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A prediction model of landslips
CATENA
VOL. 3, 215 - 230
I
GIESSEN 1976
[
A PREDICTIONMODELOF LANDSLIPS
Herbert Neuland c/o Lehrstuhl fur Hydrologie im FB Geowissenschaften Universit~t Bayreuth, Postfach 3008, D - 8580 Bayreuth SUMMARY
I t is the intention of this investigation to develop a prime relation that makes a prediction of landslips possible. Therefore 250 stable and unstable objects were obtained all over the BRD and 31 variables were determined. They consist of morphometric, soil-mechanic, material and stratification characteristics. The principle component-analysis explains the relations between the variables. An F-test shows that nine independent variables exist. A bivariate discriminant analysis chosen with 150 at random objects yields a prediction function which efficiency later is tested by a procedure of the Euklid distance. From seven tested variable combinations that one was selected which sho~ed that the slope gradient HN (o), the watershed to the considered object FL (km~) and a parameter measuring the density of the soil EIND are sufficient to predict a landslip. The significant discriminant function is T = 0,9222 • IO-5HN2 + 0,7926 Ig(FL + 10) - 0,6098 Ig(EIND + 10) with the centroids 0,1934 for slips 0,1781 for stable objects. The quality of the model was demonstrated by 100 objects from which 94 were predicted in the right group. ZUSAMMENFASSUNG Mit der Arbeit wurde das Ziel verfolgt, eine m~glichst einfache Beziehung zu finden, die es erlaubt, Rutschungen an B~schungen vorherzusagen. Dazu werden an 250 Rutschereignissen und stabilen Stellen 31 ausgesuchte Variablen ermittelt, die einfach zu bestimmen und aussagekr~ftig sind. Sie setzen sich zusammen aus morphometrischen, bodenmechanischen und das Material und die Lagerung kennzeichnenden Gr~Ben. Mittels der Hauptkomponentenanalyse wird das Beziehungsgeflecht zwischen den Variablen gekl~rt. Ein F-Test im Rahmen der Diskriminanzanalyse legt neun unabhNngige Variablen zur Modelleichung fest. Die nach einer Zufallstafel ausgesuchten Ereignisse ergeben mit einer eindimensionalen Diskriminanzanalyse eine Vorhersagegleichung, deren GUte anschlieBend mit zurUckbehaltenen Ereignissen durch ein Zuordnungsverfahren nach dem Euklidischen Abstand getestet wird. Aus den verschiedenen m~glichen Variablenkombinationen ergibt sich, dab Hang215
neigung (HN), Einzugsgebiet oberhalb einer Untersuchungsstelle (FN) und ein Dichteparameter (EIND) ausreichen, an einer B~schung bzw. an einem Hang Rutschungen vorherzusagen. Die signifikante Diskriminanzfunktion dazu, die ein Ereignis den beiden vorgegebenen Gruppen Rutschung bzw. Nicht-Rutschung zuordnet, lautet: T = 0,9222 • IO-5HN2 + 0,7926 Ig(Fl + 10) - 0,6098 Ig(EIND + 10) mit den Centroiden fur Rutschung (CR = 0,1934) und Nicht-Rutschung (CNR = 0,1781). Von 100 getesteten Ereignissen werden 94 der richtigen Gruppe zugewiesen. 1. 1.1.
INTENTIONOF THE INVESTIGATION AND PROBLEMS Intention
Thirty years ago N. SBRESNY(1941, 25 - 26) described the situation of the research about landslips. One group of scientists tries to solve the slip process by using the morphometric and soil-mechanical characteristics, while the other group prefers the theoretic physical consideration. For both methods you need big laboratories and much time, and there is not any objective and proved procedure putting out those variables influencing the slip process. For finding out the important variables K. TERZAGHI and R.B. PECK (1961, 391) even give the advise to ask experienced engineers. Therefore G. BLESSING (1966, 218 - 220) proposes to use those soil mechanical characteristics which are predicatory and very simple to determine. The opinions of K. BACKOFEN(1957, 125 - 130) and G. TRAUZETTEL (1962, 40) are important, too. They content that there cannot be only one factor responsible for a landslip. Always a combination of a group of appointed characteristics cause a landslip. The mechanical principles of slips are well known. You have to discern between the forces setting the process in motion and stopping i t . The determination of the forces is complicated by the natural slip-surfaces which are very d i f f i c u l t to accomodate to theoretic ones. The mechanical principles you will find in K. KNOBLICH (1967, 286 - 299), W. FELLENIUS (1947, 1 - 48) and H. NEULAND(1975, 9 - 13). F. WEIDENBACHremarks that i t is not possible t i l l today to state exactly the causing variables or to predict a landslip. My investigation reflects just to this opinion and tries to develop a method, which on one side discovers those variables that influence a landslip on and on. On the other hand I want to try to raise a model which is able to predict a landslip. The model is based on variables that had been determined on unstable and stable objects. The necessary foundations are taken from the mathematical multivariate s t a t i s t i c . There are some methods for predicting a landslide published by A. LANGEJAN (1965, 500 - 502), N. SAITO (1965, 537 - 541) and R. NONVEILLER (1965, 522 - 525). The disadvantage of the methods is that only one variable is considered as the causing one. Not in all cases the selected variable is the reason for the slip, hence i t follows that the probability of a computed stability-coefficient which marks stable and unstable equilibrium is only small. 1.2.
Task
The term landslide in the following text follows K. TERZAGHI and K.B. PECK (1961, 109) who consider the break of earth-masses in a slope as a landslide. The next thing to do is to select those variables out of the publications that characterize the slip-process and can be a possible reason for i t . The measurement of the variables in the f i e l d and in the laboratory should be very simple. H. KARRENBERG(1963, 9 - 10) and D.G. DECOURSEY(1973) have refered to this fact° For building the model, 250 stable and unstable slopes all over the BRD 216
Ubersichtsskizze der a u f Rutschungen u n t e r s u c h t e n
Raume
(see figure I) were selected. On each slope the s t r a t i f i c a tion was ascertained with a boring instrument and a soil sample was taken out of each layer. In the f i e l d and in the laboratory 31 variables explained in chapter 2. were determined. For f i l t e r i n g and finding out those variables causing a landslip and which are later necessary for raising the model, you have to use the principle axe analysis. I t also shows the correlation between the predictorvariables and makes i t possible to find independent predictors.
--
Autobahn
~
BundesstroFle
~
Untersuchungsraum Einzelrutschung
Mansteb I 4MiLl Quelle Shell Au~o(:ltlas72
The discriminant analysis is able to attach an object described by variables to a characteristic group. R. HERRMANN (1974), E. SCHRIMPFF (1975) used i t for predicting the runoff of rivers and D.G. DECOURSEY (1973) had success in predicting the water damage on dams.
In my special case you have to use a bivariate discriminant analysis with both the groups Figure 1: General sketch of landslips and stable slopes. By this way i t is render possible to attach n objects described by p predictorvariables to both groups mentioned above. The mathematical procedure is to find a linear combination of the predictors which separates the two groups best. A detailed description of the discriminant analysis you can find in R. HERRMANN(1974, 381 - 385). 2.
THE PREDICTORVARIABLESAND THEIR MEASUREMENT
2.1o Morphometric variables Slope gradient (HN) - Exposition (EXPO) - Length of a slope (HANG) A lot of publications cited the slope gradient as a causing variable. K. KNOBLICH (1967) and M.J. CROZIER (1973) add to the slope gradient the variable exposition of a slope and its length. The moisture of a slope that indicates the reduction of the frictional index of the soil material and the drying up of a slope are influenced by the exposition. The slope length is used as a variable to get some information about the pressure of the soil material. 2.2.
Foundation of the selected points
Depth of roots (WT) - Depth of weathering (VT) Except for certain circumstances you do not expect a slip in rocks. But when the weathering has built a layer of weather-worn material, i t tends to instability. The depth of weathering is ascertained with a boring instrument and i t also can be ascertained with geological, hydrogeological and geomorphological maps. 217
Strengthening the soil material, the roots stabilize a slope, further more some appointed sorts reduce the soil moisture. Therefore the relation between vegetation and soil material is described by the depth of roots which is tested by boring or digging. 2.3.
Watershed to a control point (FL)
G. TRAUZETTEL (1962), A. WATZNAUER(1965) and Q. ZARUBAand V. MENCL(1969) again and again found out that there is a relation between the landslip and the area above i t . This opinion could be confirmed during the work in the fields. The areas with compact ways and furrows are inclined in direction of the slip, so that the water drains to this area. First the watershed had been determined in the f i e l d by charting in following contour line and later by planimetry. I t should be a test value for the water which takes an active hand in landslips. For points where there had not been any possibility to determine a watershed, its natural size had been estimated. Also the vegetation in the areas influence the s t a b i l i t y of a slope (see figure 2).
road
slip
watershed(FL)
l { I
I I l
road slip i i
watershed(FL) I i
furrow
i
__
'.
0
h '
^cA
*
>,.
Figure 2: Watershed to a landslip 2.4.
Soil mechanical characteristics
H. KOHN-VELTEN (1963), G. TRAUZETTEL (1962) and W. LIPPMANN (1960) propose for a detailed investigation of landslips the determination of the soil mechanical parameters. Because of simplification and rationalization the following parameters had been selected, reflecting that the humid conditions of the soil accelerate the instability of a slope. 2.4.1.
