Arch. Min. Sci., Vol. 54 (2009), No 4, p. 725–737

Arch. Min. Sci., Vol. 54 (2009), No 4, p. 725–737
725
Electronic version (in color) of this article is available: http://mining.archives.pl
VIOLETTA SOKOŁA-SZEWIOŁA*
METHOD OF PREDICTION THE PROBABILITY OF A STRONG TREMOR ON THE BASIS
OF OBSERVED CHANGES OF MINING GROUND SUBSIDENCES
METODA PREDYKCJI PRAWDOPODOBIEŃSTWA WYSTĄPIENIA SILNEGO WSTRZĄSU
NA PODSTAWIE OBSERWOWANYCH ZMIAN OBNIŻEŃ TERENU GÓRNICZEGO
The method of prediction the probability of a strong tremor on the basis of the observed changes in
the size of the mining subsidences was presented. The method uses the logit model, on the basis of which
the probability of a strong tremor in time T with the energy exceeding some limit is calculated. The indices
used for the description of the seismicity of a rock mass are the number of tremors N recorded in a given
period of time and total seismic energy E recorded in this period in a specific region. Observed changes
in mining ground subsidences are characterized in the present method cross-section area of the subsiding
trough Pw along the observation line parallel to the situated advancing longwall front.
The work show the algorithm of the method and the examples of its use in practice in the region of
conducted in the conditions of high seismic activity of the exploited longwall.
Keywords: prediction of seismic activity, induced seismic activity, mining surface deformations
Zagrożenie sejsmiczne jako jedno z najważniejszych występujących w polskim górnictwie stanowi
wciąż temat licznych badań. Ich wyniki mają doprowadzić do jego zmniejszenia i jednocześnie do ograniczenia niekorzystnych skutków zjawisk sejsmicznych. Ograniczenie to jest możliwe dzięki wykorzystaniu
możliwości prognozowania potencjalnego zagrożenia sejsmicznego.
W pracy przedstawiono propozycję metody predykcji prawdopodobieństwa wystąpienia silnego
wstrząsu na podstawie obserwowanych zmian obniżeń terenu górniczego wykorzystującej model logitowy.
Metodę opracowano na podstawie wyników badań zależności pomiędzy wielkościami powierzchni terenu
oraz aktywnością sejsmiczną górotworu, które zrealizowano w latach 2004-2007 w Instytucie Eksploatacji
Złóż Politechniki Śląskiej. Istotnym rezultatem badań było wprowadzenie nowego wskaźnika Pw charakteryzującego zmiany obniżeń terenu górniczego. Wskaźnik ten jest powierzchnią poprzecznego przekroju
niecki obniżeniowej wzdłuż linii obserwacyjnej usytuowanej zgodnie z kierunkiem postępującego frontu
ścianowego (Rys. 1). Założono, że powierzchnia ta lepiej opisuje rzeczywiste ruchy górotworu niż objętość
pustki tworzącej się w górotworze wskutek prowadzonej eksploatacji. Wskaźnik wykorzystano do opisu
zmian obniżeń terenu w prezentowanej metodzie. Poziom sejsmiczności indukowanej przyjęto charakteryzować za pomocą liczby wstrząsów zarejestrowanych w danym przedziale czasu (N) oraz sumarycznej
energii sejsmicznej wstrząsów zarejestrowanych w danym przedziale czasu (E).
*
SILESIAN UNIVERSITY OF TECHNOLOGY, UL. AKADEMICKA 2, 44-100 GLIWICE, POLAND
726
Celem prezentowanej prognozy jest określenie prawdopodobieństwa wystąpienia silnego wstrząsu
na postawie narastającej powierzchni Pw oraz wyzwolonej sumarycznej energii sejsmicznej.
W metodzie poszukiwana jest zależność, pozwalająca na uzyskanie prawdopodobieństwa wystąpienia
silnego wstrząsu o energii sejsmicznej rzędu powyżej pewnego progu energetycznego na podstawie wyznaczonego modelu logitowego, który opisuje zależność między częstościami występowania poszczególnych
wariantów zmiennej zależnej, a wybranymi zmiennymi niezależnymi.
