Analisis Estadistico de Frcturas Mit 1984
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Transcript of Analisis Estadistico de Frcturas Mit 1984
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8/10/2019 Analisis Estadistico de Frcturas Mit 1984
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ST TISTIC L N LYSIS
OF
ROCK M SS
FR CTURING
Gregory B. Baecher
l
STR CT
Over the pa s t ten years c o ns id e ra b le e m p ir ic a l
work
has
been
repor ted
on
th e
s tochas t ic
des-
c r i p t i on o f
rock mass
f r a c tu r ing and on
the
s t a t i s t i c a l design
o f
j o i n t surveys This
has led to
cons i s ten t
conc lus ions cn the
d i s t r i b u t i o n a l p ro pe rt ie s o f such di scon-
t i nu i t i e s
and i s beginning to lead to
improved
survey
des igns
Two
o f
the s t ronge s t conclu-
s ions appear to
be
the Exponent ia l i ty o f
th e d i s t r ibu t ion
o f spacings between
d i s -
con t i nu i t i e s
when
measured
by
t h e i r
i n t e r sec -
t i ons with sampling l i nes and the lognormal i ty
o f
d i s c on t inu i ty
t race leng ths as observed in
outcrops Cons is tent
conclus ions on
th e
form
o f
o r ie n t at io n d i st ri b u ti o n s
appear
more
e lus ive
Sampling b iases in j o i n t surveys now
seem more
pervas ive than was e a r l i e r
thought
In
addi t ion
to the
wel l
known o r ie n ta tio n b ia s in
sampling
from two dimensional outc rops propor t iona l
leng th
b ia s
in
which
la rg e r d i sc o n ti n ui ti es a re
sampled with increased proba b i l i t y
and censor ing
b iases in which l a rge r
d i scon t i nu i t i e s
a re
of ten
only
p a r t i a l l y observed complicate s t t ~ s t i l
infer .ences . These r e su l t s a re reviewed aga ins t
a
r e c e n t s tudy
involving some
15 000 da ta
1
Associa te
Professor of Civ i l Engineering
Massachuset ts
In s t i t u t e of Technology Cambridge 02139
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INTRODU TION
J o i n t s u r v e y s a re
a n
i n t e g ra l component o f s i t e charac te r i za t ion
s tud ies in
r o c k
engineer ing
b e c a u s e
the s t r e ng th
d e f o r m a t i o n
a n d flo w b e h a v i o r o f r o c k m a s s e s a re s tro ng ly inf luenced by
th e g e o m e t r y and
engineer ing
proper t i e s o f r o c k
mass
d i scon t in -
u i t i e s F o r many d e c a d e s th e c o l l e c t i on o f g e o m e t r i c data
on
r o c k
mass
j o in t i ng h a s b e e n
r e c o g n i z e d
a s
a
p r o b l e m
o f
s t a t i s -
t i c a l s am p l i n g and
begining
in t h e m i d - 1 9 6 s many w o r k e r s
have
d e v o t e d e f f o r t to d e v e l o p i n g
sound
s u r v e y
p r o c e d u r e s a nd
to i n t e r -
pre t ing
the empi r ica l
da ta
base
The
p u r p o s e o f
t h i s
p a p e r i s to summarize empi r ica l
r e s u l t s
w i t h sp e c i a l re fe re nce to
work
a t
MIT
[6]
an d to
d iscuss
th e
in f luence
o f presen t
c o n c e p t s o f
j o in t i ng geometry
on
s a m p l i n g
p r o c e d u r e s .
G OM TRY O JOINTING
In t h i s sec t ion th e
g e o m e t r i c
p ro pe rt ie s o f j o in t i ng obser -
v a b l e
in cornmon s u r v e y s
are
d i sc r ibed W h i l e these
g e o m e t r i c
proper t i e s seem na tura l ly to f a l l
in to
d i s t i n c t
g e o m e t r i c
c l a s s e s
in
r e a l i t y
t h e y
a re
o n l y
face ts
o f o ther more f u n d a m e n t a l ways
o f d es crib in g
j o i n t
g e o m e t r y .
I t
i s t he r e fo r e im po r t a n t when
i n t e rp r e t i ng t h e impl ica t ions o f s u r v e y r e s u l t s
fo r
pred ic t ing
a g g r e g a t e
r o c k mass
behavior
t h a t these o b se rv ed g eo m et ric pro-
pe r t i e s b e viewed
as
s t rongly in terdependent
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, /
In j o i n t surveys ,
th ree
geometr ic proper t i e s a re commonly
o f i n t e r e s t :
Densi ty
e . g . ,
spac ing ,
f requency , s ize
e .g . ,
t race l eng th , a rea , and
o r i en t a t i on
e .g ., s tr ik e and
dip
o f an
approximat ing plane ,
d i r ec t i on
cos ines o f the
po le .
The
measures
adopted here a re
spac ing ,
t r ace l eng th ,
and po la r d i r ec t i on
cos ines .
Spacing
measured
by
the
separ ta ion o f th e i n t e r s ec t ions o f
adjacent j o i n t
t races with
a sampling l in e , e ith er
fo r in div id ua l
se ts o f
sub-para l l e l j o i n t s o r fo r a l l j o i n t s Fig . 1 . Trace
l eng th
i s
t yp ica l ly
measured
as the
l i n e a r
d is tance
between
the
end
po in t s o f th e i n t e r s ec t ion o f a j o i n t
with
an exposed su r face .