Liquid limit (FLG) - saturation de~ree (SATT)- water-plasticity (H~PL) ~ ~t~ ~uTi~g.-s~r~tTo~ Tn t~e_-u~p~r--l~r~ ~_)-a~d-i~f~rlor - Ta~s~(~G~)~
The variables cannot be determined by physical equations. Each one has to be determined by a laboratory procedure which is published in W.H. B~LLING (1971, 44 47) and E. SCHULTZEand H. MUHS(1967, 384 - 388). Because of rationalization a procedure developed by H. MATSCHAKand A. RIETSCHEL (1965, 135 - 139) was used here. The information of the soil mechanical parameters is limited, because on one side you have to work with disturbed samples, on the other side you only can
218
use the part of grain size ~ 0,4 mm from the sample. This portion does not seem to be enough to give back the natural properties of the soil material. 2.4.2.
Rollin~-out limit LAUS~ - Plasti£ityLindex_(PL~LShrinkage-index LSCHI
The dry soil, too, furthers a slope. The dry soll conditions are described by the indexes above. Drying soil yields crevasses which disaggregate the soil and makes i t possible for the water to i n f i l t r a t e very easily. The conclusion is the soaking and saturation of the soil material which together with other components is responsible for i n s t a b i l i t i e s . Laboratory procedures are published in W.H. BULLING (1971, 49 - 53), G. SCHULTZEand H. MUHS(1967, 392 - 395). 2.4.3.
pF-soil characteristic for describing soil physic conditions TO~I~ ~G~,-O~3-~ ~GT,-O~5-, UGT,--U~2~~G~,-U~4-, ~G~)
All scientists have regarded the water as an important variable but nowhere you find a notice for an exact measuring, except for the extraordinary measurement of the pore water-pressure. The soil moisture and water balance can be described by the pF-characteristic, the theoretical foundations and measurements of which are published in A. RODE (1959, 335 - 349), DON KIRKHAM (1964, 1 - 22), F. SCHEFFERand P. SCHACHTSCHABEL (1966, 226 - 254), R. HERRMANN(1971, 27 - 30), K.H. HARTGE (1971, 71 - 84), E. SCHLICHTING and H.P. BLUME(1966, 65 - 70), J.S. McQUEENand R.F. MILLER (1971, 521 - 527). The soil-suction (pF) render i t possible to compare the behaviour of the water in different soils with the help of equivalent soil moisture contents. A diagram of the pF drawn against the soil moisture content gives each soil a proper characteri s t i c (see figure 3). The pF-characteristic is proved suitable to describe soil profiles from which you can deduct soil physic conditions.
II"c,ay pore,
PF
"
(u) 0,~-
102 600
0 I0 20
, VoI%
Figure 3: pF - characteristic of different soil
7
L
6 4 2
2
FOGS
:~--OG20G1
4' - ~ UG3 2- ~ U G 2 O , 0 I0 20
,o
UGI
£ Vol%
Figure 4 : Gradients for describing the soil physics conditions demonstrated by a pF - characteristic of a soil from a landslip
Following F. SCHEFFERand P. SCHACHTSCHABEL(1966, 95) fixed pF-domains correspond with one pore domain that fixes the soil physic conditions. For pointing them out, to every pore or pF-domain a differential gradient (see table 1 and figure 4, OGi for the upper; UG~ for the inferior layer) which stands for the soil physics information for fixed soil moisture contents, had been appropriated.
219
Table 1: PORE-SCOPEOF DIFFERENT pF BELONGINGTO THE GRADIENTSWHICH DESCRIBE THE SOIL PHYSIC CONDITIONS gradient of the pore size (0) pF-domain upper layer inferior layer 50 0 - 0,8 OG1 UG1 50 0,8 1,77 OG2 UG2 50 - 10 1,77 - 2,54 OG3 UG3 10 - 0,2 2,54 - 4,20 OG4 UG4 0,2 4,20 - 7 OG5 UG5 2.5.
Sizing the structure of the material
2.5.1.
Stratified structure of the up~e~_~SA_O~and inferior__(S_AU)__layer
F. WEIDENBACH (1965, 3) drew attention to the fact that in a slip process always two layers become effective. This makes i t necessary to get some information about the s t r a t i f i e d structure. The quotient of all gradients of a pF-characteri s t i c shown in 2.4.3. represents a typical number for each layer: SAO OGI/OG2/OG3/OG4/OG5 (1) SAU = UGI/UG2/UG3/UG4/UG5 (2) There is a big number for clays, a small number for sand with all possibilities in between. :
2.5.2.
Upper boundary_of the bed ~OG)_-_inferior boundary of the bed ~UG)
For defining the boundary of the bed i t is assumed that there is low pressure in the unstable case. Therefore a differential quotient on the pF-characteristic between the fluidy index (Vol % water content) and the saturation marks the boundary of the bed. 2.5.3.
Clay content of the uppe~TO__N] and inferior layer_(TON_U~
Clay accelerates a slip and its influence indirectly is included in the pF-charact e r i s t i c . Measuring its influence the clay content of all layers had been estimated by a rule of thumb following a table from E. SCHLICHTINGand H.P. BLUME (1966, 21). 2.5.4.
De~ree of firmness and densi~y_(EIND~
There is used a probe for finding out the compressed layers which stop the draining water. As a simple probe you can use a Purkhauer boring-instrument with a height of f a l l of 1 m for the 5-kg-hammer. Probes developed by Kaschins~ show similiar results with ve~ small deviations. You have to measure the number of beats on the probe and the depth of penetration belonging to. Both test results are drawn in a co-ordinate system and an equilized curve is developed (see figure 5). A differential quotient on the natural bounda~ of the bed represents firmness and density of the stratified structure. Following K.H. HARTGE (1971, 153) who also describes probes besides P. SIEDECKand R. VOSS (1960, 86 - 88) the measurement result can also be considered as shear-strength.
0
'
. . . .
~ ~ ( a )
0
~
l
l
.5 I0
l
l
-
l
l
)beats
-
-
~
I((b) b
/ EIND
=
I dS
Z,
Figure 5: Determining the density parameter EIND in the unstable (a) and the stable case (b)
3.
ROUGHCOPYOF THE MODELAND STATISTICAL METHODS
3.1.
Conception of the model
All over the BRD 250 points were controlled for slips, on each point soil samples had been taken and 31 variables mentioned in chapter 2. had been estimated. From the 250 objects 150 had been selected by a random table. The remaining 100 objects are used to reexamine the model. The variables are tested for normal distribution by Kolmogorow Smirnow-Test on the 5 % level. Not normal distributed variables are transformed appropriately. The next step is to find out the correlation between the 31 variables. Making use of the principle axe analysis, i t is advantageous to find on one side the correlation and on the other side you always get one group of similiar structured variables on a component. The predictor-variables that had to be independent and had to show a good separation are selected by a special procedure with the discriminant analysis. All possible variable combinations you can form as a result of the principle axe analysis are carried out to a discriminant analysis. A computed F-value of each variable distinguishes its discriminant power. High F-values means good, low F-values bad discriminant power. When all F-values are known, those with the best F-value are considered as predictor variables. A combination out of this group yields the discriminant function. The best function shows the Wilks-Lambda which tests the separation of the used predictors. The significance of the discriminant function is checked by a chi-square-test. The model is b u i l t when the best predictor combination is found out. Next the efficiency of the model has to be tested with the remaining 100 objects. The equation of the discriminant function computes a discriminant score for each object. While building the model, for each group (landslides and stable slopes) a characteristic value was set up, called a centroid. The procedure of the Euklid distance compares the result of the discriminant function with both centroids. A tested object belongs to the group where the Euklid distance shows the smallest value. I f the features of the object agree with those of the group, the predicting succeeded. Figure 6 shows the position of different objects in the discriminant space. l
CENTROID 2 ( s t a b l e ) 22
2
2
O. 160
2
O. 170 i •
o r i g i n group
2 A
2
~
2
O. 180 -2
J
2222
CENTROID1 ( u n s t a b l e ) 2
~ m - - I
i
I
1
I
O. 190
I Iii
i i
I
O. 200
group a f t e r p r e d i c t i o n
Figure 6: Some hand-picked objects in the discriminant space 3.2.
Statistical methods
3.2.1. Principle axe anal~sis A comprehensive book about principle axe analysis was written by K. OBERLA (1968). Reduced statements are published in R. HERRMANNand E. SCHRIMPFF (1975) and H. NEULAND(1975).
221
3.2.2.
Discriminant analysis and Euklid distance ~rocedure
The mathematical foundation you f i n d in T.W. ANDERSON (1958) and J. BRYAN (1951). R. HERRMANN(1974) published an example where you f i n d the procedure of the Euklid distance, too. 4.
RESULTS
4.1.
p r i n c i p l e axe analysis
4.1.1.
Component I
Table 2 shows the rotated component matrice with loadings • 0,5. Component I represents soil mechanical parameters which describes the soil material with high soil moisture and saturation. These are the l i q u i d l i m i t , saturation degree, moisture content at saturation and p l a s t i c i t y index. All variables provide nearly the same information. Each variable can be computed from the other using a regression equation. This is the reason why you can f i n d a l l of them on one component.
Table 2: VERIFIEDPRINCIPLEAXEMATRICEAFTERVARIMAXROTATION I
II
III
IV
V
VI
VII
VIII IX
X
1HN 2 EXPO
3 WT 4 VT 5 HANG 6 FL 7 FLG 8 SAIT 9 H2OPL 10 WG 11WGU 12 AUS 13 PL 14 SCH 15 OG1 16 OG2 17 OG3 18 OG4 19 OG5 20 OS 21UG1 22 UG2 23 UG3 24 UG4 25 UG5 26 US 27 SAO 28 SAU 29 TON 30 TONU 31EIND
XI
XII
XIII XIV
XV
XVI
XVII XVIII
0,94 0,98
0,98 0,99 0,92 0,92
0,94
0,99
0,89
0,60
0,85
0,92
0,94
0,97 0,96 0,99 0,99
0,88 0,92 0,78 0,96 0,94 0,69
0,98 0,99 0,99
0,99 0,97
0,79
0,82
0,98
variance 18,9 11,1 10,3 9,3 7,4 5,3 4,8 4,4 3,8 3,2 3,2 2,8 2,7 2,3 2,2 1,8 1,3 1,2
4.1.2.
