wartości rzeczywistych,
a także prognozowane wartości tych deformacji różnią się od wartości, które rzeczywiście wystąpią, ze
względu na brak możliwości „bezbłędnego” opisu tych cech i zjawisk, zarówno na podstawie wyników
pomiarów, jak i na drodze modelowania matematycznego. Stwierdza się systematyczne i losowe rozbieżności pomiędzy wynikami oszacowań i prognoz a wynikami obserwacji zachodzących zjawisk co powoduje,
że ocena zagrożenia budynków wpływami deformacji podłoża od projektowanej eksploatacji górniczej
nie może być w pełni wiarygodna. Szczególnie niepewna i niejednoznaczna jest taka ocena w przypadku
posługiwania się kategoriami terenu górniczego i kategoriami odporności budynków.
Badania zaprezentowane w publikacji (Ostrowski, 2006) wykazały, że jednoznaczna ocena zagrożenia
budynku wpływami deformacji ciągłych powierzchni może zostać dokonana poprzez porównanie prognozowanej wartości tzw. nieprzekraczalnego odkształcenia poziomego z wartością odkształcenia poziomego
wyznaczającego tzw. odporność krytyczną budynku. Wartości tych odkształceń otrzymuje się na podstawie
prognozy deformacji powierzchni według teorii Knothego oraz oceny odporności budynku metodą punktową z uwzględnieniem systematycznych i losowych składowych nieadekwatności zastosowanego modelu
teoretycznego i metody oceny do rzeczywistości. Porównanie odporności krytycznej budynku z wartością
nieprzekraczalnego odkształcenia poziomego w miejscu jego posadowienia pozwala na jednoznaczne
stwierdzenie, czy uszkodzenia budynku przekroczą nieodczuwalny stopień uciążliwości użytkowania
(Kwiatek, 2002) czy też stopień ten będzie większy. W pierwszym przypadku budynek można uznać za
niezagrożony i wyłączyć go z procedury dalszej, szczegółowej oceny. Powyższa metoda została nazwana
metodą bezpiecznego szacowania zagrożenia budynku uszkodzeniami (Ostrowski, 2006). Przewidywany
stopień uszkodzenia budynku można ocenić na podstawie wartości ilorazu nieprzekraczalnych odkształceń
poziomych i odkształceń wyznaczających odporność krytyczną, nazwanego wskaźnikiem uszkodzenia.
Istotną informacją charakteryzującą przewidywane uszkodzenie budynku jest prawdopodobieństwo
wystąpienia takiego zdarzenia. Ze względu na trudności w wyznaczaniu wartości odchylenia standardowego odkształcenia poziomego charakteryzującego odporność budynku na deformacje ciągłe podłoża,
proponowane dotychczas metody nie są efektywne. Mankament ten jest jednak możliwy do wyeliminowania, jeżeli wykorzysta się wyniki obserwacji skutków oddziaływania eksploatacji górniczej w budynkach,
które znalazły się w zasięgu jej wpływów. Na podstawie wyników obserwacji można zbudować logitowy
model prawdopodobieństwa wystąpienia zdarzenia losowego, jakim jest uszkodzenie budynku na skutek
oddziaływania deformacji podłoża.
W modelu logitowym zmienna objaśniana ma zakres wartości [0,1], co oznacza uszkodzenie lub
nieuszkodzenie budynku. Model scharakteryzowany jest przez trzy składowe:
− składową losową, będącą wektorem utworzonym przez niezależne obserwacje o rozkładach
z wykładniczej rodziny rozkładów,
− składową systematyczną, będącą wektorem nazwanym predyktorem liniowym η o współrzędnych,
które są liniowymi funkcjami zmiennych objaśniających (w budowanym modelu są to cechy
techniczne budynku oraz wskaźniki charakteryzujące deformacje podłoża),
− funkcję wiążącą składową losową i systematyczną.
Model buduje się używając kanonicznej funkcji wiążącej, przypisującej liniowemu predyktorowi
naturalny parametr, jakim jest logit. Parametry modelu logitowego estymuje się metodą największej
wiarogodności, a ocenę dobroci dopasowania modelu dokonuje się na podstawie odchylenia, obliczonego
jako podwojony logarytm statystyki ilorazu wiarogodności. Prawdopodobieństwo p (w zbudowanym
modelu jest to prawdopodobieństwo uszkodzenia budynku) związane jest z wartością liniowego predyktora η zależnością:
p
e
1 e
Jakość prognozy prawdopodobieństwa uszkodzenia budynku ocenia się na podstawie wskaźników:
czułości prognozy, specyficzności prognozy i dokładności całkowitej. Wskaźniki te wyznacza się na
podstawie prognozowanych ilości budynków uszkodzonych i nieuszkodzonych w porównaniu do ilości
budynków faktycznie uszkodzonych i nieuszkodzonych.
Budowa modelu logitowego wymagała opracowania prognozy deformacji powierzchni oraz oszacowania odporności budynków na etapie oceny zagrożenia tych budynków wpływami projektowanej eksploatacji górniczej, a następnie zebrania danych dokumentujących zaistniałe deformacje oraz uszkodzenia
budynków po dokonaniu eksploatacji. Odpowiednie dane zostały uzyskane z czterech rejonów eksploatacji
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pokładów węgla kamiennego, dokonanych na głębokościach od 280 m do 570 m, w warstwach złoża
o wysokości od 1,4 m do 3,2 m. Oddziaływaniu tych eksploatacji poddane zostały budynki o różnych
kategoriach odporności na wpływy górnicze, różnym stanie technicznym (od 0,04 do 0,98) i różnym
wieku (od 1 roku do 115 lat). Na skutek oddziaływania eksploatacji w budynkach wystąpiły uszkodzenia
o stopniach zróżnicowanych od 0,0 do 0,5.