For j o in t s t h a t a re
s t rong ly
non-p lanar , o ther measures
are
some-
t imes used.
I f both
ends o f a
t r a ce a re
no t o bs er va ble , th e l ength
recorded
i s a
censored
l eng th .
EMPIRI L
T
The da ta
base fo r the
MIT
s tudy
comprised
j o i n t survey
da ta
from
seven
cons t ruct ion
and mining
s i t e s o f varying geology
Table 1 . These da ta
were p rin c ip al ly c ol le cte d f or
engineer ing
purposes ,
inc luding
foundat ion and s lope
des ign,
and al though
co l lec ted using
a
va r i e ty o f survey procedures , these procedures
were documented
so t h a t
s t a t i s t i c a l conclus ions could
be
drawn
from the
da ta s e t s . Ind iv idua l surveys recorded
spac ings , t race
l eng ths ,
or i en t a t i ons , phy sic al c on ditio ns
e . g . , ex ten t
o f
weather -
i ng , censor ing , type o f
t e rmina t ion e . g . ,
whether
a j o i n t ended
aga ins t ano ther j o i n t , e t c . , and occas iona l ly o th er f ea tu re s.
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Typical
r e su l t s fo r
spacing
d i s t r i bu t ions
are shown in
Fig . 2,
p lo t t ed
aga ins t
exponen t i a l cumula t ive
dens i ty
funct ions
cd f ,
F s
= exp{-As}. For 8
o f
the spacing d i s t r i bu t ions
ana lyzed
the ~ x p o n n t i l model f a i l ed
to
sa t i s fy Kolmogorov-Smirnov
c r i t e r i a fo r 5
Type
I
e r ro r .
This
seems
to
ver i fy the appl i ca
b i l i
ty
o f the exponent ia l model .
The
re , la t ion between
sample
means and
th e s tan dard devia t ions
i s shown in
Fig .
3. In
the
e xp on en tia l c as e mean
and
s tandard
devia t ion should be
equa l .
While average spacing var ies with or i en ta t ion o f the sampling
- i ne ,
exponent ia l i ty
does not . This can be seen in Fig .
4,
which
presents examples
o f
mean
spacings and coe f f i c i en t s
o f
va r i a t ion alon non-coplanar sampling
l i n e s .
Trace length d i s t r i bu t ions
do
no t
exh ib i t
the
cons i s t en t
cha r a c t e r i s t i c s
t ha t spacings do; how ever, in 82 of
the samples ,
t r ace
l engths
s a t i s f i ed
5
goodness -of - f i t
t e s t s
fo r
lognormal i ty .
In
th e e xc ep tio na l
cases ,
the d i s t r i bu t iona l forms f a i l ed to
sa t i s fy
X
2
o r K-S t e s t s a t
5
fo r e i t he r lognormal , gamma, normal ,
o r exponent ia l d i s t r i bu t ions ; al though, l ike l ihood r a t i o t e s t s
p lace
these
e xc ep ti on al c as es
c lose r
to lognormal i ty
than to
the
o the r d is tr ib u tio n s te s t ed
Fig.
5 , and
decreas ing
the Type
I
e r ro r to
al lowed
ce r t a in
of these
d is tr ib u tio n s t o
be
accepted
Table 2 .
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For a l imi ted
number
o f
case s , da ta
a re a va ila ble on t r ace
lengths observed on or thogona l
planes
F ig . 6 ) .
For
most o f
t hese ,
length
p df s e xh ib it
littl
di f fe rence
between
s t r i ke
and apparent d ip d i r ec t ions , when i nd iv idua l j o in t s se t s are
considered separa te ly . When data
a re
no t
separa ted
by j o i n t
s e t ,
the or i en t a t i on of the sampled face
inf luences the
r e l a t i ve pro
por t ions
of
jo in ts
from
i f f r ~ n t
se t s be ing sampled,
and
thus the
t r ace length
d i s t r i bu t i on s .
Much l e s s success was enjoyed
in
f i t t i ng ana ly t i ca l forms
to or i en t a t i on d i s t r i bu t i on s .
In
a l l , 22 data se t s were ana lyzed,
and each of th e fo llowin g d is tr ib u ti on s t es te d by X
and l i k e l i -
hood r a t i o
methods: Fishe r , b iv a ri at e F is he r, Bingham, bivar -
i a t e normal , and Uniform. Data were sor ted
in to
c lu s t e r s and
maximum l ike l ihood es t ima tes
made of
d i s t r i bu t i on parameters .
Resul t s are shown in Table
3.
For many data se t s no
ana ly t i ca l
form
provided a sa t i s f ac to ry
it
based on X
Based on log
l i ke l ihood r a t i o s , th e
Bingham and b iva r i a t e
Fishe r
appear
to
pro vid e the
be t t e r f i t s .