Components I I and I I I
Both components show soil physic conditions described by the gradients OG4, OG5 and UG4, UG5, component IT of the i n f e r i o r and component I l l of the upper slipped layers. The gradients explain the s i t u a t i o n during saturation (OG5, UG5) and nearl y saturation (OG4, UG4). This soil property was considered as a prerequisit f o r i n s t a b i l i t i e s because i t activates other physical parameters. 222
On this component you can also find the boundary of the bed described by a gradient in the same saturation domain. 4.1.3.
Components IV, V and VII
The status of a dry soil are projected on the components IV, V and VII. On component IV the shrinkage index, clay content of the i n f e r i o r layer, r o l l i n g - o u t l i m i t show high loadings, because clay content influences the shrinkage index and the r o l l i n g - o u t l i m i t . Soil physic conditions of a dry soil are represented by component V for the infer i o r layer and by component Vll for the upper layer. 4.1.4.
The remainin~ components
The remaining components represent only one variable and stand for i t s information. Soil engineers always looked for a simple description of the soil according to the soil moisture. The measurement of the pF-characteristic and the computed gradients in appointed pore domains obviously characterize all soil physic conditions Further the principle axe analysis shows high correlations between a l l soil mechanical parameters. Therefore i t is out of place to use a l l those intercorrelated variables in one investigation, because you w i l l not get any f u r t h e r information. By this way time-consuming measurements became unnecessary. ~.1.5.
Selection of the ind_ep_endent predictors
Selecting the independent predictors R. HERRMANN(1974, 371) proposes a procedure. The problem is to find a combination of steering variables which are not intercorrelated and which have the optimal discriminatory power. This power is checked by Wilks-~, the power for a single steering variable is found by an F-test. The results of the F-tests are shown in table 3. Variables marked by + are used for building the model. There e x i s t three groups of predictor variables. (a) f i r s t
group (13,0000 < F < 48,000)
The highest F-va]ue shows the predictor EIND with 47,7094. The task of the pred i c t o r had been to measure the density in the s o i l . These types of consolidations stop the draining water. That means that the f r i c t i o n angle in the layers decreases. Moreover, a slope is endangered by the pressure of the earth masses above the compressed layers. The pressure is passed over by the water which is not able to escape in the soil material. An appropriate high water content given, these circumstances lead to sliding o f f . The characteristic value for the water mass, influencing a s l i p , is demonstrated by the watershed FL (F : 13,51). FL describes also the vegetation and a l l the other ecological conditions. (b) second group (8,5 < F < 9,5) Slope angle (HN), depth of weathering (VT) and the depth of the roots (WT) form the second group of predictor variables. Their mode of action is mentioned in chapter 2.1., 2.2.. (C) t h i r d group (4,0 < F < 7,5) Here you w i l l find F-values between 7,4 (Shrinkage index, SCH) and 4,11 (water p l a s t i c i t y index, H2OPL). Between them there are UG3 (F = 6,18) and OG3 (F = 5,64) All are s i g n i f i c a n t on the 5 % level. The predictors OG3 and UG3 too confirm the statement that i t is possible to describe soil physic conditions with one value. Both (OG3, UG3) characterize the 223
domain where the soil water content approaches the value of the l i q u i d l i m i t and l a t e r on is saturated. Both suppositions had been considered necessary during the s l i p process. Table 3: Jariable
F-VALUESOF THE VARIABLES CLASSIFIED BY PRINCIPLE AXIS F-value
Prob (%)
component
chosen variable
FLG SATT WG PL
0,0458 0,0673 2,0599 2,5174
0,8255 0,7916 0,1495 0,1107
I
UG4 UG5 US
0,3749 0,3877 2,9694
0,5489 0,5417 0,0831
II
OG4 OG5 OS
0,0295 1,4371 0,4044
0,8582 0,2305 0,5330
III
AUS SCH TONU
4,2189 7,4000 2,0523
0,0392 0,0073 0,1370
IV
UGI UG2 UG3
1,1266 0,3796 6,1810
0,7117 0,7474 0,0179
V
H2OPL
4,1101
0,0480
VI
OGI OG2 OG3
0,1385 0,0942 5,6367
0,7117 0,7574 0,0179
VII
HANG
0,1479
0,7030
VIII
SAU
1,4237
0,2327
IX
13,5187
0,0006
X
0,3108
0,5850
XI
9,3111
0,0031
XII
+
47,7094
0,0000
XIII
+
FL EXPO WT EIND
+
+ +
+
+
HN
9,3441
0,0030
XIV
+
VT
8,6641
0,0041
XV
+
SAO WGU
0,5211 2,1136
0,5251 0,1452
XVI XVII
TON
0,0004
0,9826
XVIII
4.2. 4.2.1.
The discriminant analysis as a Prediction Model The d i f f e r e n t combinations of the Predictor variables
The next task is to find out the most favourable discriminant function of the nine independent predictors. In the f i r s t part of table 4 the possible combinations with the F-ratio and i t s significance are shown, in the second part you find the Wilks-Lambda and the chi-square (significance of the discriminant function) to each combination. The last column consists of a number of impacts i . d . 224
of those objects that had been predicted in the r i g h t group. Table 4:
POSSIBLE INDEPENDENTPREDICTORCOMBINATIONS
Predictors
prob. (%)
I
II
III
IV
V
VI
Vll
47,7094 13,5187 9,3441
0,00 0,06 0,30
+ + +
+ + +
+ + +
+ + +
+ + +
+ + +
+ + +
4 WT 5 VT 6 SCH 7 UG3 80G3 9 H2OPL
9,3111 8,6641 7,4000 6,1810 5,6367 4,1101
0,31 0,41 0,73 1,34 1,79 4,18
+
+ +
+ + +
+ + + +
+ + + + +
+ + + + + +
combination
Wil ks-~
prob. (%)
I II III IV V Vl Vll
0,674 0,634 0,634 0,632 0,610 0,589 0,579
1EIND 2 FL 3 HN
F-value
0,00 0,00 0,00 0,00 0,00 0,00 0,00
x2 57,86 66,47 66,39 66,43 71,46 76,23 78,40
prob. (4) 0,00 0,00 0,00 0,00 0,00 0,00 0,00
prediction(%) 86 87 87 87 88 89 90
(a) combination V I I , VI The maximum separation is obtained by combination VII (~ : .579) with a l l nine predictor variables. 90 % of the observations for testing the accuracy had been predicted in the r i g h t group. The disadvantage of this combination is that the discriminant function seems to be too complex, which means that you spend too much time for measuring the variables. The same reason is applicable to combination VI in which the p l a s t i c i t y index (H2OPL) is not considered a predictor variable anymore. Therefore the accuracy of the model becomes 89 %. (b) combinations V, IV The separation decreases when in combination V OG3 and in combination IV UG3 are removed from the discriminant function. The big change of Wilks-Lambda here shows once more, that i t is possible to use the pF-characteristic for describing soil physic conditions. In both cases the prediction becomes worse the degree of 1%. (c) combinations I I I ,
II
There is nearly any change of Wilks-Lambda and of accuracy, when you use combination I I I and I I , where the depth of weathering (VT) and shrinkage l i m i t (SCH) are taken away. Both variables can be used as predictors, but they do not improve the prediction. The reason can be seen in the way of determinating and measuring the variables. For finding the depth of weathering exactly, you need neutron probes or geoelectrical measuring apparatus. The information of the shrinkage l i m i t is changed because in the measuring procedure disturbed samples were used. Therefore i t cannot be sure that they reproduce the natural shrinkage.
225
(d) combination I The big change of Wilks-Lambda (0.04) demonstrates, that the roots of the vegetation are able to influence a slip process. They stabilize the slope, promoting the solidity. They are in touch with the soil water, they absorbe. This entails that the soil moisture content is reduced. The reduction of the soil water restricts the activity of other parameters that promote a slip. A slope becomes more stable under those conditions. 4.2.2.