W pracy przestawiono szczegółowo algorytm metody. Treść wywodu prowadzącego do uzyskania
modelu logitowego zaprezentowano w postaci wzorów od (1) do (8). Ostateczną postać wyznaczonego
logitu przedstawiono w postaci wzoru (9). Dokładność modelu ustala się jest poprzez wyznaczenie
odchylenia standardowego reszt S(u) (wzór 10). Poziom dopasowania danych teoretycznych do empirycznych ustala się poprzez wyznaczenie współczynnika determinacji R2 (wzór 12). Prognoza logitu
w czasie T obliczana jest z wykorzystaniem opracowanego modelu logitowego (wzór 14). Obliczany
jest bezwzględny i względny błąd prognozy logitu (wzory 15, 16). W przypadku uzyskania wartości
pozwalających na stwierdzenie, że wyznaczone prognozy logitu są dopuszczalne obliczana jest prognoza
prawdopodobieństwa wystąpienia silnego wstrząsu w czasie T (wzór 17). Po upływie czasu prognozy
wykonywana jest analiza ex post (wzór 18).
W pracy przeprowadzono ocenę zdolności predykcyjnych modelu dla obszaru eksploatacji ścianowej
prowadzonej przez KWK „Halemba”. Model logitowy wyznaczono na podstawie wyników badań deformacji powierzchni terenu w rejonie eksploatacji prowadzonej ścianą w pokładzie 415/1. Powierzchnię Pw
określono na podstawie okresowych pomiarów geodezyjnych na linii obserwacyjnej (Rys. 2). Narastające
pole powierzchni przekroju niecki obniżeniowej i sumaryczną energii sejsmicznej w czasookresach cykli
pomiarów niwelacyjnych oraz prawdopodobieństwo empiryczne wystąpienia wstrząsu o energii sejsmicznej E ≥ 5 · 104 [J] w analizowanym rejonie w funkcji narastającego pola powierzchni Pw przedstawiono
na rysunku 3. Na podstawie danych z pierwszych 8 miesięcy prowadzonej eksploatacji i wyników 33
cykli pomiarów niwelacyjnych ustalono model logitu w postaci (19). Odchylenie standardowe reszt dla
modelu oraz współczynnik determinacji. wskazuje, iż model można, zastosować do prognozowania prawdopodobieństw wystąpienia silnych wstrząsów. Prognozę logitu L*T obliczono wykorzystując zależność
(14) dla okresu obejmującego następnych 8 miesięcy eksploatacji. Błędy względne ex ante w badanych
czasookresach prognozowania nie przekraczały 5% co oznacza, że wyznaczone prognozy logitu są dopuszczalne. Prognozę prawdopodobieństwa p*T wyznaczono wykorzystując zależność (17). Na rysunku 4
przedstawiono wykresy rzeczywistych oraz prognozowanych prawdopodobieństw w okresie prognozy.
Metodę można zastosować do predykcji silnych wstrząsów w rejonie obejmującym obszar aktualnie
prowadzonej eksploatacji. Estymację parametrów wyznaczonego modelu logitowego należy każdorazowo
połączyć z badaniem poprawności modelu. Metody nie można zastosować, gdy liczba zarejestrowanych
zjawisk sejsmicznych jest mała lub gdy nie posiadamy odpowiedniej ilości danych pozwalających na
ustalenie wskaźnika Pw. Metoda wymaga przeprowadzenia jej weryfikacji w warunkach prowadzenia
eksploatacji w innych rejonach.