IMPLICATIONS EMPIRICAL FINDINGS
The
empi r i ca l
f ind ings fo r j o i n t spac ing
and t r ace length
are
im ila r to
those
repor ted
elsewhere
in
the
l i t e r a t u r e
Table
4 ) ,
al though perhaps more broadly ve r i f i ed . The common observat ion
t h a t
j o i n t spacing a re e x po n en ti al ly d i s t r ibu ted along
any
l i ne t rough
the rock
mass
and
t h a t
a ve ra ge spa ci ng s
a long non-pa ra l l e l l i ne s
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fo llow simple
t r i g o n o m e t r i c r e l a t i o n s
may
be
i n t e r p r e t e d t o
imply
t h a t
j o i n t s
a r e
randomly
and
independen t ly
p o s i t i o n e d in
space
i . e .
t h e i r i n t ersec t ions
w i t h
a rb it r ar y l i n e s a r e oisson
Whether spacing
o r
o t h e r geometr ic
p r o p e r t i e s
a re a ct u a l l y
independent from one j o i n t t o th e next has y e t t o be answered
f o r t h e g e n e r a l c a s e . Var iograrns have been e s t i m a t e d f o r j o i n t
p r o p e r t i e s
[ 1 2 , 1 3 ] , b u t
s p a t i a l
c o r r e l a t i o n s
a r e
d i f f i c u l t t o
v e r i f y . Because
s p a t i a l
c o r r e l a t i o n s would
imply
e i t h e r
c l u s t e r
ing
p o s i t i v e )
o r d i s p e r s i o n n e g a t i v e ) ,
they
would
be
expected
t o modify t h e e x p o n e n t i a l i t y o f
spacing
p d f s . However t h e s e
d i f f e r e n c e s
might
be
masked by
sampling
v a r i a t i o n s .
wo
examples
o f t y p i c a l
a u t o c o r r e l a t i o n
f u n c t i o n s
f o r j o i n t s e t s e X h i b i t i n g
s p a t i a l
s t r u c t u r e a r e
shown in F i g . 7.
The
i m p l i c a t i o n
o f lognormal i ty
o f t r a c e l e n g t h s i n u n c l e a r .
Lognormal
d i s t r i b u t i o n s
o f
geometr ic
p ro p e rt i e s a re
common
o b s e r -
v a t o i n s
in
geology b u t
may
merely be
an a r t i f a c t o f
sampling
b i a s e s a s d is c us s e d below [ 2 ] . I f t h i s i s t r u e , then more r e f i n e d
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s t a t i s t i c a l work o r more c r e a t i v e d a t a c o l l e c t i o n schemes
w i l l
be r e q u i r e d t o
c h a r a c t e r i z e j o i n t s i z e
d i s t r i b u t i o n s .
one i m p l i c a t i o n o f
a n a l y t i c a l
forms seldom
p r o v i d i n g
s a t
i s f a c t o r y
approximat ions t o
o r i e n t a t i o n
d a t a
i s t h a t a t t e n t i o n
p ro ba bly s ho uld
be swi tched t o n o n - p a r a m e t r i c a n a l y s i s .
MO LS OF JOINTING
To d ev el op s am p lin g
p l a n s and
i n t e r p r e t
t h e i r
r e s u l t s , a
concept
o r
model
o f
t h e
geometry
o f
j o i n t i n g
i s
needed.
I d e a l l y ,
such
a model would
be
s p e c i f i e d by a l i m i t e d number o f
p a r a m e t e r s ,
and
be
s imple
enough t o be i d e n t i f i e d
from
normal f i e l d o b s e r
v a t i o n s .
S e v e r a l models
o f
f r a c t u r e geometry
have
been
proposed
i n
v a r i o u s l i t e r a t u r e s , b u t t o
t h e a u t h o r s
knowledge only two
have
found
use
i n
j o i n t
s u r v e y s .
These
a r e
t h e
random disk
model
] and th e Poisson
f l a t
model [19] . The f i r s t has
t h e
b e n e f i t
o f f l e x i b i l i t y
and
perhaps r e a l i s m ; t h e second h a s
t h e
b e n e f i t
o f mathemat ical t r a c t a b i l i t y and power. N e i t h e r , however
i s
p r e d i c a t e d on
a
m e c h a n i s t i c
concept o f
j o i n t
development.
The r a n d o m - d i s k model i d e a l i z e s
j o i n t s
a s bounded p l a n a r
f e a t u r e s
o f
random
s i z e
and
o r i e n t a t i o n ,
randomly
p o s i t i o n e d
t h r e e d i m e n s i o n a l
s p a c e . The shape o f t h e s e f e a t u re s
may
be
f i x e d
e . g . ,
c i r c l e s ) o r
a l l o w e d
t o v a r y
w i t h i n r e s t r i c t e d
f a m i
l i e s . e .g . , e l l i p s e s ) . F o r
c e r t a i n
a p p l i c a t i o n s o n l y t h e assump
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8
t ion
of
convexity i s
requi red
[17]
Fig
8) . The model i s
speci f ied
by
an
in tens i ty
measure
(e .g . , number
o f
j o i n t
centers
per
rock
volume ,
or ien ta t ion
d is t r ibu t ion
parameters , and s iz e d is tr ib u ti o n
parameters.
- _ 0 _
.--
.
_.0
...
_
. __
r _ _ _
; 1
The
Poisson f l a t model idea l i ze s
j o in t s
as convex polygonal
fea tures
formed by
the in te r sec t ions o f a Poi s son- l i ne process
on
pianes which a re themselves the r ea l i z a t i ons o f
a
Poisson
piane
process
in
space
R
3
Fig
9
) .