Electing the combination for the prediction model
The discriminant function which is considered as the prediction model has to satis fy the following requirements. (i) only a few variables in the discriminant function ( i i ) measurement and determination of the variables must be simple and quickly to be carried out. ( i i i ) the discriminant function must guarantee a practicable prediction. The discriminant functions of the combinations V, VI and VII show a useful prediction, but from the practical view they are too complex. Therefore you have to refer to those combinations, which are only unessentially worser. The most favourable of them is combination II because i t is able to predict a slip with four predictors as well as combination IV with six predictors. The discriminant function is: T = .1140 IO-4HN2 - .2048 IO-2WT2/3 + .8119 Ig(FL+lO) - .5838 Ig(EIND+IO)
(3)
When this function seems to be too complex for the practical use, you should choose the function of combination I: T = .9222 IO-5HN2 + .7926 Ig(FL+lO) - .6098 Ig(EIND+IO) (4) The correlation matrice in table 5 shows that there are high correlations between the variables of equation (4) and the discriminant dimension. This points out that the variables have main portion of the separation. Table 5: CORRELATIONBETWEENPREDICTORSAND DISCRIMINANT DIMENSIONS
1. EIND 2. FL 3. HN 4.2.3.
-.8666 .5081 .4280
Accurac_y_test
Table 6 shows the centroid matrice for the slipping problem. Table 6: CENTROIDMATRICE Cgr g r
= number of groups (here 2) = discriminant dimension (here 1) .1934 Cgr = ( .1781 ) for equation (3) .2332 Cgr = ( .2154 )
for equation (4)
In a bivariate discriminant analysis you get only one discriminant score computed by equations (3) or (4). The smallest difference between the discriminant score T 226
and the centroids for the two groups (see table 6) determines to which group a sample belongs. The result of the 100 samples used for testing shows the table 7. Table 7: PREDICTIONOF TESTEDOBJECTS
combination I
original group
predicted to SL ST
SL ST
59 10
4 27
SL ST
60 10
27
SL ST
60 9
28
VI
SL ST
60 8
3 29
VII
SL ST
61 8
2 29
II,
III,
IV
V
3 3
SL = group of slips ST = group of stable slopes Two slips and eight stable slopes in combination Vll are not predicted in the r i g h t groups. This can mean that the eight stable cases (now considered as slips) l a t e r became unstable ones. The accuracy of the prediction model then would be better than i t is shown now. These eight objects are predicted as slips in all combinations. One of these objects could be checked l a t e r and i t turned out that the slope had become unstable. One other object turned out to be unstable before the evaluation of the datas. The s l i p process took place in more parts and reached catastrophic dimensions a f t e r a hard rain. The reason had been a draining furrow with a watershed of .75 km2. Further a schoal lithosol lay upon a layer of regosol which had been very firm and impenetrable. A prediction with counter-measurements in due time would perhaps have prevented this damage. I f you presume that the remaining six objects which had been stable at the time of measuring now are or w i l l become unstable, the prediction model following equation (4) is able to predict r i g h t 94 objects (= 94 %) of the testing set. Two slips and two stable slopes could not be predicted. One reason was the decreasing e f f i c i e n c y of the separation while l e t t i n g out predictors. Another was that a slope had been considered unstable but r e a l l y had been stable. One of the stable slopes showed layers with sand material where a fine-pored sand lay between gravel. The soil water in the fine-pored sand, the slope-gradient and the pressure of the upper soil-material probably lead to the i n s t a b i l i t y . Other objects in sand material with an impenetrable and firm layer are predicted as slips. The benefits of this model are to be seen in i t s s i m p l i c i t y and quickness. With a few values, which can be measured very easily, and a simple mathematical procedure you can predict slips an a slope. An example of the procedure is shown in table 8.
227
Table 8: EXAMPLEFOR A PREDICTION CSL = centroid for slips
} out of table 6
CST = centroid for the stable case using equation (4) predictor HN (o)
sample 1
sample 2
34
14
FL (km2)
.55
.20
EIND
.08
.60
.2098
.1760
/T-CsL/
.0164
.0274
/T-CsT/
.0317
.0021
T
The results are: the slope where sample 1 was taken is unstable. Sample 2 stands for the stable case. The introduced model is time invariant. This means that i t is impossible to pred i c t the moment when a s l i p w i l l take place. I t is possible to predict a s l i p taking place or not. I t works in all soil materials except for sands where slips cannot be very well predicted. The range of application of the model you w i l l find in constructing streets and highways where new slopes are b u i l t . Here i t is important to know very quickly i f slips are to be expected or not. Therefore you can use function (4). I f the application of the model would predict a landslip, appropriate counter-measures could be started. Another result of the model had been that the depth of the roots influences a s l i p . You should use equation (3) i f there is vegetation on a slope or i f you know the sort of vegetation which is planned to grow on a slope. With the application of the suggested model the s l i p process is not solved, because the problem is too complex to find an exact solution. LITERATURE ANDERSON, T.W. (1958): An Introduction to Multivariate S t a t i s t i c a l Wiley: New York, 374 pp.
Analysis.
BACKOFEN, K. (1957): Klassifikation der Rutschungen? Geologie und Bauwesen, 23, 125 - 130. BLESSING, G. (1966): Die Sicherung rutschgef~hrdeter H~nge. StraBen- und Tiefbau, 20, 218 - 220. BULLING, W.H. (1971): Bodenkennziffern und Klassifizierung von B~den. Springer, New York, 192 pp. BRYAN, J. (1951): The generalized discriminant function: mathematical foundation and computation routine. Harvard Educational Review, 21, 90 - 95. 228
CROZIER, M.J. (1973): Techniques for the morphometric analysis of landslips. Zeitschr. f. Geomorphologie, N.F. 1_]_7,78 - 101. DECOURSEY, D.G. (1973): An Application of Discriminant Analysis in Design Review. Water Resources Research, 9, 93 - 102. DIN 18 121Bestimmung des Wassergehaltes durch Ofentrocknung DIN 18 122 Zustandsgrenzen (Konsistenzgrenzen) FELLENIUS, W. (1947): Erdstatische Berechnungen. Dritte unver~nderte Aufl., von Wilhelm Ernst Verlag, Berlin, 48 pp. GOSSEFELD, J. (1973): Fortran IV-Programm JGNVT (unver~ffentlicht). HARTGE, K.H. (1971): Die physikalische Untersuchung von B~den. Enke Verlag, Stuttgart, 166 pp. HERRMANN, R. (1971): Zur regionalhydrologischen Analyse und Gliederung der nordwestlichen Sierra Nevada de Santa Marta (Kolumbien). GieBener Geogr. Schriften Nr. 23 (Sonderheft 1), Giessen, 88 pp. HERRMANN, R. (1974): Ein Anwendungsversuch der mehrdimensionalen Diskriminanzanalyse auf die AbfluBvorhersage, Catena 1, 367 - 385. HERRMANN, R. und SCHRIMPFF, E. (1975): Zur Vorhersage des AbfluBverhaltens in tropischen Hochgebirgen West- und Zentralkolumbiens. 40. Deutscher Geographentag, Innsbruck (im Druck). KARRENBERG, H. (1963) in: Geologische und bodenmechanische Ursachen von Rutschungen, Gleitungen und BodenflieBen. Forschungsberichte des Landes NordrheinWestfalen, Nr. 1138. Westdeutscher Verlag, Opladen, 63 pp. KIRKHAM, Don: Soil Physics, in: Ven te Chow (Hrsg.): Handbook of Applied Hydrology. Selection 5, 1 - 22, Mc Graw H i l l , New York 1965. KNOBLICH, K. (1967): Mechanische Gesetzm~Bigkeiten beim Auftreten von Hangrutschungen. Zeitschr. f. Geomorphologie, N.F. 11, H. 3, 286 - 299. KDHN-VELTEN, H. (1963) in: Geologische und bodenmechanische Ursachen von Rutschungen, Gleitungen und BodenflieBen. Forschungsberichte des Landes NordrheinWestfalen Nr. 1138. Westdeutscher Verlag, Opladen, 63 pp. LANGEJAHN, A. (1965): Some Aspects of the Safety Factor in Soil Mechanics: Considered as a Problem of Probability. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2, Montreal, 500 - 502. LIPPMANN, W. (1960): Messungen von Bodenbewegungen infolge zeitlicher Schwankungen im Wasserhaushalt. Diss. 1960, 72 pp. MATSCHAK, H. und RIETSCHEL, A. (1965): Vereinfachte FlieBgrenzenbestimmung an bindigen Bodenarten fur ingenieurgeologisch bodenmechanische Untersuchungen. Zeitschr. f. angew. Geologie, Bd. 11, 3, 135 - 139. MCQUEEN, J.S. und MILLER, R.F. (1974): Approximating Soil Moisture Characteristics From Limited Data: Empirical Evidence and Tentative Model. Water Resources Research, Vol. 10, No. 3, 521 - 527. 229
NEULAND, H. (1975): Zur Vorhersage von Hang- und B~schungsrutschungenmit Hilfe der Diskriminanzanalyse. Diss. K~In 1975, 55 pp. NONVEILLER, E. (1965): The S t a b i l i t y Analysis of Slopes with a Slip Surface on General Shape. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2 Montreal, 522 - 525. RODE, A. (1955): Das Wasser im Boden. Akademie'Verlag, Berlin, 464 pp. SAITO, M. (1965): Forecasting the Time of Occurence of a Slope Failure. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2. Montreal, 537 - 541. SBRESNY, N. (1941): Grundlegendes Uber den Stand der Untersuchung einer Rutschung Geologie und Bauwesen, 3, 25 - 26. SCHEFFER, F. und SCHACHTSCHABEL,P. (1966): Lehrbuch der Bodenkunde, Enke Verlag, S t u t t g a r t , 473 pp. SCHLICHTING, E. und BLUME, H.P. (1966): Bodenkundliches Praktikum. Parey Verlag, Hamburg, 209 pp. SCHOFIELD, R.K. (1935): The pF of Water in Soil. Trans. Intern. Congr. Soil Sc, 3rd, 2, Oxford, 37 - 48. SCHRIMPFF, E. (1975): Ein mathematisches Modell zur Vorhersage von AbfluBereignissen im Bereich der Anden Kolumbiens- SUdamerika. Diss. K~In 1975 (im Druck). SCHULTZE, E. und MUHS, H. (1967): Bodenuntersuchungen fur Ingenieurbauten. Springer, Heidelberg, 631 pp. SIEDECK, P. und VOSS, R. (1960): Die BodenprUfverfahren bei StraBenbauten, DUsseldorf, 124 pp. TERZAGHI, K. und PECK, R.B. (1961): Die Bodenmechanik in der Baupraxis. Springer, Heidelberg, 431 pp. TRAUZETTEL, G. (1962): Die Rutschungen im wUrttembergischen Knollenmergel. Arbeiten aus dem Geol.-Pal~ontol. I n s t i t u t der TH Stuttgart, Neue Folge (seit 1953) Nr. 3__22,143 pp. OBERLA, K. (1968): Faktorenanalyse. Springer, Heidelberg, 399 pp. WATZNAUER, A. (1965): Die Rutschung von Nieder-Tenzel (1941) - eine Korrektur. Zeitschr. f. angew. Geol., Bd. I i , 12, 667. WEIDENBACH, F. (1965): Die geologischen Voraussetzungen fur die Entstehung von Rutschungen. StraBen- und Tiefbau, 12, 1 - 8. ZARUBA, Q. u. MENCL, V. (1969): Landslides and Their Control. Elsevier Verlag, Amsterdam, 214 pp.