Estymacja prawdopodobieństwa uszkodzenia budynków została dokonana na podstawie zmiennych
objaśniających, spośród których wyróżniono zmienne jakościowe (kategoria terenu górniczego i kategoria odporności budynku) oraz zmienne ilościowe (prognozowane według teorii Knothego odkształcenie
Kn
np
poziome │ε max│, nieprzekraczalne odkształcenie poziome │εmax│ , odkształcenie poziome graniczne εgr
wynikające z oceny odporności budynków metodą punktową, stan techniczny budynku st, wiek budynku
kr
tb, odporność krytyczna ε max oraz wskaźnik uszkodzenia Ink). Badanie zależności prawdopodobieństwa
uszkodzenia budynku od zmiennych objaśniających wykonano przyjmując założenie, że każdy badany
model powinien zawierać zmienne reprezentujące prognozowane deformacje ciągłe podłoża budynku
oraz cechy techniczne budynku związane z jego odpornością na deformacje.
W wyniku dokonanych badań stwierdzono, że najbardziej odpowiedni do oceny prawdopodobieństwa uszkodzenia budynku poddanego oddziaływaniu deformacji podłoża, wynikających z wpływów
górniczych, jest model w postaci:
η = +5,190 – 8,704 . st + 0,170 . Ink
Wartość predyktora η decydującego o wartości prawdopodobieństwa uszkodzenia budynku zależy od
stanu technicznego budynku st oraz wskaźnika uszkodzenia Ink.. Analiza rozkładu prawdopodobieństwa
uszkodzenia budynku według tego modelu wykazuje, że model bardziej zdecydowanie reaguje na zmiany
stanu technicznego st budynku niż na zmiany wskaźnika uszkodzenia Ink. Waga stanu technicznego st jako
zmiennej objaśniającej uszkodzenie budynku jest dużo wyższa niż waga drugiej zmiennej objaśniającej.
Wynika z tego, że budynki w złym stanie technicznym są relatywnie dużo bardziej zagrożone uszkodzeniami niż budynki w dobrym stanie, niezależnie od wartości spodziewanych deformacji podłoża.
Zaproponowana w publikacji metoda oceny prawdopodobieństwa uszkodzenia budynków na terenach
górniczych może być stosowana w przypadku projektowania eksploatacji podziemnej z zawałem stropu
oraz budynków o konstrukcji tradycyjnej.
Słowa kluczowe: deformacje powierzchni terenu, poziome odkształcenia, prawdopodobieństwo uszkodzenia budynku, model logitowy
1. Introduction
One of the most important issues touching in mining terrains is the protection of
building structures from the harmful impact of mines on their foundations. A preliminary
assessment of the threat to buildings will based on a prognosis of the deformation of
the land surface as well as an estimation of those technical features of a building, which
affect the resistance of the building structure to continuous deformation of its foundations. The degree of the threat to a building structure will be clarified by a comparison
of both characteristics. The validity this of assessment, and further action based on it,
depend on the reliability of data on both characteristics. Underestimation of the threat
brings us too close to the safety limit for the building, or will lead to increased repair
costs. An overestimation of the threat can unnecessarily increase the mining costs and
(or) preventive building measures, albeit guaranteeing a high level of safety.
In practice, a “faultless” assessment of the threat to a building caused by continuous
deformation of its foundations, is not possible because of the huge degree of complexity
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of the impact as well as the contribution of random factors to the deformation process.
Mathematical models describing these impacts are always a mere approximation to
reality and describe “average” states resulting from the influence of the determining
factors. The influence of random factors is usually defined by standard deviation from
the modelled characteristics of the phenomenon.
Methods for assessing the threat to buildings caused by continuous deformation
demand a knowledge of the predicted values of the horizontal strain or curvature of
the surface of the terrain, as well as the limiting values of the horizontal strain (in short
“limiting strain”), which determine the resistance of the building. These values are
calculated by the application of deterministic models, as average values. Depending on
the method, standard deviations of individual characteristics are taken into account or
not. Methods of “one-dimensional” analysis (Batkiewicz et al., 1977; Sroka et al., 1994)
estimate the threat based on the predicted values of the horizontal strains of the surface
with a definite standard deviation and an error-free estimation of the values of limiting
horizontal strains. Methods of “two-dimensional” analysis additionally take into account
the standard deviations of indices characterizing the resistance of the building (Sroka et
al., 1994; Kwiatek, 2004). The method for predicting the costs of repair to buildings in
mining terrains, with the application of a logit model is a particular example of “twodimensional” analysis (Kaszowska, 2002). A disadvantage of the “one-dimensional”
analysis method is the assumption that the assessment of building resistance is without
errors, which is not possible in practice. However “two-dimensional” analysis demands
a knowledge of the value of the standard deviation of the index describing the resistance
of the building structure. This value can be achieved only based on premises resulting
from the experience of the researcher.
This article presents a proposal for the assessment of the probability of building
damage in mining areas by the application of a logit model. The logit model is suitable
in the analysis of such a problem, because it considers the limitability of dependent
variables and also assumes the application of explanatory variables both quantitative
as well as qualitative and their interaction as predictors. Unlike in competing models
which use quantile functions of other than logistic distribution (e.g. the probit model), the
interpretation of the estimated parameters is easy and does not depend on the sampling
method (retrospective or prospective).
2. Formulating of the problem
Assessing the threat of damage to buildings resulting from deformation their foundations, which is related to mining can be done in the following ways:
1) a comparison of the categories of building resistance to continuous deformation (table 1) with the categories of mining area (table 2) in which the buildings are situated,
2) it can be based on the expert opinions of mining and construction specialists.
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The first method is applied in the assessment of numerous groups of buildings of
traditional construction and it is compulsory when so-called activity mining plans are
prepared (Rozporządzenie…, 2002, part 2, paragraph 2.26, annex 1 and 2). Second
method is applied in the case of buildings of particular significance, be it social, cultural,
historical, architectural, or with high vulnerability to the impact of mining.
The application of each of these methods demands a determination of the predicted
surface deformation indexes around the buildings as well as the nature of their resistance
to continuous deformation. Currently, a prognosis of surface deformation is in line with
Knothe’s theory (Knothe 1984). The assessment of the resistance of a building to continuous deformations of its foundations, i.e. the category of building resistance, is made
based on a score method (Zasady…, 1979; Kwiatek, 2002). The idea of this method is to
assign of a definite number nc of points to individual technical features of the building
and to define its building resistance category (KO) based on the sum of points (tab. 1).