The
shape
o f
the
fea tures
a re f ixed only to
the
ex t en t
o f be ing polygons and may have an
average ob l iqu i ty
d i f f e r en t
from 1 .0 ,
Within
a
l a rge volume
o f rock
a number o f j o i n t s can be co-p l ana r . The
model i s spec i f i ed by
a dens i ty
o f
planes
in
R
3
a
dens i ty
of
l ine s
in
R
2
o r i en t a t i on
d i s t r i bu t i on parameters fo r the
planes
.
and
l i ne s ,
and
a
co lor ing
ra t io
which randomly
ass igns
po ly -
gons as
j o in t s
o r
as
rock br idges . Within
t h i s
model both spac ing
and t r ace
l ength pd f s a re
necessar i ly
Exponent ia l .
The
importance o f
these
models
to s t a t i s t i c a l sampling
and
in fe rence
t ha t
they
provide
an
o rg an iz in g re fe re nc e w ith in
which
to
i n t e r p r e t da ta , and
a
l imi t ed
number
o f
parameters
with
which
to
summarize
i n fe rences .
.
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S MPLIN PL NS
While
the design
o f
j o in t
surveys
i s
s t ra ight forward
s t a t i s t i c a l problem in prac t i c e
s t rong geomet r i c
b iases m y
be
in t roduced by sampling procedures . I f these b iases a re
no t
accounted for , survey plans m y be s tr o ng ly non -r epr e sen ta ti ve ,
with the r e su l t
t h a t
data a re Gi f f i cu l t to
i n t e r p r e t .
Thus th e
pr inc ipa l
concern
in des igning
j o i n t
survey
s t r a t eg ie s i s
recog-
niz ing sampling b iases and c orr ec tin g f or them.
This sec t ion focuses
on types
o f
sampl ing b ia se s .
Most such
b iase s are
eas i ly cor rec ted ,
if recognized .
Sampling
theory
r e su l t s , inc luding procedures , es t ima to rs , and p re cis io ns a re
summarized
in
Ref. [ ]
Spacing
and Trace Length
M ost work to da te has cons idered sampl ing inferences fo r
spacing t r ace l ength , and or ien ta t ion as mutual ly
independent
fo r an
except ion ,
see
[14] . This i s in
f ac t
no t the case ,
bu t
s impl i -
f i e s
mathemat ics .
S ta t i s t i c a l
aspec t s
o f
in fe rences
o f
j o i n t spac ing a re
f a i r ly rou t ine , and exponent ia l sampling theory i s wel l developed.
Presuming j o in t
l oca t ions
to be
random
and independent average
spacings a lo n g n o n c op la na r sampling
l i nes
a re simply
r e l a t ed ,
and
the
combined
sample can be used
to
in c re as e e stim at e prec i s ions .
Sampling
in fe re nc es fo r j o i n t t r a ce
le ng th a re
much
l ess
simple
s ince they a re
complicated
by geometr ic b iases l ead ing
to
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11
w i t h i n th e
r oc k mass ( e .g . ,
u s in g rh e r an d om - di sk m o d el ), the
sample
i s ,t he re fo re , l ik e ly to b e quadra t ica l ly biased .
Th e
e f f ec t
of a l i nea r
bias
i s shown s c h e m a t i c a l l y in Fig . 1 2 .
, .
The
p robab i l i t y
o f a t r ace l ength 1 appear ing in the
s am p l e
_
i s
the produc t o f th e p ro b ab il i ty o f t appear ing
on th e o utcro p,
f (1 )d , a nd
the cond i t i ona l
p robab i l i t y o f t in te r sec t ing t h e
s a m p l i n g
l i ne t
d o e s
a p p e a r
on
th e ou tc rop ,
k i ,
f ( )d i
= k 1 , f 1 ) d l
s
9 )
in which
k i s a n o r m a l i z i n g constant . Any h igher
order bias in t ro
d u ces
th e cond i t iona l
probabi l i ty
k in , in which k e q u a l s th e
rec ip roca l o f th e n th cen t r a l
moment
of f ( i )
.
in te res t ing pr ope r t y. o f
th e
length
bias
t h a t
t serves a s a f i l t e r
t h a t
t r a n s f o r m s
many common
.
d is t r ibu t ions
f 1 )
i n to
a p p r o x i m a t e l y
l o g n o rm al
forms
[2] .
I n
the
sense o f
cornmon
goodness -o f - f i t t e s t s t hese t r a n s f o r m e d
pd f s
a re in d is tin g u is h ab le
from
l o g n o rm al
pd f s a t r e a l i s ti c
s am p l e
s i ze s .
T h i s i s d e m o n s t r a t e d in Fig . 13 i n w h i c h l i n e a r ly
biased
e x p o n e n t i a l an d
l o g n o rm al
f 1)
I
S a re t e s ted aga ins t
be s t
t
lo gn orm als an d
shown
to s a t i s fy
K-S c r i t e r i a a t
th e 5
l e ve l .
S i n c e s ize
b i ases
a re common i n geo log ica l s a m p l i n g [ I I ] , t i s
in te re s t ing to sp ec ula te t h a t th e common obse rva t ion
o f l o g No rm al
pd f s fo r g e o m e t r i c prope r t i e s
i s
pr imar i l y
a n
a r t i f a c t o f
s a m p l i n g procedures .