Słowa kluczowe: predykcja zagrożenia sejsmicznego, sejsmiczność indukowana, deformacje terenu
górniczego
1. Introduction
The prediction of the probability of a strong tremors is one of the most important
tasks during the exploitation in the conditions of induced seismicity. The importance
of the issue results mainly from the relation between tremors and rock burst. The aim
of the prediction can be formulated as the assessment of time, energy and place of the
occurrence of a strong tremor. The prediction (forecast) is the statement concerning
specific future, which we formulate on the basis of possessed data from the past and
the knowledge of the fragment of reality which was the subject of the forecast (Greń,
727
1978). The prediction we obtain on the basis the assessment with particular reliability,
at least one of the parameters of the phenomena forecasted in order to receive the lowest possible error of estimation. On this basis the operations are conducted in order to
reduce disadvantageous results of the occurrence.
There is a large number of methods by means of which the evaluation of the probability of seismic activity is determined. There are statistic, analytic methods, and qualitative
and quantitative methods (inter alia Lasocki, 1981, Takuska-Węgrzyn, 2008).
In statistic methods it is usually assumed that there is a given quantitative model
based on the empirical relationship and theoretical conception, which describes the process of the emergence of tremors. As a result probability functions describing the data are
received. The estimation of the parameters of these functions is based on the empirical
data. As a result we we receive the probability of a strong tremor in the region at the
time of assumption. The dynamics of the process is described using the variations of the
parameters of the function in time and the process itself can be perceived as a stationary in a given constant period of time. The work (Głowacka, 1991) shows one of the
methods of evaluation based on the estimation of the variable in time risk on the basis
of the relationship between the total seismic energy and the exploited volume.
Taking into consideration the overt assumption base and the possibility of its verification and the possibility of making comparisons with the results of the forecasts achieved
using other methods, the work shows the method of prediction of seismic activity based
on the observed changes in the mining ground subsidences using logit model. The possibility of characterizing the level of the induced seismicity by taking into consideration
the indices which describe the deformation processes of the tremor-inducing strata of
a rock mass and ground surface has been indicated in a number of works (Wanior, 1982;
Goszcz, 1988; Drzęźla et al., 1994; Bańka, 1996).
The presented method was made on the basis of the results of the research conducted
in years 2004-2007 in Institute of Mining of the Silesian University of Technology
(Sokoła-Szewioła, 2006, 2008a, 2008b; Sokoła-Szewioła et al., 2007). As a result of
these works changes in mining ground subsidences were described using the area of the
cross-section of the trough subsiding along the observation line situated in accordance
with the direction of the advancing longwall front, approximately in the middle of the
longwall area. Induced seismic activity is described using the number and the energy
of the tremors recorded in the researched region. The method calculates the probability
a strong tremor with the energy exceeding energetic threshold in time T. The energetic
threshold is dependent on the local conditions of mining. The assessment of predicting
of the calculated logit model was conducted for the area of the longwall mining in the
region of the “Halemba” hard coal mine.
728
2. Indices which characterize induced seismicity
The coal mine geophysical stations calculate the coordinates of the tremor focuses
and seismic energies as standard, hence the choice of the indices:
N — the number of tremors recorded in a given period of time,
E — total energy of tremors recorded in a given period of time.
The analysis of the stronger seismic phenomena should always be conducted before the use of the results of the record in order to eliminate possible events of regional
character.
3. Index which characterizes the changes of mining ground
subsidences
For describing the changes of the mining ground subsidences resulting from mining
the Pw cross-section area of the subsiding trough along the observation line situated
according to the direction of an advancing longwall front was assumed in a described
method.
The index was established after Kijko (1985) who assumed that there is dependence
between the volume of the formed cavern in the rock mass and the observed seismic
activity of the rock mass. It was thus obvious that the relationship between the volume
Vw, of the subsiding trough forming on the ground, and seismic emission of the rock
mass, as the volume of troughs is dependent on the volume of the formed cavern. At
the same time it was assumed that the volume of observed trough better corresponds to
the true movements of the rock mass than the volume of the trough forming as a result
of the exploitation of the cavern in the rock mass.