The assessment of the threat to buildings of mining-induced surface deformation is
based on the results of a comparison the predicted (according to Knothe’s theory) value
of horizontal strain ε (as the maximum value this index at the terrain local to the building
(KTG)
location ε Kn
max or category of the mining terrain │ε max │) with the value of the limiting
horizontal strain expressed by the category of building resistance │ε(KO)
.
lim │
TABLE 1
Qualification table of score method
TABLICA 1
Tabela kwalifikacyjna metody punktowej
Category of building
resistance
(KO)
1
4
3
2
1
0
Limiting horizontal
strain
│ε(KO)
lim │ [mm/m]
Sum of nc points
According to
(Zasady..., 1979)
According to
(Kwiatek, 2002)
2
3
4
│εl │ = 9,0
(4)
≤ 20
≤3
│εl │ = 6,0
(3)
21÷27
4÷22
(2)
│εl │
(1)
│εl │
(0)
│εl │
= 3,0
28÷36
23÷40
= 1,5
37÷47
41÷57
= 0,3
≥ 48
≥ 58
However, the results of assessments prepared with this deterministic method might
not be very reliable, because they do not take into account the considerable influence of
random factors on the behaviour of the analysed phenomnon. Particularly, comparing
the category of mining terrain to the category of building resistance can lead to unsatisfactory results, for reasons shown by Ostrowski (2006).
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TABLE 2
Categories of mining terrain
TABLICA 2
Kategorie terenu górniczego
Category of
mining terrain
(KTG)
1
0
I
II
III
IV
V
Slope
[mm/m]
2
(0)
|Tmax
| = 0,5
(I)
|Tmax
| = 2,5
(II)
|Tmax| =5,0
(III)
|Tmax
| =10,0
(IV)
|Tmax | =15,0
|T (V) | > 15,0
Values of the foreseen deformation
Radius of curvature
Horizontal strain
[km]
[mm/m]
3
(0)
|Rmin
| = 40
(I)
|Rmin
| = 20
(II)
|Rmin| = 12
(III)
|Rmin
|=6
(IV)
|Rmin | = 4
|R (V) | < 4
(0)
|εmax
|
(II)
|εmax
|
(III)
|εmax|
(IV)
|εmax
|
(V)
|εmax|
|ε (0) |
4
= 0,3
= 1,5
= 3,0
= 6,0
= 9,0
> 9,0
It can be deduced from geodetic measurements, that the standard deviation of horizontal strains σε is (Popiołek & Ostrowski, 1981; Popiołek et al., 1999):
avr
0,15 0,25 max
(1)
avr
| is maximum, average value of horizontal strain.
where |εmax
It should be natural, that the categories of mining land were formulated in times, when
so-called the basic version of Knothe’s theory was applied to prognoses of deformation. In
that version the value of coefficient B binding vertical and horizontal deformations is:
B = 0,4 r
(2)
where r is the radius of the dispersion of impacts.
Later research (Popiołek & Ostrowski, 1978; Hejmanowski et al., 2005) showed that
the value of coefficient B in the exploitation of hard coal deposits and also deposits of
copper ores is smaller and its average value is:
B = 0,32r
(3)
The difference results from changes in the conditions of deposit exploitation over the
years, mainly due to carrying out the exploitation at deeper and deeper levels. From the
above a very important conclusion can be drawn: the results of prognoses of horizontal
strains ε Kn, which were made according to the basic version of Knothe’s theory were overestimated by about 25% compared to values resulting from geodetic observations.
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The result of the prognosis of horizontal strains according to the basic version of
Knothe’s theory vere overestimated by about 25% compared to values resulting from
geodetic observations.
Tche results of the prognosis of horizontal strains according to the basic version of
Knothe’s theory is influenced by systematic and random factors with definite values.
This is what causes the discrepancy between the results of prognoses and the results of
observations. Taking into consideration relation (1) it was possible to estimate that in this
case the probability of the occurrence of a horizontal strain ε larger than the predicted
εKn is Pε = 0,21 (Popiołek, 1989).
To remove this “error”, which results from systematic and random factors, from the
nexc
result of the prognosis, an not exceedable horizontal strain εmax
was defined at work
nexc
) = 0,05, is
(Ostrowski, 2006). The value this index, satisfying the condition P(ε > εmax
calculated from the formula:
nexc
Kn
max
1,2 max
(4)
Kn
where εmax
is the extreme (in point and in time) of the horizontal strain calculated according to the basic version of Knothe’s theory.
The resistance of a building to deformation of foundation, or strictly to horizontal
strain ε, it is identified with limiting horizontal strain εlim equal to the bottom limit of
the resistance category (tab. 1), suitable for a given building.
The category of resistance is established with the score method. From the accomplished analysis in the paper (Ostrowski, 2006) it can be concluded that it is possible to
cr
estimate the resistance of building as so-called critical horizontal strains εres
, which is
counted from formula:
cr
εres
= 0,65εlim
(5)
where:
cr
εres
— critical horizontal strain (critical resistance) marking the limit, exceeding
this limit means damage to the building, beyond imperceptible degree
(Kwiatek, 2002) of difficulty its use,
εlim — limiting horizontal strain marked with the interpolation method from the re(KO)
lation between the number nc of points and horizontal limiting strain |εlim
|.
In (Ostrowski, 2006) a so-called index of predictable damage to the building Ink was
defined. Its value is calculated from the formula:
I nk nexc
max
cr
res
(6)
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Index Ink determine the predicted degree of damage to the building which can be
understood as the extent of necessary repairs or preventive measures.
nexc
, the resistance
These forecasted horizontal deformations of surface εgr(KO), εKn, εmax
(KO)
cr
, εlim, εres
, and index Ink, were analysed
of buildings to continuous deformations εlim
based on the logit model of the probability of the occurrence of a random episode, in
this case damage to a building. This study is aimed at finding the significance of the
respective characteristics for the probability of damage to a building and constructing
an appropriate prediction equation.