230
VOL. 3, 215 - 230
I
GIESSEN 1976
[
A PREDICTIONMODELOF LANDSLIPS
Herbert Neuland c/o Lehrstuhl fur Hydrologie im FB Geowissenschaften Universit~t Bayreuth, Postfach 3008, D - 8580 Bayreuth SUMMARY
I t is the intention of this investigation to develop a prime relation that makes a prediction of landslips possible. Therefore 250 stable and unstable objects were obtained all over the BRD and 31 variables were determined. They consist of morphometric, soil-mechanic, material and stratification characteristics. The principle component-analysis explains the relations between the variables. An F-test shows that nine independent variables exist. A bivariate discriminant analysis chosen with 150 at random objects yields a prediction function which efficiency later is tested by a procedure of the Euklid distance. From seven tested variable combinations that one was selected which sho~ed that the slope gradient HN (o), the watershed to the considered object FL (km~) and a parameter measuring the density of the soil EIND are sufficient to predict a landslip. The significant discriminant function is T = 0,9222 • IO-5HN2 + 0,7926 Ig(FL + 10) - 0,6098 Ig(EIND + 10) with the centroids 0,1934 for slips 0,1781 for stable objects. The quality of the model was demonstrated by 100 objects from which 94 were predicted in the right group. ZUSAMMENFASSUNG Mit der Arbeit wurde das Ziel verfolgt, eine m~glichst einfache Beziehung zu finden, die es erlaubt, Rutschungen an B~schungen vorherzusagen. Dazu werden an 250 Rutschereignissen und stabilen Stellen 31 ausgesuchte Variablen ermittelt, die einfach zu bestimmen und aussagekr~ftig sind. Sie setzen sich zusammen aus morphometrischen, bodenmechanischen und das Material und die Lagerung kennzeichnenden Gr~Ben. Mittels der Hauptkomponentenanalyse wird das Beziehungsgeflecht zwischen den Variablen gekl~rt. Ein F-Test im Rahmen der Diskriminanzanalyse legt neun unabhNngige Variablen zur Modelleichung fest. Die nach einer Zufallstafel ausgesuchten Ereignisse ergeben mit einer eindimensionalen Diskriminanzanalyse eine Vorhersagegleichung, deren GUte anschlieBend mit zurUckbehaltenen Ereignissen durch ein Zuordnungsverfahren nach dem Euklidischen Abstand getestet wird. Aus den verschiedenen m~glichen Variablenkombinationen ergibt sich, dab Hang215
neigung (HN), Einzugsgebiet oberhalb einer Untersuchungsstelle (FN) und ein Dichteparameter (EIND) ausreichen, an einer B~schung bzw. an einem Hang Rutschungen vorherzusagen. Die signifikante Diskriminanzfunktion dazu, die ein Ereignis den beiden vorgegebenen Gruppen Rutschung bzw. Nicht-Rutschung zuordnet, lautet: T = 0,9222 • IO-5HN2 + 0,7926 Ig(Fl + 10) - 0,6098 Ig(EIND + 10) mit den Centroiden fur Rutschung (CR = 0,1934) und Nicht-Rutschung (CNR = 0,1781). Von 100 getesteten Ereignissen werden 94 der richtigen Gruppe zugewiesen. 1. 1.1.
INTENTIONOF THE INVESTIGATION AND PROBLEMS Intention
Thirty years ago N. SBRESNY(1941, 25 - 26) described the situation of the research about landslips. One group of scientists tries to solve the slip process by using the morphometric and soil-mechanical characteristics, while the other group prefers the theoretic physical consideration. For both methods you need big laboratories and much time, and there is not any objective and proved procedure putting out those variables influencing the slip process. For finding out the important variables K. TERZAGHI and R.B. PECK (1961, 391) even give the advise to ask experienced engineers. Therefore G. BLESSING (1966, 218 - 220) proposes to use those soil mechanical characteristics which are predicatory and very simple to determine. The opinions of K. BACKOFEN(1957, 125 - 130) and G. TRAUZETTEL (1962, 40) are important, too. They content that there cannot be only one factor responsible for a landslip. Always a combination of a group of appointed characteristics cause a landslip. The mechanical principles of slips are well known. You have to discern between the forces setting the process in motion and stopping i t . The determination of the forces is complicated by the natural slip-surfaces which are very d i f f i c u l t to accomodate to theoretic ones. The mechanical principles you will find in K. KNOBLICH (1967, 286 - 299), W. FELLENIUS (1947, 1 - 48) and H. NEULAND(1975, 9 - 13). F. WEIDENBACHremarks that i t is not possible t i l l today to state exactly the causing variables or to predict a landslip. My investigation reflects just to this opinion and tries to develop a method, which on one side discovers those variables that influence a landslip on and on. On the other hand I want to try to raise a model which is able to predict a landslip. The model is based on variables that had been determined on unstable and stable objects. The necessary foundations are taken from the mathematical multivariate s t a t i s t i c . There are some methods for predicting a landslide published by A. LANGEJAN (1965, 500 - 502), N. SAITO (1965, 537 - 541) and R. NONVEILLER (1965, 522 - 525). The disadvantage of the methods is that only one variable is considered as the causing one. Not in all cases the selected variable is the reason for the slip, hence i t follows that the probability of a computed stability-coefficient which marks stable and unstable equilibrium is only small. 1.2.
Task
The term landslide in the following text follows K. TERZAGHI and K.B. PECK (1961, 109) who consider the break of earth-masses in a slope as a landslide. The next thing to do is to select those variables out of the publications that characterize the slip-process and can be a possible reason for i t . The measurement of the variables in the f i e l d and in the laboratory should be very simple. H. KARRENBERG(1963, 9 - 10) and D.G. DECOURSEY(1973) have refered to this fact° For building the model, 250 stable and unstable slopes all over the BRD 216
Ubersichtsskizze der a u f Rutschungen u n t e r s u c h t e n
Raume
(see figure I) were selected. On each slope the s t r a t i f i c a tion was ascertained with a boring instrument and a soil sample was taken out of each layer. In the f i e l d and in the laboratory 31 variables explained in chapter 2. were determined. For f i l t e r i n g and finding out those variables causing a landslip and which are later necessary for raising the model, you have to use the principle axe analysis. I t also shows the correlation between the predictorvariables and makes i t possible to find independent predictors.
--
Autobahn
~
BundesstroFle
~
Untersuchungsraum Einzelrutschung
Mansteb I 4MiLl Quelle Shell Au~o(:ltlas72
The discriminant analysis is able to attach an object described by variables to a characteristic group. R. HERRMANN (1974), E. SCHRIMPFF (1975) used i t for predicting the runoff of rivers and D.G. DECOURSEY (1973) had success in predicting the water damage on dams.
In my special case you have to use a bivariate discriminant analysis with both the groups Figure 1: General sketch of landslips and stable slopes. By this way i t is render possible to attach n objects described by p predictorvariables to both groups mentioned above. The mathematical procedure is to find a linear combination of the predictors which separates the two groups best. A detailed description of the discriminant analysis you can find in R. HERRMANN(1974, 381 - 385). 2.
THE PREDICTORVARIABLESAND THEIR MEASUREMENT
2.1o Morphometric variables Slope gradient (HN) - Exposition (EXPO) - Length of a slope (HANG) A lot of publications cited the slope gradient as a causing variable. K. KNOBLICH (1967) and M.J. CROZIER (1973) add to the slope gradient the variable exposition of a slope and its length. The moisture of a slope that indicates the reduction of the frictional index of the soil material and the drying up of a slope are influenced by the exposition. The slope length is used as a variable to get some information about the pressure of the soil material. 2.2.
Foundation of the selected points
Depth of roots (WT) - Depth of weathering (VT) Except for certain circumstances you do not expect a slip in rocks. But when the weathering has built a layer of weather-worn material, i t tends to instability. The depth of weathering is ascertained with a boring instrument and i t also can be ascertained with geological, hydrogeological and geomorphological maps. 217
Strengthening the soil material, the roots stabilize a slope, further more some appointed sorts reduce the soil moisture. Therefore the relation between vegetation and soil material is described by the depth of roots which is tested by boring or digging. 2.3.
Watershed to a control point (FL)
G. TRAUZETTEL (1962), A. WATZNAUER(1965) and Q. ZARUBAand V. MENCL(1969) again and again found out that there is a relation between the landslip and the area above i t . This opinion could be confirmed during the work in the fields. The areas with compact ways and furrows are inclined in direction of the slip, so that the water drains to this area. First the watershed had been determined in the f i e l d by charting in following contour line and later by planimetry. I t should be a test value for the water which takes an active hand in landslips. For points where there had not been any possibility to determine a watershed, its natural size had been estimated. Also the vegetation in the areas influence the s t a b i l i t y of a slope (see figure 2).
road
slip
watershed(FL)
l { I
I I l
road slip i i
watershed(FL) I i
furrow
i
__
'.
0
h '
^cA
*
>,.
Figure 2: Watershed to a landslip 2.4.