Geodesic measurements using especially designed observation lines are performed
on the surface during the mining exploitation. One of the lines is usually situated along
an advancing longwall front. Thus we mainly possess the observations of the change
in subsidences of the points of the observation line situated along the longwall area,
which by necessity leads to the adoption of simplified assumption that the volume Vw
of the forming trough changes in time proportionally to the cross-section area Pw of the
subsiding trough, formed along the observation line. Measured changes in subsidences
of the surface points of the ground along the observation line in one measurement cycle
and the area Pw were show in Fig. 1.
729
Piont number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
w [m]
0
–0,10
–0,20
–0,30
–0,40
–0,50
–0,60
–0,70
–0,80
–0,90
Pw
Fig. 1. Measured change in subsidences of the surface points of the ground along the observation line,
Pw – the area of the cross-section of subsiding trough
4. Method of prediction the probability of
a strong tremor
The research (Sokoła-Szewioła and others, 2007) showed that there is a statistically significant relationship between the increasing total seismic energy of the tremors
E (increasing total number of tremors N) and the increasing cross-section area of the
subsiding trough Pw along the observation line situated according to the direction of
advancing longwall front, approximately in the middle of the longwall area. On this
basis and on the basis of the argument presented in chapter 3 in the method it was assumed that based on the changes in Pw and changes in observed seismic activity we
can estimate the probability of a strong tremor. The analyses of the relationship between
the seismic activity and cross-section area of a subsiding trough showed the validity of
applying the Ej quantity (total seismic energy of the tremors released in j time-period
of realized geodesic surveying) and Pwj (cross-section area of the subsiding trough in
the end of j – time-period of realized geodesic surveying) for predicting the probability
strong tremors.
The aim of the forecast is to determine the probability of a strong tremor based
on the increasing cross-section area of the subsiding trough and total released seismic
energy in time-period of realized geodesic surveying.
It was assumed that the relationship will be searched that will allow us to predict the
probability of a strong tremor with the energy of the order of above energetic threshold
on the basis of the determined logit model (Maddala, 2008). The relationship between the
occurrence frequency of the particular values of the dependent variable and the selected
independent variables was described in the model. Energetic threshold is dependent on
the local conditions of mining (Konopko, 1995).
730
In the method:
It was assumed that in j – time-period of realized geodesic surveying occurred nj
variants (tremors).
– On the basis of the formula (1) we calculate the frequency of occurrence of chosen
variant (tremor with energy not lower than the energetic threshold eg. E ≥ 5 . 105 [J])
in j – time-period of realized geodesic surveying:
pj =
yj
j ∈ {1, ..., m}
nj
(1)
where:
yj — the number of the chosen variants occurred in j – time-period of realized
geodesic surveying cycle,
pj — frequency of the chosen variant in j – time-period of realized geodesic surveying cycle,
nj — number of elements in j – time-period of realized geodesic surveying cycle
(number of tremors in a period),
m — the number of time-period of realized geodesic surveying cycle, for j ∈ {1,...,m}
the total number of tremors recorded in the whole analyzed period is N:
m
N = å nj
(2)
j=1
– The frequency pj is an empirical probability of an event occurring of chosen variants
in j – time-period of realized geodesic surveying cycle.
Let pj (probability in j – time period) be the function of independent variables X1,..., Xk
that is:
pj = f (X1j , ..., Xkj)
j ∈ {1, ..., m}
(3)
it was assumed:
X1 — cross-section of the subsiding trough Pwj [m2] in the end of j – time period
of realized geodetic surveying cycle,
X2 — total seismic energy in j – time period Ej [J] of the realized surveying cycle,
k — number of independent variables.
– It was assumed that f is a linear function.
– Monotonic transformations of the probability p ∈ (0,1) into an interval were carried
out: (–∞, +∞) through the logit transformation (4):
731
æ pj
Lj = ç
ç1 - p
j
è
æ
ç
ç
è
j Î {1,..., m}
(4)
– The form of a model was searched (5):
j ∈ {1, ..., m}
Lj = β0 + β1 X1j + ... + βk XKj + ξ j
(5)
where:
β0,β1,..., βk — unknown parameters,
X1,..., Xk — independent variables,
ξj — random factor,
k — number of independent variables.