3. Logit model – theoretical basis
The influence of explanatory variables on the dependent variable can be described
by a regression model (e.g. linear or non-linear). However, if the dependent variable is
bounded, e.g. [0,1], a classical regression model is not adequate, because without special
restrictions imposed on the ranges of explanatory values, a dependent variable can take
values beyond the established range. A proper tool in the modelling of binary responses
(dependent variables) is a logit model which is one versions of the generalized linear
model (GLZ). The generalized linear model (McCullagh & Nelder, 1989; Agresti, 1990)
is characterized by three components:
Y1 1. Random component Y M is a vector made by independent observations Yi,
Yn i = 1,…, n of distributions from the Nelder-Wedderburn exponential family of
distributions with density or mass function of the form:
f ( yi ; i , ) e
yii b (i )
c ( yi , )
ai ( )
(7)
where θi and φ are parameters, and functions ai (φ), b(θi) and c(yi, φ) are known as
smooth (differentiable twice) functions. Parameter θi of the random variable distribution Yi acts as a location parameter and can take different values for respective
observations Yi , i = 1,…, n. The parameter φ is the same for all the observations. It
can be interpreted as a scale parameter and it is treated as the nuisance parameter.
1 = M , the coordinates of which are linear
2. A systematic component is a vector η
n x11 L x1m functions of explanatory variables, i.e. η = Xβ, where X = M O M is a ma x n1 L x nm 169
1 trix of the plan of experiment (like in a classical linear model), vector β = M m is a vector of unknown parameters. Vector η is called linear predictor. Its components ηi take the form:
m
i xij j ; i=1,…,n , j =1,…,m , m < n
j 1
(8)
3. A function linking a systematic component with a random variable. If this is written as µi = E(Yi), the expected value of the variable Yi, then expected value µi is
connected with the linear predictor ηi by function g, more precisely
ηi = g(µi),
(9)
where g is a monotonic and differentiable function.
In particular, let us consider a random component Y made by independent binary
random variables Yi, i = 1,…, n of Bernoulli distributions with the mass functions
f ( y i , p i ) p iyi (1 p i )1 yi : yi = 0, 1; pi (0 ,1)
(10)
Let us take instead of the parameter p of Bernoulli distribution f (y, p) = p y(1 – p)1–y
a new parameter θ defined by the equation:
( p ) log
p
1 p
(11)
The new parameter θ called logit, is a function of parameter p, being the expected
value of a dichotomous response Y i.e. p = E(Y). Logit θ (p) is a log odds of a positive
response y = 1. Logit θ (p) is a strictly increasing smooth function of probability p,
transforming the interval (0,1) into the interval (−∞, ∞).
p
Recording densities (10) with new parameters (logits) i log i
1 pi
~
yii log(1 ei )
f ( yi , pi ) f ( yi , i ) e
(12)
it can be seen that, the Bernoulli distributions form a 1-parameter exponential family
with a logit as a natural parameter.
Using a canonical link function, attributing natural parameter θ to a linear predictor,
one can obtain the following generalized linear model (GLZ):
log
m
pi
( pi ) i xij j , i = 1,…,n , j = 1,…,m , m < n
1 pi
j 1
(13)
170
Parameters βj are estimated with a maximum likelihood method, which for GLZ models means a generalized least squares method; more precisely, an iteratively re-weighted
least squares method. Applying the asymptotic properties of the maximum likelihood
estimators one can construct asymptotic confidence intervals for the estimated parameters
and construct asymptotic significance tests for them. Testing hypothesis H0: β = β0 will
be based on the multidimensional version of Wald statistics
W = (β̂ – β0)T[cov(β̂)]–1(β̂ – β0)
(14)
The strong consistence and asymptotic normality of β̂ implies an asymptotic distribution of the Wald statistics W, which is a χ2 distribution with degree of freedoms df equal
to the rank of covariance matrix cov(β̂), which is equal to the number of nonredundant
parameters (coordinates) in vector β.
The assessment of goodness of fit of the logit model is made by the deviance statistic
φm, calculated according to the formula:
φm = 2(Ls – Lm)
(15)
where:
Lm — logarithm of the maximum of the likelihood function for the considered
model,
Ls — logarithm of the maximum of the likelihood function for the saturated
model.
Thus deviance φm is a doubled logarithm of the statistics of the likelihood ratio statistic. The saturated model is a model with no constraints related to the parameters, the
number of which equals the number of observations. The saturated model provides an
ideal fit to the data.
Standardized deviance φstd is calculated from the formula:
std nm
(16)
where:
n — the size of the analysed sample,
m — the number of parameters in the model.
The deviance allows the construction the following statistic:
0 m
0
(17)
where φ0 is the deviance of the model in which the only explanatory variable is constant.
171
This statistic (17) give the proportion of deviance explained by the model and makes an
analogue of the determination coefficient R2, which is known from the linear regression
analysis.
Deviance φm provides a base for the construction of the adjustability measures, which
are particularly useful in comparing several competing models. One such measure is the
Akaike information criterion (AIC):
AIC = φm + 2m
(18)
where the component 2m plays acts a the penalty paid for adding new parameters. Having several models, we usually choose the one which has the lowest value of AIC. Other
criteria, e.g. the Bayes information criterion by Schwartz BIC, involve a different penalty
function. Detailed considerations referring to these criteria can be found in (Hastie et
al., 2001). In the case where the sample size is very big and the number of estimated
parameters is small (such a situation occurs in this paper) the influence of the penalty
function on AIC is very small in relation to the deviance. Thus, it is enough to compare
the deviances of the considered models.
Probability p is connected with the value of a linear predictor η by the relation
1
0.8
0.6
0.4
0.2
-4
-2
0
2
4
Fig. 1. The dependence of probability p on linear predictor η
Rys. 1. Zależność prawdopodobieństwa p od liniowego predyktora η
p
e
1 e
(19)
The quality of the prognosis of the probability of damage to buildings is assessed based
on the estimated logit model, taking the arbitrary criterion of quantitative compatibility
of the prognosis with reality and plotting a 2×2 table showing four possible outcomes
of the prognosis and real building damage (table 3).