Soil mechanical characteristics
H. KOHN-VELTEN (1963), G. TRAUZETTEL (1962) and W. LIPPMANN (1960) propose for a detailed investigation of landslips the determination of the soil mechanical parameters. Because of simplification and rationalization the following parameters had been selected, reflecting that the humid conditions of the soil accelerate the instability of a slope. 2.4.1.
Liquid limit (FLG) - saturation de~ree (SATT)- water-plasticity (H~PL) ~ ~t~ ~uTi~g.-s~r~tTo~ Tn t~e_-u~p~r--l~r~ ~_)-a~d-i~f~rlor - Ta~s~(~G~)~
The variables cannot be determined by physical equations. Each one has to be determined by a laboratory procedure which is published in W.H. B~LLING (1971, 44 47) and E. SCHULTZEand H. MUHS(1967, 384 - 388). Because of rationalization a procedure developed by H. MATSCHAKand A. RIETSCHEL (1965, 135 - 139) was used here. The information of the soil mechanical parameters is limited, because on one side you have to work with disturbed samples, on the other side you only can
218
use the part of grain size ~ 0,4 mm from the sample. This portion does not seem to be enough to give back the natural properties of the soil material. 2.4.2.
Rollin~-out limit LAUS~ - Plasti£ityLindex_(PL~LShrinkage-index LSCHI
The dry soil, too, furthers a slope. The dry soll conditions are described by the indexes above. Drying soil yields crevasses which disaggregate the soil and makes i t possible for the water to i n f i l t r a t e very easily. The conclusion is the soaking and saturation of the soil material which together with other components is responsible for i n s t a b i l i t i e s . Laboratory procedures are published in W.H. BULLING (1971, 49 - 53), G. SCHULTZEand H. MUHS(1967, 392 - 395). 2.4.3.
pF-soil characteristic for describing soil physic conditions TO~I~ ~G~,-O~3-~ ~GT,-O~5-, UGT,--U~2~~G~,-U~4-, ~G~)
All scientists have regarded the water as an important variable but nowhere you find a notice for an exact measuring, except for the extraordinary measurement of the pore water-pressure. The soil moisture and water balance can be described by the pF-characteristic, the theoretical foundations and measurements of which are published in A. RODE (1959, 335 - 349), DON KIRKHAM (1964, 1 - 22), F. SCHEFFERand P. SCHACHTSCHABEL (1966, 226 - 254), R. HERRMANN(1971, 27 - 30), K.H. HARTGE (1971, 71 - 84), E. SCHLICHTING and H.P. BLUME(1966, 65 - 70), J.S. McQUEENand R.F. MILLER (1971, 521 - 527). The soil-suction (pF) render i t possible to compare the behaviour of the water in different soils with the help of equivalent soil moisture contents. A diagram of the pF drawn against the soil moisture content gives each soil a proper characteri s t i c (see figure 3). The pF-characteristic is proved suitable to describe soil profiles from which you can deduct soil physic conditions.
II"c,ay pore,
PF
"
(u) 0,~-
102 600
0 I0 20
, VoI%
Figure 3: pF - characteristic of different soil
7
L
6 4 2
2
FOGS
:~--OG20G1
4' - ~ UG3 2- ~ U G 2 O , 0 I0 20
,o
UGI
£ Vol%
Figure 4 : Gradients for describing the soil physics conditions demonstrated by a pF - characteristic of a soil from a landslip
Following F. SCHEFFERand P. SCHACHTSCHABEL(1966, 95) fixed pF-domains correspond with one pore domain that fixes the soil physic conditions. For pointing them out, to every pore or pF-domain a differential gradient (see table 1 and figure 4, OGi for the upper; UG~ for the inferior layer) which stands for the soil physics information for fixed soil moisture contents, had been appropriated.
219
Table 1: PORE-SCOPEOF DIFFERENT pF BELONGINGTO THE GRADIENTSWHICH DESCRIBE THE SOIL PHYSIC CONDITIONS gradient of the pore size (0) pF-domain upper layer inferior layer 50 0 - 0,8 OG1 UG1 50 0,8 1,77 OG2 UG2 50 - 10 1,77 - 2,54 OG3 UG3 10 - 0,2 2,54 - 4,20 OG4 UG4 0,2 4,20 - 7 OG5 UG5 2.5.
Sizing the structure of the material
2.5.1.
Stratified structure of the up~e~_~SA_O~and inferior__(S_AU)__layer
F. WEIDENBACH (1965, 3) drew attention to the fact that in a slip process always two layers become effective. This makes i t necessary to get some information about the s t r a t i f i e d structure. The quotient of all gradients of a pF-characteri s t i c shown in 2.4.3. represents a typical number for each layer: SAO OGI/OG2/OG3/OG4/OG5 (1) SAU = UGI/UG2/UG3/UG4/UG5 (2) There is a big number for clays, a small number for sand with all possibilities in between. :
2.5.2.
Upper boundary_of the bed ~OG)_-_inferior boundary of the bed ~UG)
For defining the boundary of the bed i t is assumed that there is low pressure in the unstable case. Therefore a differential quotient on the pF-characteristic between the fluidy index (Vol % water content) and the saturation marks the boundary of the bed. 2.5.3.
Clay content of the uppe~TO__N] and inferior layer_(TON_U~
Clay accelerates a slip and its influence indirectly is included in the pF-charact e r i s t i c . Measuring its influence the clay content of all layers had been estimated by a rule of thumb following a table from E. SCHLICHTINGand H.P. BLUME (1966, 21). 2.5.4.
De~ree of firmness and densi~y_(EIND~
There is used a probe for finding out the compressed layers which stop the draining water. As a simple probe you can use a Purkhauer boring-instrument with a height of f a l l of 1 m for the 5-kg-hammer. Probes developed by Kaschins~ show similiar results with ve~ small deviations. You have to measure the number of beats on the probe and the depth of penetration belonging to. Both test results are drawn in a co-ordinate system and an equilized curve is developed (see figure 5). A differential quotient on the natural bounda~ of the bed represents firmness and density of the stratified structure. Following K.H. HARTGE (1971, 153) who also describes probes besides P. SIEDECKand R. VOSS (1960, 86 - 88) the measurement result can also be considered as shear-strength.
0
'
. . . .
~ ~ ( a )
0
~
l
l
.5 I0
l
l
-
l
l
)beats
-
-
~
I((b) b
/ EIND
=
I dS
Z,
Figure 5: Determining the density parameter EIND in the unstable (a) and the stable case (b)
3.
ROUGHCOPYOF THE MODELAND STATISTICAL METHODS
3.1.
Conception of the model
All over the BRD 250 points were controlled for slips, on each point soil samples had been taken and 31 variables mentioned in chapter 2. had been estimated. From the 250 objects 150 had been selected by a random table. The remaining 100 objects are used to reexamine the model. The variables are tested for normal distribution by Kolmogorow Smirnow-Test on the 5 % level. Not normal distributed variables are transformed appropriately. The next step is to find out the correlation between the 31 variables. Making use of the principle axe analysis, i t is advantageous to find on one side the correlation and on the other side you always get one group of similiar structured variables on a component. The predictor-variables that had to be independent and had to show a good separation are selected by a special procedure with the discriminant analysis. All possible variable combinations you can form as a result of the principle axe analysis are carried out to a discriminant analysis. A computed F-value of each variable distinguishes its discriminant power. High F-values means good, low F-values bad discriminant power. When all F-values are known, those with the best F-value are considered as predictor variables. A combination out of this group yields the discriminant function. The best function shows the Wilks-Lambda which tests the separation of the used predictors. The significance of the discriminant function is checked by a chi-square-test. The model is b u i l t when the best predictor combination is found out. Next the efficiency of the model has to be tested with the remaining 100 objects. The equation of the discriminant function computes a discriminant score for each object. While building the model, for each group (landslides and stable slopes) a characteristic value was set up, called a centroid. The procedure of the Euklid distance compares the result of the discriminant function with both centroids. A tested object belongs to the group where the Euklid distance shows the smallest value. I f the features of the object agree with those of the group, the predicting succeeded. Figure 6 shows the position of different objects in the discriminant space. l
CENTROID 2 ( s t a b l e ) 22
2
2
O. 160
2
O. 170 i •
o r i g i n group
2 A
2
~
2
O. 180 -2
J
2222
CENTROID1 ( u n s t a b l e ) 2
~ m - - I
i
I
1
I
O. 190
I Iii
i i
I
O. 200
group a f t e r p r e d i c t i o n
Figure 6: Some hand-picked objects in the discriminant space 3.2.
Statistical methods
3.2.1. Principle axe anal~sis A comprehensive book about principle axe analysis was written by K. OBERLA (1968). Reduced statements are published in R. HERRMANNand E. SCHRIMPFF (1975) and H. NEULAND(1975).
221
3.2.2.
Discriminant analysis and Euklid distance ~rocedure
The mathematical foundation you f i n d in T.W. ANDERSON (1958) and J. BRYAN (1951). R. HERRMANN(1974) published an example where you f i n d the procedure of the Euklid distance, too. 4.
RESULTS
4.1.
p r i n c i p l e axe analysis
4.1.1.
Component I
Table 2 shows the rotated component matrice with loadings • 0,5. Component I represents soil mechanical parameters which describes the soil material with high soil moisture and saturation. These are the l i q u i d l i m i t , saturation degree, moisture content at saturation and p l a s t i c i t y index. All variables provide nearly the same information. Each variable can be computed from the other using a regression equation. This is the reason why you can f i n d a l l of them on one component.