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12
The t r a ce l ength
da ta
of Fig . 14
were
col lec ted as
area
samples i . e .
every j o i n t
with in a very
l a rge
sampling f i e ld
was measured)
a t
the ground
sur face
(litop
of
rock ) and on the
f lo o r o f a 2 m
(60 ) deep
e ~ c a v a t i o n ( bot tom
The
j o i n t
popula t ions
a re fo r
presen t
purposes
e s se n ti al ly i d en t ic a l
and y e t the
bottom
o f rock sample has
a
somewhat lower mode and
a
much
t h i nne r
upper
t a i l .
The
reason
i s
t h a t
many
of
th e
t races
observed
in
the
excavat ion
run
o ff
in to the rock wal l s
and can
no t b e o bse rve d in
t h e i r
en t i r e t y . Since
t h i s censor ing
occurs
with propor t iona l ly higher
p ro ba bi l i ty to
longer
t r a c e s
the
sample
i s
biased toward
shor t e r
l engths and the extreme
upper
tail disappea rs
comple te ly .
Censoring
i s a wel l
known
s amp ling p roblem in l i f e t e s t ing
and
other
f i e l d s
of
s t a t i s t i c s .
For
th e t r a d i t i o n ~ l problem in
which th e po in t
of
censor ing i s
cons tan t i . e .
a l l
t r aces
longer
than
1
c
a re censored and a l l shor t e r than
c
a re observed
com-
p l e t ~ l y
a
l a rge l i t e r a t u r e
of
both
f r equen t i s t and
Bayesian
methods has been d eveloped .
Pr imar i ly t h i s
l i t e r a t u r e
deal s
wi th Exponent ial d i s t r ib u t i o n s [ 3,
J b u t r e s u l t s a lso
e x i s t
fo r o th e r forms
[ e . g . 8 9] The q ue st io n o f f ix ed p oin t
censor ing fo r
j o i n t surveys has been cons ide red
by
Cruden [ ~ J
Baecher and Lanney
[2
J and
P r i e s t
and Hudson [15J.
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13
Unless
th e
sampl ing program
f o r j o i n t surveys s
c o n s t r a i n e d
such
t h a t
j o i n t s
l o n g e r
than
a
f i x ed l e ng th
1
c
a r e
n o t
measured
even they i n
f a c t
could
b e , t h e
problem o f censor ing becomes
more
d i f f i c u l t .
I n
p a r t i c u l a r , t h e p o i n t
o f
censor ing s i t s e l f
a
random v a r i a b l e . The o b s e r v a t i o n s recorded
a r e
1
a
s e t
o f
complete ly
observable t r a c e s ,
=
{1
x
, 1 , . , 1
x
, r } : and 2
a s e t
o
t r a c e s f o r
which
only one o r
n e i t h e r end
i s
o b s e r v a b l e , t
=
- z
{ l
Z
, 1 , , 1
z
,t}
The
l i k el i h oo d o f
( - ,I t s
CD
t
n f 1 . I e )
n
f 1 Jle d1
1 x ~
1
z,
~
Rz,j
10
i n
which
= th e parameters o f t h e
t r a c e l e n g t h
pdf c o r r e c t e d
f o r
o t h e r b i a s e s ) . The
secon d term i n t h e r i g h t hand s i d e i s
t h e
p r o b a b i l i t y t h a t a
censored t r a c e
would be lo ng er than t h a t
observed. C l e a r l y ,
c l o s e d form maximizat ion o f Eq. 10 with r e s -
p e a t t o
e s only p o s s i b l e f o r p d f s having a n a l y t i c a l cumulat ive
p r q b a b i l i t y d i s t r i b u t i o n s .
Truncat ion
b i a s
I n c o l l e c t i n g
j o i n t
d a t a
a d e c i s i o n s
u s u a l l y made n o t t o
record
t r a c e s
s h o r t e r
than
some
c u t - o f f
l e n g t h .
This
d e c i s i o n
i s
made
e i t h e r out
o f expediency
o r
because s h o r t
t r a c e s a r e
d i f f i
c u l t t o d i s t i n g u i s h , a s f o r example i n photographs.
S e v e r a l
w orkers ha ve
noted
t h a t
t h i s form o f t r u n c at i o n in tr o du ce s b ia s
i n t o the sampling p l a n , i n c r e a s i n g the
sample.mean [ 2 , 5 , 1 5 ] .
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14
F i g . 15
shows
th e b i a s i n th e sample mean r e s u l t i n g
from
t r u n c a t i n g
a t
a
given
f r a c t i o n o f
th e
mean t r a c e
le n g t h , f o r
an
e x p o n e n t i a l
pdf o f
t r a c e
l e n g t h . This
b i a s
s m a l l e r f o r
d i s t r i b u t i o n s , l i k e th e lognormal with zero d e n s i t y a t the
o r i g i n . The f i g u r e shows t h a t th e e f f e c t o f
t r u n c a t i o n ~ s
on
e sti m a te s o f c e n t r a l tendency o f the t r a c e
l e n g t h
pdf
i s
s m a l l ,
u n l e s s t h e chosen
t r u n c a t i o n
l e v e l
i s
l a r g e e . g . ,
>1 o f t h e
mean . In
most
c a s e s
t h i s b i a s may be s a f e l y i g n o r e d .