172
In table 3 a means the number of buildings which were actually damaged in line
with the prognosis. Number b is the number of buildings which were actually damaged
where the prognosis had not indicate the possibility of damage. Number c is the number
of buildings which were not damaged where the prognosis had indicated the possibility
of damage. Number d is the number of buildings which were not damaged in line with
the prognosis.
TABLE 3
Typical 2×2 table of sample. Entries are number of sample data
TABLICA 3
Typowa tablica próbki z zestawieniem danych wejściowych
The predicted damage
(P+)
The predicted lack of damage
(P–)
a
b
c
d
The damaged buildings
(F+)
The intact buildings
(F–)
The quality of the assessment of the probability of damage to buildings can be judged
provided the database is large enough, by an estimation of:
• the sensitivity of the forecast, defined as conditional probability P(P+ |F+) i.e.
the probability that the model predicts the damage to the building (P+) given the
building was actually damaged (F+), is estimated by the ratio:
Sens a
ab
(20)
• the specificity of the forecast, defined as conditional probability P(P– |F–) i.e. the
probability that the model does not predict the damage to the building, given the
building was not actually damaged, is estimated by the ratio:
Spec d
d c
(21)
• total accuracy, defined as probability P((P+ ∩ F+) ∪ (P– ∩ F–)) that the prognosis
is in line with what actually happens, estimated by the ratio:
Ta ad
abcd
(22)
A good prognostic system should be characterized by high values of these indicators.
173
4. Research material
The analyzed research material consists of geological-mining documentation of the
accomplished exploitation of coal beds, the results of geodetic measurements of surface
deformation as well as assessments of building structures and records of damage to
buildings which can be links to this mining activity. The data were collected from four
regions: “Bierun” (region No. 1), “Byczyna” (region No. 2), “Kostuchna” (region No. 3)
and “Laziska” (region No. 4). Mining exploitation underneath these regions was carried
out by a wall system with fall of roof, to a depth H = 270÷560 m, in deposit layers of
thicknesses g = 1,4÷3,2 m. The speed of the progress of exploitation was v = 8÷28 m/
week. So, it was at small or average speed. Thus, it was possible to ignore the influence
of speed in the growth of surface deformation on damage to buildings.
TABLE 4
Basic statistical data of the characteristics of surface deformation of the terrain
as well as buildings in the analyzed regions
TABLICA 4
Podstawowe dane statystyczne charakterystyk deformacji terenu i budynków w analizowanych rejonach
FEATURE
Database
BIE
REGIONS
Database
Database
BYC
KOS
Database
LAZ
Database
ALL
1
2
3
4
5
6
Number
KO = 1
KO = 2
KO = 3
KO = 4
KTG = 0
KTG = I
KTG = II
KTG = III
KTG = IV
KTG = V
su (0 – 0,1)
su (0,1 – 0,2)
su (0,2 – 0,5)
Age of building tb
[years]
61
8
20
30
3
8
10
3
28
12
0
38
12
11
464
45
140
222
57
78
87
192
107
0
0
335
81
48
85
8
31
20
26
4
6
4
23
38
10
52
24
9
43
1
16
11
15
8
3
3
28
1
0
19
10
14
653
62
207
283
101
98
106
202
186
51
10
444
127
82
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
(0,07÷0,85) (0,04÷0,96) (0,15÷0,98) (0,19÷0,95) (0,05÷0,98)
380-430
500-570
280-320
350-370
280-570
H [m]
3,1-3,2
2,1-2,8
1,7-2,2
1,4
1,4-3,2
g [m]
KO − category of building resistance, KTG – category of mining terrain,
su – degree of damage, st – technical state
st
174
Surface deformations the terrain local to every building was defined by theoretical
modeling with the application of Knothe theory, and assigning representative values to
process characteristics and the parameters of the model, which are based on the results of
measurements in individual regions. This procedure facilitated a determination of the most
reliable values of deformation indexes, particularly the extreme instantaneous values of
horizontal strains ε Kn
max, at the location of every analyzed building. The calculations were
made according to the basic version of the theory, which made it possible to determine
the category of mining terrain by a widely applied interpretation of these values.
Buildings of traditional construction, such as those in small towns, suburban areas
and countryside, were influenced by exploitation with fall of roof. The resistance of
buildings to the influence of continuous deformation was estimated by the score method
(Zasady..., 1979 – encl. 18), by a team of high class building experts from the AGH
University of Science and Technology in Cracow (Projekt badawczy, 1997). Every building was classified into a category of resistance to the influences of continuous surface
deformations.
Damage to buildings, such as scratches on walls, was classified by a degree of damage
su in a scale from 0.0 to 1.0 according to the classification in the literature (Wodyński &
Kocot 1996). Also, technical state st, the degree of technical wear sz as well as degree
of natural wear sn of every building were assessed. Basic statistical information, characterizing mining exploitation and deformation of the foundations of analyzed buildings
as well as building assessments in individual regions are listed in table 4 (5 databases
were made). These arise from the set of results of calculations and estimations put in
table 4, supplemented by detailed values of surface deformation indexes. They contained
information characterizing individual regions (base BIE, database BYC, database KOS,
base LAZ) as well as all the regions together (database ALL).
Preliminary analysis of the sets contained in the created databases (BIE, BYC, KOS
and LAZ) showed that there were features of homogeneity between them. Thus basic
modeling was made on the total set ALL, and the obtained results were checked on the
component databases.
5. Analysis of estimated models
The estimation of building damage probability was done based on the data sets presented in table 4 by use of the Generalized linear and nonlinear models – a logit model
manager module from STATISTICATM software. Qualitative variables (category of
mining terrain and category of building resistance) as well as quantitative variables (all
other indicators) were distinguished among the explanatory variables. Qualitative and
quantitative variables are the components of linear predictor η determined in formula (8).
Statistical analysis of the logit model allows one to determine values of parameters βj,
describing how much the particular explanatory variables affect the probability of build-
175
ing damage. Probability P(Ui = 1) of damage of i-th building may be calculated based
on a determined value of predictor ηi in line with formula (19).