Thus, the model of multiple regression was searched (6):
L = Xβ + ξ
(6)
where:
β — unknown parameters vector,
X — observation matrix created from the observations of the independent variables in the each time-period,
ξ — vector of random factor.
The generalized least squares method was applied. Assuming that the matrix X is full
type and m > k + 1 estimator of the logit parameters vector has a following form (7):
β̂ = (X TŴ –1X )–1X TŴ –1L
(7)
where:
é wˆ 1
ê
.
ê
Wˆ = ê
ê
ê
ë0
.
0 ù
ú
ú
ú
ú
.
ú
wˆ m û
wˆ j =
1
n j pj (1 - pj )
for
jÎ {1,..., m}
^
Based on the empirical observations we determine the value b of estimator β in the
logit model receiving (8):
^
L j = b0 + b1 X1 j + ... + bk Xkj
j ∈ {1, ..., m}
(8)
thus taking into consideration the variables assumed in the model receiving (9):
^
L j = b0 + b1 Pwj + b2Ej
j ∈ {1, ..., m}
(9)
732
– in order to investigate the accuracy of the model we calculate the standard deviation
of remainder S(u) according to the formula (10):
S (u) = S 2 (u)
(10)
where:
S(u)2 — remainder variance of the model calculated according to the formula (11):
S 2 (u ) =
Ù -1
1
uT W u
m - k -1
(11)
where:
u — calculated vector of the remainder of logit model
– we calculate the value of the factor of the determination R2 in order to calculate the
level of match of theoretical and empirical data (12):
m
R2 = 1 -
Ù
å ( Lj - Lj ) 2
j =1
m
-
å ( L j - L)
j Î {1,..., m}
(12)
2
j =1
where:
–
L — mean arithmetic value of empirical logits,
L^j — theoretical values of the logits calculated from the model (9),
Lj — value of empirical logits.
If the model turns out to be precise we begin the forecasts. The forecasts of logit are
determined using forecast principle as in linear econometric models.
*
*
,..., X k,T
are the values of independent varia– If T is the time of the forecast, and X 1,T
bles in the forecasted time, the forecast of logit L*T forecasted in time T is as follows
(13):
LT* = b0 + b1 X 1*,T + ... + bk X k*,T
(13)
where:
T = m + 1,..., m + h, h > 0 (h – horizon of the forecast)
Using determined model (9) logit forecast is as follows (14):
LT* = b0 + b1 PwT* + b2 ET*
T = m + 1 , ..., m+h
(14)
733
where:
Pw*T — cross-section of the subsiding trough in the end of period of “T ” [m2],
E *T — total seismic energy in the period of “T ” [J].
– We calculate the absolute error of the forecast of the logit ex ante using following
formula (15):
D ( L*T ) = S (u ) 1 + c T ( X T W -1 X ) -1 c
(15)
where:
*
cT = (1, X 1,T
,... X *k,T )
S(u) — standard deviation of the remainder.
– The relative error of the ex ante forecast was calculated according to the following
formula (16):
V ( L*T ) =
D (LT* )
100%
L*T
T = m + 1 , ..., m+h
(16)
– The forecast of the probability pT* in time T is determined using formula (4) is (17):
*
pT*
=
10 LT
T = m + 1 , ..., m+h
*
1 + 10 LT
(17)
After the forecast ex post analysis is performed – forecasted and true probabilities
are compared.
For this reason the following data are calculated:
– the mean squared error ex post of forecasts calculated in forecasted time is (18):
S p2 =
1
h
m+h
å( p
T= m +1
*
T
- pT ) 2
where:
pT — true probability in the forecasted time,
pT* — forecasted probability.