O r i e n t a t i o n
Geometric
b i a s e s
i n j o i n t s u r v e y s , s p e c i f i c a l l y f o r
j o i n t
o r i e n t a t i o n , were brought
t o t h e a t t e n t i o n o f the e n g i n e e r i n g
l i t e r a t u r e by R. Ter zagh i [ 18 ] a lt ho ugh Sander e t a l . [16] and
o t h e r s had e a r l i e r c o n s i d e r e d
r e l a t e d
problems
w i t h t h in s ec ti o n s.
The
problem Terzaghi p oin te d o ut i s t h a t j o i n t s more o r l e s s p a r -
a l l e l t o an outcrop
a r e
sampled with p r o b a b i l i t y approaching
zero
i . e . , t h e r e i s
a
b l i n d zone
f o r any
p a r t i c u l a r
outcrop
o r
b o r i n g ) .
Since
o u t c r o p s
may
form
along j o i n t
s u r f a c e s , t h e r e i s
o f t e n a
s t r o n g p o s s i b i l i t y
t h a t an
e n t i r e s e t
o f j o i n t s i s being
s y s t e m a t i c a l l y
under r e p r e s e n t e d
i n the survey r e s u l t s .
The
c or r e ct i o n f o r
t h i s
b i a s
i s
s imple .
O r ie n ta tio n d ata
can be weighted
i n
i n v e r s e p r o p o r t i o n
t o t h e i r
p r o b a b i l i t y o f
appearing
i n
t h e
sampled
p o p u l a t i o n . Con sid er in g only
outcrop
sampling
th e p r o b a b i l i t y
o f a j o i n t
o f
s p e c if ic o r ie n ta ti o n
i n t e r s e c t i n g an
outcrop
can be seen from F i g . 1 6 t o be p r o p o r t i o n a l
~ . .
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5
to
L s in a
d
Thus observed da ta should be ad jus ted by th e weight ing f a c t o r
W
a
s in
a
Terzahgi
gives
t h i s
weight ing func t ion
on equa l areas
ne t s .
For
sampling from
mul t ip le
o utcro ps th e weigh t ing func t ion
fo r some
or i en t a t i on J { A ~ V }
becomes p ropo r t i ona l to
in which B
i
i s the dimension o f outcrop Qi={ m n} i s the po le to
the be s t f i t t i n g plane to
th e o utc ro p
and Jn
= the
do t p ro
duc t .
CONCLUSIONS
The r e su l t s
of th e p re s en t ana ly s i s o f
j o i n t
survey
da ta
appear
to
be c on sis te nt in
impor tan t
ways
wi th r e su l t s pre sen ted
e l sewhere
in th e l i t e r a t u r e and in
c on ju n ct io n w i th
those o the r
r e su l
t s
appear to
j u s t i f y th ree
emp ir ic a l c oncl us ion s
on
the
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16
d i s t r i bu t i ona l
proper t i e s
o f
o on j o i n t
survey measurements
Spec i f ica l ly
j o i n t sp ac in gs as measured a long sampling l i ne s
o r in bor ings
are
very
f requent ly
d i s t r i bu t ed exponent ia l lYi
t r ace len gth s as
measured in outcorps a re
of t en d i s t r i bu t e d
lognorrnallYi
and
p lan a r
or i e n t a t i ons o f j o i n t s
do
no t
appear
wel l modelled
by
common a na ly t i c a l forms
S t a t i s t i c a l
analys is o f j o i n t survey
procedures both
in
th e presen t s tudy and
as
repor ted e lsewhere in th e l i t e r a t u r e
lead
to
th e
conclusion
t h a t
g eometr ic s amplin g
b iases
a re
o on
in t r ad it io n al sampling plans and
must
be guarded
aga ins t . The
most f requen t of these a re th e w ell known
or i en t a t i on
b ia s i n
which j o i n t s s ub pa ra lle l to an
outcrop
a re unde r rep re sen ted in
samples co l l ec ted
on
the outc roPi s ize b i as i n which l a rge r
j o i n t s
a re
sampled with g r ea te r p r ob a b il it y than sma l l e r j o i n t s
are i
and
censoring
b i a s
in
which
l a rge r
j o in t s
a re
more
f requent ly
masked by overburden o r
excavat ion l imi t s than
smaller j o in t s
a re .
The impl ica t ions o f these conclus ions fo r the
des ign
o f
j o i n t surveys and fo r th e use o f survey data in engineer ing models
o f rock
masses
a re aga in o f
two types .
F i r s t c og en t r ea so n
ex i s t s
fo r adopt ing c e r t a i n
d i s t r i b u t i o n a l forms when
deve loping
s tochas t i c
models
o f f rac tu red media fo r
s t r e ng th
deformat ion
and flow ana lys i s and when designing survey procedures . Second
th e
sampling
theory
o f
j o i n t
surveys i s not
s t r a i g h t
forward and
na ive s t a t i s t i c a l es t imato r s
may
be s t rong ly
b iased .
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17
KNOWLEDGMENTS
This
work was sponsored by
the
U S Bureau o f Mines
under
Cont rac t
J 27S lS
The
au thor wishes the th an k N ic ho la s A
Lanney
and W ill iam D ers ho w it z who con t r ibu ted
s ub sta n tia lly to
the
ana lys i s
o f l ength spac ing and or i e n t a t i on respec t ive ly . The
work was
j o in t l y
d i rec ted
by Herber t
Ein s t e in and h is
comments and suggest ions are gra te fu l ly
acknowledged
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REFERENCES
Baecher,
G.B., N.A. Lanney, and H.R. Einstein, 1977, Sta t is t ica l
description of rock f racturs and
sampling,
18th u.S. Symposium on
Rock
Mechanics.