The examination of the dependence of building damage probability on explanatory
variables was performed by assuming that each tested model should encompass some
variables representing continuing deformation of building foundations as well as those
technical features which play a part in its resistance to such deformation. Damage or
lack of damage depends on buildings reaction to the influence of deformations of foundations. Therefore, an explanation stating that possible damage can be caused solely
by deformation of foundations or only by specific features of the building may not be
correct, as will be suggested below.
Categories of mining area KTG as well as horizontal strains ε were taken into consideration to describe surface deformation characteristics. On the other hand, categories
of building resistance to continuous deformation KO and limiting horizontal strains εlim
resulting from an evaluation of building resistance by using a score method, were analyzed as explanatory variables pertaining to building characteristics. Building age tb as
well as technical state st were distinguished as independent explanatory variables too.
Even though these features affect building resistance to continuous deformation only
to a certain degree, the accuracy of its value determination is much higher than those
of limiting horizontal strains εlim and categories of building resistance KO. According
to the classification (Wodyński & Kocot 1996) used for building damage evaluation in
the examined areas, those buildings with a degree of damage su ≤ 0.05 were considered
as intact.
In order to obtain the most reliable model for building damage probability it was
necessary to conduct a large number of analyses whose results were not always satisfactory. In the case of the model where the category of mining terrain KTG as well as the
category of building resistance KO were used as quantitative explanatory variables, the
significance levels obtained was within α = 0,24÷0.65 (for KTG) and α = 0,13÷0,32 (for
KO), which ruled this model out. More satisfactory models for assessing the significance
to the probability of building damage of explanatory variables are discussed below.
Models based on surface deformation indexes modeled according to the Knothe
theory as well as on evaluation of building resistance using a score method are presented
first.
MODEL No. 1
Explanatory variables: extreme instantaneous horizontal strain according to the basic
version of Knothe theory | ε Kn
max | and category of building resistance to continuous deformation (4 categories: KO1, KO2, KO3, KO4 shown in table 1) were considered.
Linear predictor η:
Kn
1,861 I 1[KO ] 0,347 I 2[KO ] 0,065 0,170 max
0,833 I 3[KO ] 1,375 I4[KO ]
(23)
176
where the category of resistance indicator is defined by
1, when KO k
I k[KO ] 0, when KO k
(24)
Naturally, a given building shows only one category of resistance, therefore only one
of the four predictor components pertaining to category of resistance is active (non-zero)
in predictor (23). Wald tests confirm the significance of all No. 1 model parameters. All
p-values were ≤ 0,001, except for the parameter corresponding to the second resistance
category which shows a p-value equal to 0,04. The diagnostic indicators of the model
are:
φstd
1,217
Sens
63,6%
Spec
71,6%
Ta
67,4%
MODEL No. 2
Explanatory variables: limiting horizontal strain εlim, resulting from an evaluation
of building resistance using the score method as well as the category of mining terrain
KTG (6 categories were considered and marked with symbols 0, I, II, III, IV, V in accordance with table 2).
Linear predictor η:
1,44 0,38 lim 0,40 I 0[KTG ] 0,15 I I[KTG ] 0,07 I II[KTG ] 0,50 I III[KTG ] 0,10 I IV[KTG ] 0,13 I V[KTG ]
(25)
All p-values for various categories of mining terrain (0,076; 0,498; 0,705; 0,098;
0,724, respectively) exceeded the assumed significance level α = 0,05, and only the
influence of horizontal limiting strain εl (p ≤ 0,001) was statistically significant. Model
No. 2, therefore, described in formula (25) is not reliable.
MODEL No. 3
Explanatory variables: limiting horizontal strain εlim, resulting from an evaluation of
building resistance using the score method as well as extreme instantaneous horizontal
strain according to the basic version of the Knothe theory | ε Kn
max |.
Linear predictor η:
Kn
1,090 0,368 lim 0,143 max
(26)
The influence of both variables on building damage probability remains significant
(p-values for both parameters are ≤ 0.001). The model diagnostic indicators are:
177
φstd
1,265
Sens
69,0%
Spec
60,0%
Ta
64,6%
MODEL No. 4
Explanatory variables: technical state st of building as well as extreme instantaneous
horizontal strain according to the basic version of the Knothe theory | ε Kn
max |.
Linear predictor η:
Kn
5,410 9,317 st 0,172 max
(27)
The influence of both variables on building damage probability remains significant
(p-values for both parameters are ≤ 0.001). Model diagnostic indicators
φstd
0,976
Sens
63,6%
Spec
79,0%
Ta
77,2%
MODEL No. 5
Explanatory variables: building age tb as well as extreme instantaneous horizontal
strain according to the basic version of the Knothe theory | ε Kn
max |.
Linear predictor η:
Kn
1,948 0,037 tb 0,164 max
(28)
The influence of both variables on building damage probability remains significant
(p-values for both parameters are ≤ 0,001). The model diagnostic indicators are
φstd
1,235
Sens
65,6%
Spec
68,4%
Ta
66,9%
Apart from these models, some other models based on not exceedable horizontal
nexc
strain εmax
(see formula 4) as well as on horizontal strain determining critical building
cr
resistance εres (see formula 5) were also tested. Models for pairs of explanatory variables,
similar to those in models No. 3, No. 4 and No. 5, were created.
MODEL No. 6
nexc
as well as horizontal
Explanatory variables: not exceedable horizontal strain εmax
cr
strain determining critical building resistance εres .
Linear predictor η:
cr
nexc
1,090 0,566 res
0,119 max
(29)
178
The values of p-values of explanatory variables, standardised deviation φstd and
statistics Sens, Spec and Ta are equal to the values of those parameters defined in model
No. 3.
MODEL No. 7
nexc
Explanatory variables: not exceedable horizontal strain εmax
as well as technical
state st of a building.
Linear predictor η:
nexc
5,410 9,317 st 0,143 max
(30)
The values of p-values of explanatory variables, standardised deviation φstd as well
as statistics Sens, Spec and Ta are equal to the values of those parameters defined in
model No. 4.