Table 2: VERIFIEDPRINCIPLEAXEMATRICEAFTERVARIMAXROTATION I
II
III
IV
V
VI
VII
VIII IX
X
1HN 2 EXPO
3 WT 4 VT 5 HANG 6 FL 7 FLG 8 SAIT 9 H2OPL 10 WG 11WGU 12 AUS 13 PL 14 SCH 15 OG1 16 OG2 17 OG3 18 OG4 19 OG5 20 OS 21UG1 22 UG2 23 UG3 24 UG4 25 UG5 26 US 27 SAO 28 SAU 29 TON 30 TONU 31EIND
XI
XII
XIII XIV
XV
XVI
XVII XVIII
0,94 0,98
0,98 0,99 0,92 0,92
0,94
0,99
0,89
0,60
0,85
0,92
0,94
0,97 0,96 0,99 0,99
0,88 0,92 0,78 0,96 0,94 0,69
0,98 0,99 0,99
0,99 0,97
0,79
0,82
0,98
variance 18,9 11,1 10,3 9,3 7,4 5,3 4,8 4,4 3,8 3,2 3,2 2,8 2,7 2,3 2,2 1,8 1,3 1,2
4.1.2.
Components I I and I I I
Both components show soil physic conditions described by the gradients OG4, OG5 and UG4, UG5, component IT of the i n f e r i o r and component I l l of the upper slipped layers. The gradients explain the s i t u a t i o n during saturation (OG5, UG5) and nearl y saturation (OG4, UG4). This soil property was considered as a prerequisit f o r i n s t a b i l i t i e s because i t activates other physical parameters. 222
On this component you can also find the boundary of the bed described by a gradient in the same saturation domain. 4.1.3.
Components IV, V and VII
The status of a dry soil are projected on the components IV, V and VII. On component IV the shrinkage index, clay content of the i n f e r i o r layer, r o l l i n g - o u t l i m i t show high loadings, because clay content influences the shrinkage index and the r o l l i n g - o u t l i m i t . Soil physic conditions of a dry soil are represented by component V for the infer i o r layer and by component Vll for the upper layer. 4.1.4.
The remainin~ components
The remaining components represent only one variable and stand for i t s information. Soil engineers always looked for a simple description of the soil according to the soil moisture. The measurement of the pF-characteristic and the computed gradients in appointed pore domains obviously characterize all soil physic conditions Further the principle axe analysis shows high correlations between a l l soil mechanical parameters. Therefore i t is out of place to use a l l those intercorrelated variables in one investigation, because you w i l l not get any f u r t h e r information. By this way time-consuming measurements became unnecessary. ~.1.5.
Selection of the ind_ep_endent predictors
Selecting the independent predictors R. HERRMANN(1974, 371) proposes a procedure. The problem is to find a combination of steering variables which are not intercorrelated and which have the optimal discriminatory power. This power is checked by Wilks-~, the power for a single steering variable is found by an F-test. The results of the F-tests are shown in table 3. Variables marked by + are used for building the model. There e x i s t three groups of predictor variables. (a) f i r s t
group (13,0000 < F < 48,000)
The highest F-va]ue shows the predictor EIND with 47,7094. The task of the pred i c t o r had been to measure the density in the s o i l . These types of consolidations stop the draining water. That means that the f r i c t i o n angle in the layers decreases. Moreover, a slope is endangered by the pressure of the earth masses above the compressed layers. The pressure is passed over by the water which is not able to escape in the soil material. An appropriate high water content given, these circumstances lead to sliding o f f . The characteristic value for the water mass, influencing a s l i p , is demonstrated by the watershed FL (F : 13,51). FL describes also the vegetation and a l l the other ecological conditions. (b) second group (8,5 < F < 9,5) Slope angle (HN), depth of weathering (VT) and the depth of the roots (WT) form the second group of predictor variables. Their mode of action is mentioned in chapter 2.1., 2.2.. (C) t h i r d group (4,0 < F < 7,5) Here you w i l l find F-values between 7,4 (Shrinkage index, SCH) and 4,11 (water p l a s t i c i t y index, H2OPL). Between them there are UG3 (F = 6,18) and OG3 (F = 5,64) All are s i g n i f i c a n t on the 5 % level. The predictors OG3 and UG3 too confirm the statement that i t is possible to describe soil physic conditions with one value. Both (OG3, UG3) characterize the 223
domain where the soil water content approaches the value of the l i q u i d l i m i t and l a t e r on is saturated. Both suppositions had been considered necessary during the s l i p process. Table 3: Jariable
F-VALUESOF THE VARIABLES CLASSIFIED BY PRINCIPLE AXIS F-value
Prob (%)
component
chosen variable
FLG SATT WG PL
0,0458 0,0673 2,0599 2,5174
0,8255 0,7916 0,1495 0,1107
I
UG4 UG5 US
0,3749 0,3877 2,9694
0,5489 0,5417 0,0831
II
OG4 OG5 OS
0,0295 1,4371 0,4044
0,8582 0,2305 0,5330
III
AUS SCH TONU
4,2189 7,4000 2,0523
0,0392 0,0073 0,1370
IV
UGI UG2 UG3
1,1266 0,3796 6,1810
0,7117 0,7474 0,0179
V
H2OPL
4,1101
0,0480
VI
OGI OG2 OG3
0,1385 0,0942 5,6367
0,7117 0,7574 0,0179
VII
HANG
0,1479
0,7030
VIII
SAU
1,4237
0,2327
IX
13,5187
0,0006
X
0,3108
0,5850
XI
9,3111
0,0031
XII
+
47,7094
0,0000
XIII
+
FL EXPO WT EIND
+
+ +
+
+
HN
9,3441
0,0030
XIV
+
VT
8,6641
0,0041
XV
+
SAO WGU
0,5211 2,1136
0,5251 0,1452
XVI XVII
TON
0,0004
0,9826
XVIII
4.2. 4.2.1.
The discriminant analysis as a Prediction Model The d i f f e r e n t combinations of the Predictor variables
The next task is to find out the most favourable discriminant function of the nine independent predictors. In the f i r s t part of table 4 the possible combinations with the F-ratio and i t s significance are shown, in the second part you find the Wilks-Lambda and the chi-square (significance of the discriminant function) to each combination. The last column consists of a number of impacts i . d . 224
of those objects that had been predicted in the r i g h t group. Table 4:
POSSIBLE INDEPENDENTPREDICTORCOMBINATIONS
Predictors
prob. (%)
I
II
III
IV
V
VI
Vll
47,7094 13,5187 9,3441
0,00 0,06 0,30
+ + +
+ + +
+ + +
+ + +
+ + +
+ + +
+ + +
4 WT 5 VT 6 SCH 7 UG3 80G3 9 H2OPL
9,3111 8,6641 7,4000 6,1810 5,6367 4,1101
0,31 0,41 0,73 1,34 1,79 4,18
+
+ +
+ + +
+ + + +
+ + + + +
+ + + + + +
combination
Wil ks-~
prob. (%)
I II III IV V Vl Vll
0,674 0,634 0,634 0,632 0,610 0,589 0,579
1EIND 2 FL 3 HN
F-value
0,00 0,00 0,00 0,00 0,00 0,00 0,00
x2 57,86 66,47 66,39 66,43 71,46 76,23 78,40
prob. (4) 0,00 0,00 0,00 0,00 0,00 0,00 0,00
prediction(%) 86 87 87 87 88 89 90
(a) combination V I I , VI The maximum separation is obtained by combination VII (~ : .579) with a l l nine predictor variables. 90 % of the observations for testing the accuracy had been predicted in the r i g h t group. The disadvantage of this combination is that the discriminant function seems to be too complex, which means that you spend too much time for measuring the variables. The same reason is applicable to combination VI in which the p l a s t i c i t y index (H2OPL) is not considered a predictor variable anymore. Therefore the accuracy of the model becomes 89 %. (b) combinations V, IV The separation decreases when in combination V OG3 and in combination IV UG3 are removed from the discriminant function. The big change of Wilks-Lambda here shows once more, that i t is possible to use the pF-characteristic for describing soil physic conditions. In both cases the prediction becomes worse the degree of 1%. (c) combinations I I I ,
II
There is nearly any change of Wilks-Lambda and of accuracy, when you use combination I I I and I I , where the depth of weathering (VT) and shrinkage l i m i t (SCH) are taken away. Both variables can be used as predictors, but they do not improve the prediction. The reason can be seen in the way of determinating and measuring the variables. For finding the depth of weathering exactly, you need neutron probes or geoelectrical measuring apparatus. The information of the shrinkage l i m i t is changed because in the measuring procedure disturbed samples were used. Therefore i t cannot be sure that they reproduce the natural shrinkage.
225
(d) combination I The big change of Wilks-Lambda (0.04) demonstrates, that the roots of the vegetation are able to influence a slip process. They stabilize the slope, promoting the solidity. They are in touch with the soil water, they absorbe. This entails that the soil moisture content is reduced. The reduction of the soil water restricts the activity of other parameters that promote a slip. A slope becomes more stable under those conditions. 4.2.2.