(18)
734
5. The prediction of the probability of a strong tremor
in the region of “Halemba” hard coal mine
Logit model was calculated on the basis of the results of the research of ground
surface deformation has been conducted in the area of longwall exploitation carried out
by “Halemba” hard coal mine on the seam 415/1, at the average depth of 600 m. The
length of the longwall amounted to 285 m and the face advance was 1070 m. Extraction
was carried out with use of longwall system, with fall of roof, at the height of 3.5 m
leaving coal on the 0.9 m-wide floor. The seam in the area is classified as the 3rd degree
of rock burst threat. The details of the research were presented in the work (SokołaSzewioła et al., 2007). In the object part in the period of the observation 339 tremors
with the seismic strength from 103 to 106 [J] were recorded. Maximum seismic energy
of the tremor was 3 . 106 [J]. The prediction the probability of the tremor with energy
E ≥ 5 . 104 [J] was assumed.
The area Pwj was calculated on the basis of the periodic geodesic measurements:
leveling and angular-linear on the observation line 1. Figure 2 shows out lines of the
observed longwall and the draft of the observation line in the range used for the prediction of the probability of a strong tremor.
Obs
1
5
erv
Point of observation line
ed
long
wal
l
10
15
Obs
20
erva
tion
line
(1)
25
30
34
36
Fig. 2. The outlines of the exploitation performed on observed longwall, draft of the observation line
Figure 3 shows total recorded seismic energy and empirical probability of a tremor
with energy E ≥ 5 . 104 [J] in time-periods of the cycles of the leveling measurements
in the function of increasing surface area of the subsiding trough in the period used to
calculate the model.
In the analysed set of observation including the period of the first 8 months of mining according to the algorithm presented in point 4 the following function relation was
calculated (19):
^
L j = 0,002607587 – 0,00093676 Pwj – 0,000000485748 Ej
(19)
735
1,8
Ej /10^6
pj
1,6
Ej /10^6 [J], p j
1,4
1,2
1,0
0,8
0,6
0,4
0,2
701,5
676,1
643,7
627,0
607,0
580,7
537,2
485,1
418,8
364,6
316,9
292,3
222,7
123,8
78,5
47,7
0
Pwj [m 2]
Fig. 3. Increasing surface area Pwj of the cross-section of the subsiding trough
and total seismic energy Ej and the probability pj of the tremor with the energy E ≥ 5 . 104 [J]
in time-periods of the cycles of the leveling measurements
The model was calculated using the data on 216 recorded tremors and the results of
the and the results of the observation of change of size from 33 measuring cycles.
− The calculated standard deviation of the remainder for the model is S(u) = 0,592.
The calculated index R2 = 0,528.
The forecast of the logit LT* was calculated using the dependence (14) for the period
of the next 8 month of conducted exploitation.
− Relative errors ex ante in the researched time-periods of the forecast did not
exceed 5%, which means that the calculated forecasts of the logit acceptable.
The forecast of the probability of a strong tremor p*T was then calculated using the
dependence (17). Figure 4 shows the charts of true and predicted probabilities. Ex post
mean squared error according to the formula (18) was 0,12.
736
1,0
0,9
PT*
PT
0,8
0,7
PT, PT*
0,6
0,5
0,4
0,3
0,2
0,1
0
810
854
945
1010
1063
1121
1198
1262
1316
PwT* [m 2]
Fig. 4. True pT and forecasted pT* probabilities of a strong tremor in the forecasted time
6. Summary and conclusions
− The article discusses the method of prediction the probability of a strong tremor
devised on the basis of the calculated logit model.
− The method assumes that there is the dependence between the volume Vw of
the subsiding trough forming on the area and the observed seismic activity of
a rock mass. The volume Vw of the forming trough changes in time proportionally to the area Pw of the cross-section the subsiding trough, formed along the
observation line. On this basis the increasing area of the cross-section Pw of the
subsiding trough along the observation line situated parallel to the direction of
the advancing longwall front was assumed as an index of changes in size of the
mining surface.
− Total seismic energy of induced tremors and the number of tremors recorded in
a given period of time were assumed as the indices of the induced seismic activity
in the method.
− The method can be applied to the prediction of strong tremors in the region of
the current exploitation.
− The estimation of the parameters of the calculated logit model should be each
time connected with the research on the correctness of the model.