MODEL No. 8
nexc
Explanatory variables: not exceedable horizontal strain εmax
and building age tb.
Linear predictor η:
nexc
1,948 0,037 t b 0,137 max
(31)
The values of p-values of explanatory variables, standardised deviation φstd and
statistics Sens, Spec and Ta are equal to the values of those parameters defined in model
No. 5.
Comparison of models where horizontal strain is modeled according to the basic
version of the Knothe theory and the comparison of resistance estimates using a score
method with the models featuring not exceedable horizontal strain and critical resistance,
proves that there are no significant differences. These models vary only in the values
of some parameters β, but this has no effect on standardised deviation φstd and statistics
Sens, Spec and Ta .
Moreover two models of building damage probability were examined, taking into
consideration damage index Ink (see formula 6) which was used to determine probable
degree of building damage.
MODEL No. 9
Explanatory variables: building age tb and damage index Ink.
Linear predictor η:
1,640 0,031tb 0,234 Ink
(32)
179
The influence of both variables on building damage probability remains significant
(p-values for both parameters are ≤ 0.001). The model diagnostic indicators are:
φstd
1,215
Sens
62,0%
Spec
72,3%
Ta
66,9%
MODEL No. 10
Explanatory variables: technical state of building st as well as damage index Ink.
Linear predictor η:
5,190 8,704 st 0,170 Ink
(33)
The influence of both variables on building damage probability remains significant
(p-values for both parameters are ≤ 0.001). The model diagnostic indicators are:
φstd
0,978
Sens
75,5%
Spec
79,4%
Ta
77,3%
6. Test result interpretation
Table 5 contains the values of the most significant statistics which feature in the above
models, i.e. standardized deviations φstd and total accuracy Ta of the prognosis.
TABLE 5
List of statistical values for the evaluation of logit models
TABLICA 5
Zestawienie wartości statystyk do oceny liniowych predyktorów η modeli logitowych
nr 1 (|
Model
|
ε Kn
max ,
KO)
nr 2 (εlim, KTG)
nr 3 (εlim, | ε Kn
max|)
cr
nexc
nr 6 (εres
, |εmax
|)
nr 4 (st, | ε Kn
max|)
nexc
nr 7 (st, |εmax
|)
nr 5 (tb, | ε Kn
max|)
nexc
nr 8 (tb, |εmax
|)
nr 9 (tb, Ink)
nr 10 (st, Ink)
Ta [%]
φstd
1,217
67,4
–
–
1,265
64,6
0,976
77,2
1,235
66,9
1,215
0,978
66,9
77,3
180
It appears from the data contained in table 5 that models No. 4 (see formula 27),
No. 7 (see formula 30) and model No. 10 (see formula 33) have the highest matching
quality and the best total accuracy of the prognosis. The estimation of building damage
probability according to models No. 4 and No. 7 renders identical results, which may be
nexc
explained by the rather small influence of horizontal strains | ε Kn
max | or | εmax | as compared
to influence of building technical state st .
Since values of standardized deviance φstd as well as total accuracy Ta for models
No. 4, No. 7 and No. 10 are equal, these models may be considered to be equally reliable.
The lack of significant differences between these models motivates the use one of them
based on premises other than those which result from purely evaluation. Since model
No. 10 contains indicator Ink, used to define probable degree of building damage su its
application to building damage probability prognosis is recommended.
Table 6 contains a list of building damage probability values calculated according to
formula 33 for determined ranges of technical state st and expected damage index Ink.
TABLE 6
Probability of building damage according to model No. 10 (formula 33)
TABLICA 6
Rozkład prawdopodobieństwa uszkodzenia budynku według modelu nr 10 (wzór 33)
Technical
state st
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,0
0,99
0,97
0,94
0,87
0,74
0,54
0,33
0,17
0,08
0,03
2,0
0,99
0,98
0,95
0,89
0,77
0,58
0,37
0,20
0,09
0,04
Damage index Ink
3,0
4,0
5,0
1,00
1,00
1,00
0,98
0,99
0,99
0,96
0,96
0,97
0,91
0,92
0,93
0,80
0,82
0,85
0,62
0,66
0,70
0,40
0,45
0,49
0,22
0,25
0,29
0,11
0,12
0,14
0,05
0,06
0,07
6,0
1,00
1,00
0,97
0,94
0,87
0,73
0,53
0,32
0,17
0,08
7,0
1,00
1,00
0,98
0,95
0,89
0,76
0,58
0,36
0,19
0,09
When analyzing building damage probability distribution according to table 6, it
may be noted that this model reacts significantly better to building technical state st
changes than to changes in damage index Ink. The weight of technical state st taken as an
explanatory variable in building damage is much higher than the weight of the second
explanatory variable. It should be remembered, and this is significant in the evaluation
of probability values shown in table 6, that a building is considered to be damaged if
damage degree su > 0,05 in accordance with the (Wodyński & Kocot, 1996) scale.
It should be stated that only one technical feature of building influencing its resistance to continuous deformation of foundations is represented in model No. 10. No other
181
features are taken into account when using the score method for building resistance
evaluation. However, those models where other features are included in the explanatory
variables, i.e. building resistance category KO (model No. 1) and limiting horizontal
strain εlim (model No. 3), appeared to be less reliable due to the concomitant values of
standardized deviations φstd and total accuracy Ta. Based on the results obtained a building resistance evaluation using categories is not very accurate. A building’s reaction to
horizontal strains of its foundations is largely dependent upon its technical state. All
other technical features which influence a building’s resistance to continuous surface
deformation are reflected indirectly in the damage index Ink. However, the relatively
low influence of this explanatory variable on building damage probability leads to the
conclusion that other technical features of building, taken individually, may be less
significant to the damage of that building than st , its overall technical state.
The building damage probability model according to formula 33 was verified using
sets contained in the component bases BIER BYC, KOS and LAZ. Quality measures
of the prognosis, such as Sens, Spec and Ta were determined for each set. The results
are shown below.