2.
Baecher,
G.B. and N.A. Lanney, 1978, Trace length biases
in
jo int
sur
veys,
19th
u.S. Symposium on Rock Mechanics.
3. Bartholomew D.J. , 1957,
A
problem
in
l i fe testing, Journal American
Stat is t ical
Association, 52:350-55.
4. Barton,
C.M. 1978, Analysis of jo in t
t races,
19th u.S. Symposium on
Rock Mechanics.
5.
Cruden, D.M.
1977, Describing the size of discontinui t ies, International
Journal
of Rock Mechanics and Mining Science,
14 :
133-37.
6. Einstein, H.H., G.B. Baecher and D
Veneziano, 1978,
Risk
analysis
of
rock
slopes in
open p it mines, Final
report
to the U.S. Bureau of Mines,
Contract J02750l5.
7. Epstein, B., 1954,
Truncated l i fe
tests
in
the exponential
case,
Annals
of Mathematical Stat is t ics 25: 555 f f .
8. Fisher, R.A.,
1931, The truncated Normal
distribution, British Associa
tion for
the
Advancement Science, Mathematical
Tables,
I: XXXIII-XXXIV.
9.
Hald,
A., 1949,
Maximum
likelihood
estimators
of
Normal distr ibut ion truncated a t a known point ,
32: 119.
the parameters of an
Skand. Aktuar Tidsk,
10. Jewell,
W.S., 1977 , Bayesian
l i fe test ing using th e to ta l
Q on
tes t in
Tsokos, C.P. and I.N.
Shimi
Eds.) , Theory
and
Application
Reliabi l i ty,
v. 1.
11.
Kaufman G.M. 1963,
Stat is t ical
Decision and
Delated
Techniques
in
Oil
and Gas
Exploration,
Prentice-Hall.
12. LaPointe,
P.R., 1980, Analysis of
th e
spatial
variation
in
rock
mass
properties through geostat is t ics , 21st u.s . Symposium on Rock Mechanics.
13
.Miller,
S.M., 1979 ,
Geostatis t ical
analysis
for eva luat ing
spatial
dependence in f ra ctu re se t characteris t ics , thesis submitted in parial
fulfillment
of the
requirements
for
the degree Master
of
Science,
Univer
sity of Arizona.
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14.
Pahl P J
1981
Estimating th e mean
len gth o f discontinuity
t races
International Journal o f Rock Mechanics and Mining
Sciences
18: 221-228.
15.
Priest
S D
and
J .
Hudson
1981
Estimation
of d iscont inui ty
spacing
and trace
length
using scan1ine surveys International
Journa1.2i
Rock
Mechanics and
Mining Science 18:183-199.
16. Sander B. 1926 Zur petrographisch-tektonisschen Analyse I I I Jahrb.
d.
geol.
Bundesanst. Wien
17 . Santalo L. 1976
Stochastic
Geometry and Integral Calculus Addison-Wesley.
18. Terzaghi R. 1964 Sources
of
errors in
jo in t surveys
eo technique
15 : 287-304/
19.
Veneziano
D.
1978
probabilistic
model of joints
in
rock M T working
paper.
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F i g u r e Number
1
2 .
3 .
4 .
5.
6 .
7
8 .
9 .
1 0 .
FIGURE TITLES
Ti t le
T yp ic al c ha lk
sampling
l ine
on
an
o u t c r o p ,
showing def in i t ions
o f jo in t
sp acin g
A
and t race
le n g th 1 ).
Jo in t
sp acin g frequency dis t r ibu t ions f o r
s ix 6)
s i t e s Sample s izes between and
2000; plot ted
d a t a fo r f i f teen in tervals o f
each
d a t a
se t
Mean sp acin g s and s t a n d a r d
d e v i a t i o n s
o f
sp acin g f o r in te rsec t ions
o f
jo in ts
w i t h
sampling
l i nes D i f f e r e n t symbols refer to di f ferent s i t e s
F or
e x p o n e n t i a l
dis t r ibu t ions
th e mean
and
s t a n d a r d
d e v i a t i o n
s hould be th e same, as
shown
by
45
l i ne
Mean sp acin g s among jo ints and coeff ic ients
of
varia t ions
stan d ard deviation/mean) f o r
in te rsec t ions sampled
a lo ng n on -c op la na r
sampling l ines w ith in th e sarne rock mass. F or
e x p o n e n t i a l
dis t r ibu t ions
th e
Cov s hould eq u al
1 0
Cumulative
dis t r ibu t ions
o f j o in t
t race l e n g t h
fo r one s i t e having tw o d i s t i nc t s u b p a r a l l e l
se ts
Trace
l e n g t h s
measured
in
h o r i z o n t a l
o u tcro p s
i . e .
s
t r ike
and
in
ver t i ca l o u tcro p s
approximately
para l l e l
to d i p s i . e . IIdi
ps
l l
B e s t f i t t ing cumulative dis t r ibu t ion f u n c t i o n s
fo r
s t r ike
and
d ip
l e n g t h s
o f jo in t
t races
a t
same
s i t e
A utocovariance f u n c t i o n s f o r
jo in t
d i p , showing
l i m i t e d alth o u g h def in i te
spat ia l
corre la t ions
Random
d i s k
model fo r
j o in t s
P ois s on
f l a t model
f o r
jo in t s
Prof i le view o f j oi nt s i nt er se ct in g
an
o u t c r o p .