− The methods cannot be applied when the number of recorded phenomena is small
and when we do not have enough data making it possible to calculate the index Pw.
− Conducted assessment of predicting potential of the calculated logit model was
positive. Relative errors ex ante in the researched time-periods the forecasts did not
737
exceed 5%, which means that the calculated forecasts of the logit are acceptable.
Ex post mean squared error was 0,12. The value of the error is connected with
the lower seismic activity observed in period of the forecast than had expected.
− The presented method require to conduct the assessment of the predicting in the
conditions of the exploitation in other regions.
References
Bańka P., 1996. Metoda prognozowania czasowych zmian aktywności sejsmicznej w oparciu o deformacje górotworu
wywołane eksploatacją górniczą. Praca doktorska. Politechnika Śląska, Gliwice.
Drzęźla B., Białek J., Jaworski A., Bańka P., Kołodziejczyk P., 1994. Badanie związków sejsmiczności indukowanej
eksploatacją górniczą z parametrami opisującymi deformacje ośrodka skalnego. Sprawozdanie końcowe z realizacji
projektu badawczego KBN nr 903759101. Gliwice, praca niepublikowana.
Głowacka E., 1991. Ocena zagrożenia sejsmicznego górotworu z uwzględnieniem przebiegu eksploatacji. Praca doktorska, Warszawa.
Goszcz A., 1988. Wpływ gradientu prędkości obniżania się powierzchni pod wpływem robót górniczych na stan zagrożenia
wstrząsami górniczymi. Wydawnictwo AGH, Kraków, ZN AGH, seria: Górnictwo, z. 141, 67-73.
Greń J., 1978. Metodologiczne aspekty prognozowania ekonomicznego. Przegląd statystyczny, nr 1, 3-4.
Kijko A., 1985. Theoretical model for relationship between mining seismicity and excavation area. Acta Geoph. Pol.
Vol.33, 231-239.
Lasocki S., 1981. Statystyczna metoda oceny zmian aktywności mikrosejsmologicznej rejonu przyeksploatacyjnego.
Archiwum Górnictwa. 26, z. 1, s. 131-160.
Konopko W., 1995. Doświadczalne podstawy kwalifikowania wyrobisk górniczych w kopalniach węgla kamiennego do
stopni zagrożenia tąpaniami. P.N. GIG, nr 795. Katowice.
Maddala G.S., 2008. Ekonometria. PWN, Warszawa.
Sokoła-Szewioła V., 2006. Obserwowane zmiany obniżeń punktów powierzchni terenu a rejestrowana aktywność sejsmiczna górotworu. XIII Międzynarodowa Konferencja Naukowo-Techniczna nt. Górnicze zagrożenia naturalne
2006. GIG, 319-328. Katowice.
Sokoła-Szewioła V., Białek J., Bańka P. i in., 2007. Zależności pomiędzy wielkościami deformacji powierzchni terenu
i aktywnością sejsmiczną górotworu. Projekt badawczy KBN 4T12A 032 26. Gliwice, praca niepublikowana.
Sokoła-Szewioła V., 2008a. Models of dependences between induced seismic activity and observed deformations of surface
in the area of the conducted longwall exploitation. Gospodarka Surowcami Mineralnymi, t. 24, z. 31, 159-172.
Sokoła-Szewioła V., 2008b. Jakościowe oraz ilościowe zależności pomiędzy sejsmicznością indukowaną i prędkościami
obniżeń punktów obserwacyjnych na terenie górniczym. Przegląd Górniczy, 11-12, 32-37.
Takuska-Węgrzyn E., 2008. Application of Statistical Methods for Evaluation of Rock-Burst Risks in Copper ore Mine
Conditions. Arch. Min. Sci., vol. 53, iss. 1, p. 23-30.
Wanior J., 1982. Metoda prognozowania wstrząsów i tąpnięć w oparciu o wyniki pomiarów geodezyjnych. PTPNOZ.
Częstochowa.
Received: 16 July 2009