Region
Bierun
Byczyna
Kostuchna
Laziska
Sens
61,7%
70,6%
68,8%
63,3%
Spec
64,3%
73,5%
64,9%
61,5%
Ta
62,3%
72,2%
67,0%
62,8%
The results of this verification may be considered satisfactory since they confirm the
reliability of model No. 10 (see formula 33) when applied to an estimation of the building damage probability in those units which are exposed to the influence of horizontal
strains on foundations.
7. Conclusion
The method of assessing the probability of damage to buildings in mining land according to logit model No. 10 (formula 33) can be applied to planning underground
hard coal exploitation with fall of roof (in 1.5 m to 3.5 m thick layers of coal beds at the
depth range of 250 m to 600 m) under buildings of traditional construction of any age
and technical state. The model can easily be updated based on current results on surface
deformation and adapted to deal with particular mining and geological features which
obtain in land areas each of which may have its own characteristics.
This research work financed by statutory investigations KOTGGiGG AGH
nr 11.11.150.650
182
REFERENCES
A g r e s t i A., 1990. Categorical Data Analysis. Wiley.
B a t k i e w i c z W., C h o c z e w s k i B., P o p i o ł e k E., 1977. Ocena zagrożeń obiektów budowlanych na terenach
górniczych z uwzględnieniem rozproszenia statystycznego wskaźników deformacji. Ochrona Terenów Górniczych,
nr 41.
M c C u l l a g h P., N e l d e r J.A. 1989: Generalized Linear Model. Chapman & Hall.
H a s t i e T., T i b s h i r a n i R., F r i e d m a n n J., 2001. The Elements of Statistical Learning. Springer.
H e j m a n o w s k i R., K w i n t a A., P a t y k o w s k i G., 2005. Metoda wyznaczania współczynnika proporcjonalności
przemieszczeń poziomych B dla rejonu O/ZG „Lubin”. [w:] Materiały Konferencji Naukowo-Technicznej: „VIII Dni
Miernictwa Górniczego i Ochrony Terenów Górniczych”. GIG Katowice, Ustroń, 15-17 czerwca 2005, 324-333.
K a s z o w s k a O., 2002. Metoda prognozowania kosztów usuwania szkód w budynkach na terenach górniczych. [w:]
Materiały Konferencji: „Problemy Ochrony Terenów Górniczych”, Główny Instytut Górnictwa, Ustroń-Zawodzie,
18-19 kwietnia 2002, 211-216.
K n o t h e S., 1984. Prognozowanie wpływów eksploatacji górniczej. Śląsk, Katowice.
K w i a t e k J., 2002. Obiekty budowlane na terenach górniczych. Wydawnictwo Głównego Instytutu Górnictwa, Katowice.
K w i a t e k J., 2004. Problemy współdziałania obiektów budowlanych z podłożem na terenach górniczych. [w:] Materiały II Konferencji Naukowo-Technicznej: „Problemy projektowania i ochrony obiektów budowlanych na terenach
górniczych”. Instytut Techniki Budowlanej, Oddział Śląski, Rudy Raciborskie, 27-28 maja 2004, 85-96.
O s t r o w s k i J., 2006. Deformacje powierzchni a zagrożenie uszkodzeniami budynków na terenach górniczych
w ujęciu probabilistycznym. AGH Uczelniane Wydawnictwa Naukowo-Techniczne. Rozprawy i monografie 160,
s. 120, Kraków.
P o p i o ł e k E., 1989. Ochrona terenów górniczych. Skrypt uczelniany nr 1172, AGH, Kraków.
P o p i o ł e k E., O s t r o w s k i J., 1978. Zależność między nachyleniami a przemieszczeniami poziomymi terenu
w ostatecznie wykształconych nieckach obniżeniowych. Ochrona Terenów Górniczych, nr 46.
P o p i o ł e k E., O s t r o w s k i J., 1981. Próba ustalenia głównych przyczyn rozbieżności prognozowanych i obserwowanych poeksploatacyjnych wskaźników deformacji. Ochrona Terenów Górniczych, nr 58.
P o p i o ł e k E., O s t r o w s k i J., N i e d o j a d ł o Z. et. al., 1999. Wstępne wyniki analizy dotychczasowych geodezyjnych badań wpływów eksploatacji górniczej na deformacje powierzchni terenu w LGOM. [w:] Materiały Konferencji
Naukowo-Technicznej: „V Dni Miernictwa Górniczego i Ochrony Terenów Górniczych”. Szczyrk, 29.09-1.10. 1999.
Wydawnictwo Głównego Instytutu Górnictwa, Seria: Konferencje nr 30, 96-107.
Projekt badawczy, 1997. Losowość pogórniczych deformacji terenu i odporności obiektów powierzchniowych w świetle
wyników pomiarów geodezyjnych i obserwacji budowlanych oraz jej wpływ na wiarygodność prognoz górniczych.
Projekt Badawczy KBN zrealizowany w Katedrze Ochrony Terenów Górniczych AGH w Krakowie pod kierownictwem E. Popiołka, niepublikowany.
Rozporządzenie..., 2002. Rozporządzenie Ministra Spraw Wewnętrznych i Administracji z dnia 14 czerwca 2002 r.
w sprawie planów ruchu zakładów górniczych (Dz. U. z 2002 r. Nr 94, poz. 840).
S r o k a A., P o p i o ł e k E., H e j m a n o w s k i R., 1994. Probabilistyczna metoda oceny stopnia zagrożenia obiektów na
skutek podziemnej eksploatacji górniczej. Bezpieczeństwo Pracy i Ochrona Środowiska w Górnictwie, nr 1 (9).
W o d y ń s k i A., K o c o t W., 1996. Metodyka oceny stanu technicznego budynków o tradycyjnej konstrukcji zlokalizowanych na terenach górniczych. Przegląd Górniczy Nr 7-8. Katowice, 1996.
Z a s a d y ..., 1969. Zasady stosowania budownictwa zastępczego na terenach górniczych. Praca zbiorowa. Śląsk, Katowice.
Z a s a d y ..., 1979. Zasady sporządzania programów ochrony terenów górniczych. ZG SITG, WUG, Katowice.
Received: 28 January 2008