Note
tha t
larger jo in ts have a p r o p o r t i o n a t e l y
g r ea te r p robab il it y
o f s t r ik ing
o u tcro p than do
s m a l l
j o in t s
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Table
ata ase fo r Empir ical
o i n t
S t a t i s t i c s
SITE
PURPOSE
GEOLOGY
GREEN COUNTY
NUCLEAR
POWER FOLDED SEDIMENTA
SITE
A
NUCLEAR
POWER
HIGH
GRADE
t ETAMORPH
ICS
SITE B
NUCLEAR
POWER
SHALLOW
WATER
SEDIMENTS
BLUE HILLS
STUDY
AREA GRANITE PORPHYRY
AND
VOLCANICS
PINE
HILL
STUDY
AREA GRANITE
AND
VOLCANICS
DUVAL
MINE
COPPER PORPHYRY
PINCOCK ALLEN
MINES VARIOUS
MAINLY
AND HOLT--VARIOUS
COPPER
PORPHYRI
SITES
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Table
2
Resul t s of Chi Square Goodness o f f i t
Tests fo r
Trace
Length
is t r ibu t ions
SITE
EXPONENTIAL
MM LO NORM L
Site A
top fail
fail
fail
Site
A
bottom
fail
fail
fail
Site A
sides
fail
fai l
pass
Site
A
sides
fail
fail
pass
Greene
Co.
fail
fai l .
pass
trench A
Greene Co.
fail
fai
1 pass
trench B
Greene
Co.
fail
fail
pass
trench
C
Greene
Co.
fail pass
pass
trench T
Site
fa il fa il pass
Blue
Hills fail
fai l pass
Pine
Hills
fail fail pass
all
significance
levels
set
at 5
except
where
in-
dicated
by
*) which were
set at
1 .
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Table 3
Goodness o f t r e su l t s
fo r
or i en t a t i on da ta
JOINT
SET
Fisher
ivar ia te
Fisher
Bingham
x
2
t es t
A
558
420
480 none
B 80
8
none
lC
294
144
107 none
A
127
32 47 none
B
72
60
282 none
C 442
323 283
none
D
131
91
326
none
2E 29
none
2F
89
69
51 none
3A
125
122
114
a l l
3B
20 none
3C
20 20 91
none
4 568
555 517 none
SA
445 288
272
none
5B
236
142
149 Bingham
6A
79
57
37
none
6B
62
27 27
a l l
7A 383
298 280
none
7B
251
91 140 none
7C
574 575 555
none
7D
120 45
41
Bingham
ivar ia te
Fisher
8 295
144
107
none
Tota l
Bes t Fi t s
13
d i s t r i bu t i ona l
forms s a t i s fy ing
goodness
o f t
c r i t e r i on
at
5
confidence.
~
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Table 4 Best
f i t t ing
dis t r ibu t ions fo r
jo in t
spacing
and t race length, as
reported
in the
l i t e ra tu re .
Fi t t ing procedure and goodness-of-f i t t es t ing vary
from
one
source to
another.
Predominant rock
mass
geologies also vary.
SOUR E
SP ING TR E
LENGTH
OLO Y
Barton 1977)
Bridges 1976)
Call, et a1. 1976)
Cruden 1977)
McMahon 1974)
Priest Hudson 1977)
Robertson 1970)
Snow l970)
Steff-an, e t a1. 1975)-
Exp
Exp
Exp
logN
logN
Exp
Exp
logN
Exp
Exp
-Metamorphic
Metamorphic
Copper Porphyry
Various
Various
Chalk and
various
DeBeers
Mine
Crysta11ines
~ t m o r p h i c ~
and
various
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t
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en
o
X
o
g
ro J
Z J
we:
en
Q
Z
o
:
_
cn
ro
roO:
zo
Q.
o
Q J
~ e :
ic E
n
o
t
o
v
o
N
o
o
A J . 1 1 1 8 1 8 0 ~ d 3 A I J . 1 1 n W n ~
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1
8
4
6
MEAN SPACING
2
1 .. .
2
8
Z
>
6
W
Z
4
CJ
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.
.
.
5.0475
1
.5
8
9
8
7
>
60
STRIKE SET 2
.J
50
m
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-
t::
o
n
m
W
o
IC >
:E
,
Q
o
Q
.L:I H LE>N3 : : > V ~ l
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5
4
3
laJ
2
Z
1
2
5 10
SEPARATION M
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O NT
~ ~ ~ S U R F C E
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r
P F
SAMP L E
TR E LENGTH
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99
BIASED
logNormal
KOLMOGOROV SMIRNOV
BOUNDS. 5 n=100
\
BEST
FIT
logNormal
95
1
9
8
t
d
7
m
ti
30
20
o
4 5 6 7 8 9 1 20
TRACE LENGTH FT)
30
40 50 60
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....
100
_
:
:
C)
1
J
Z
o
SITE A
TOP
SITE A:
BOTTOM
1 0 L _ L L
JL
...L.
..L... _
1
1
50 90 99
100
CUMULATIVE FREQUENCY )
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LaJ
E
LaJ
:
LaJ
25 5 75 1
TRUNC TION LENGTH ME N
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~ ~