Electing the combination for the prediction model
The discriminant function which is considered as the prediction model has to satis fy the following requirements. (i) only a few variables in the discriminant function ( i i ) measurement and determination of the variables must be simple and quickly to be carried out. ( i i i ) the discriminant function must guarantee a practicable prediction. The discriminant functions of the combinations V, VI and VII show a useful prediction, but from the practical view they are too complex. Therefore you have to refer to those combinations, which are only unessentially worser. The most favourable of them is combination II because i t is able to predict a slip with four predictors as well as combination IV with six predictors. The discriminant function is: T = .1140 IO-4HN2 - .2048 IO-2WT2/3 + .8119 Ig(FL+lO) - .5838 Ig(EIND+IO)
(3)
When this function seems to be too complex for the practical use, you should choose the function of combination I: T = .9222 IO-5HN2 + .7926 Ig(FL+lO) - .6098 Ig(EIND+IO) (4) The correlation matrice in table 5 shows that there are high correlations between the variables of equation (4) and the discriminant dimension. This points out that the variables have main portion of the separation. Table 5: CORRELATIONBETWEENPREDICTORSAND DISCRIMINANT DIMENSIONS
1. EIND 2. FL 3. HN 4.2.3.
-.8666 .5081 .4280
Accurac_y_test
Table 6 shows the centroid matrice for the slipping problem. Table 6: CENTROIDMATRICE Cgr g r
= number of groups (here 2) = discriminant dimension (here 1) .1934 Cgr = ( .1781 ) for equation (3) .2332 Cgr = ( .2154 )
for equation (4)
In a bivariate discriminant analysis you get only one discriminant score computed by equations (3) or (4). The smallest difference between the discriminant score T 226
and the centroids for the two groups (see table 6) determines to which group a sample belongs. The result of the 100 samples used for testing shows the table 7. Table 7: PREDICTIONOF TESTEDOBJECTS
combination I
original group
predicted to SL ST
SL ST
59 10
4 27
SL ST
60 10
27
SL ST
60 9
28
VI
SL ST
60 8
3 29
VII
SL ST
61 8
2 29
II,
III,
IV
V
3 3
SL = group of slips ST = group of stable slopes Two slips and eight stable slopes in combination Vll are not predicted in the r i g h t groups. This can mean that the eight stable cases (now considered as slips) l a t e r became unstable ones. The accuracy of the prediction model then would be better than i t is shown now. These eight objects are predicted as slips in all combinations. One of these objects could be checked l a t e r and i t turned out that the slope had become unstable. One other object turned out to be unstable before the evaluation of the datas. The s l i p process took place in more parts and reached catastrophic dimensions a f t e r a hard rain. The reason had been a draining furrow with a watershed of .75 km2. Further a schoal lithosol lay upon a layer of regosol which had been very firm and impenetrable. A prediction with counter-measurements in due time would perhaps have prevented this damage. I f you presume that the remaining six objects which had been stable at the time of measuring now are or w i l l become unstable, the prediction model following equation (4) is able to predict r i g h t 94 objects (= 94 %) of the testing set. Two slips and two stable slopes could not be predicted. One reason was the decreasing e f f i c i e n c y of the separation while l e t t i n g out predictors. Another was that a slope had been considered unstable but r e a l l y had been stable. One of the stable slopes showed layers with sand material where a fine-pored sand lay between gravel. The soil water in the fine-pored sand, the slope-gradient and the pressure of the upper soil-material probably lead to the i n s t a b i l i t y . Other objects in sand material with an impenetrable and firm layer are predicted as slips. The benefits of this model are to be seen in i t s s i m p l i c i t y and quickness. With a few values, which can be measured very easily, and a simple mathematical procedure you can predict slips an a slope. An example of the procedure is shown in table 8.
227
Table 8: EXAMPLEFOR A PREDICTION CSL = centroid for slips
} out of table 6
CST = centroid for the stable case using equation (4) predictor HN (o)
sample 1
sample 2
34
14
FL (km2)
.55
.20
EIND
.08
.60
.2098
.1760
/T-CsL/
.0164
.0274
/T-CsT/
.0317
.0021
T
The results are: the slope where sample 1 was taken is unstable. Sample 2 stands for the stable case. The introduced model is time invariant. This means that i t is impossible to pred i c t the moment when a s l i p w i l l take place. I t is possible to predict a s l i p taking place or not. I t works in all soil materials except for sands where slips cannot be very well predicted. The range of application of the model you w i l l find in constructing streets and highways where new slopes are b u i l t . Here i t is important to know very quickly i f slips are to be expected or not. Therefore you can use function (4). I f the application of the model would predict a landslip, appropriate counter-measures could be started. Another result of the model had been that the depth of the roots influences a s l i p . You should use equation (3) i f there is vegetation on a slope or i f you know the sort of vegetation which is planned to grow on a slope. With the application of the suggested model the s l i p process is not solved, because the problem is too complex to find an exact solution. LITERATURE ANDERSON, T.W. (1958): An Introduction to Multivariate S t a t i s t i c a l Wiley: New York, 374 pp.
Analysis.
BACKOFEN, K. (1957): Klassifikation der Rutschungen? Geologie und Bauwesen, 23, 125 - 130. BLESSING, G. (1966): Die Sicherung rutschgef~hrdeter H~nge. StraBen- und Tiefbau, 20, 218 - 220. BULLING, W.H. (1971): Bodenkennziffern und Klassifizierung von B~den. Springer, New York, 192 pp. BRYAN, J. (1951): The generalized discriminant function: mathematical foundation and computation routine. Harvard Educational Review, 21, 90 - 95. 228
CROZIER, M.J. (1973): Techniques for the morphometric analysis of landslips. Zeitschr. f. Geomorphologie, N.F. 1_]_7,78 - 101. DECOURSEY, D.G. (1973): An Application of Discriminant Analysis in Design Review. Water Resources Research, 9, 93 - 102. DIN 18 121Bestimmung des Wassergehaltes durch Ofentrocknung DIN 18 122 Zustandsgrenzen (Konsistenzgrenzen) FELLENIUS, W. (1947): Erdstatische Berechnungen. Dritte unver~nderte Aufl., von Wilhelm Ernst Verlag, Berlin, 48 pp. GOSSEFELD, J. (1973): Fortran IV-Programm JGNVT (unver~ffentlicht). HARTGE, K.H. (1971): Die physikalische Untersuchung von B~den. Enke Verlag, Stuttgart, 166 pp. HERRMANN, R. (1971): Zur regionalhydrologischen Analyse und Gliederung der nordwestlichen Sierra Nevada de Santa Marta (Kolumbien). GieBener Geogr. Schriften Nr. 23 (Sonderheft 1), Giessen, 88 pp. HERRMANN, R. (1974): Ein Anwendungsversuch der mehrdimensionalen Diskriminanzanalyse auf die AbfluBvorhersage, Catena 1, 367 - 385. HERRMANN, R. und SCHRIMPFF, E. (1975): Zur Vorhersage des AbfluBverhaltens in tropischen Hochgebirgen West- und Zentralkolumbiens. 40. Deutscher Geographentag, Innsbruck (im Druck). KARRENBERG, H. (1963) in: Geologische und bodenmechanische Ursachen von Rutschungen, Gleitungen und BodenflieBen. Forschungsberichte des Landes NordrheinWestfalen, Nr. 1138. Westdeutscher Verlag, Opladen, 63 pp. KIRKHAM, Don: Soil Physics, in: Ven te Chow (Hrsg.): Handbook of Applied Hydrology. Selection 5, 1 - 22, Mc Graw H i l l , New York 1965. KNOBLICH, K. (1967): Mechanische Gesetzm~Bigkeiten beim Auftreten von Hangrutschungen. Zeitschr. f. Geomorphologie, N.F. 11, H. 3, 286 - 299. KDHN-VELTEN, H. (1963) in: Geologische und bodenmechanische Ursachen von Rutschungen, Gleitungen und BodenflieBen. Forschungsberichte des Landes NordrheinWestfalen Nr. 1138. Westdeutscher Verlag, Opladen, 63 pp. LANGEJAHN, A. (1965): Some Aspects of the Safety Factor in Soil Mechanics: Considered as a Problem of Probability. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2, Montreal, 500 - 502. LIPPMANN, W. (1960): Messungen von Bodenbewegungen infolge zeitlicher Schwankungen im Wasserhaushalt. Diss. 1960, 72 pp. MATSCHAK, H. und RIETSCHEL, A. (1965): Vereinfachte FlieBgrenzenbestimmung an bindigen Bodenarten fur ingenieurgeologisch bodenmechanische Untersuchungen. Zeitschr. f. angew. Geologie, Bd. 11, 3, 135 - 139. MCQUEEN, J.S. und MILLER, R.F. (1974): Approximating Soil Moisture Characteristics From Limited Data: Empirical Evidence and Tentative Model. Water Resources Research, Vol. 10, No. 3, 521 - 527. 229
NEULAND, H. (1975): Zur Vorhersage von Hang- und B~schungsrutschungenmit Hilfe der Diskriminanzanalyse. Diss. K~In 1975, 55 pp. NONVEILLER, E. (1965): The S t a b i l i t y Analysis of Slopes with a Slip Surface on General Shape. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2 Montreal, 522 - 525. RODE, A. (1955): Das Wasser im Boden. Akademie'Verlag, Berlin, 464 pp. SAITO, M. (1965): Forecasting the Time of Occurence of a Slope Failure. Proc. of the Intern. Conf. on Soil and Mech. Found. Eng., 2. Montreal, 537 - 541. SBRESNY, N. (1941): Grundlegendes Uber den Stand der Untersuchung einer Rutschung Geologie und Bauwesen, 3, 25 - 26. SCHEFFER, F. und SCHACHTSCHABEL,P. (1966): Lehrbuch der Bodenkunde, Enke Verlag, S t u t t g a r t , 473 pp. SCHLICHTING, E. und BLUME, H.P. (1966): Bodenkundliches Praktikum. Parey Verlag, Hamburg, 209 pp. SCHOFIELD, R.K. (1935): The pF of Water in Soil. Trans. Intern. Congr. Soil Sc, 3rd, 2, Oxford, 37 - 48. SCHRIMPFF, E. (1975): Ein mathematisches Modell zur Vorhersage von AbfluBereignissen im Bereich der Anden Kolumbiens- SUdamerika. Diss. K~In 1975 (im Druck). SCHULTZE, E. und MUHS, H. (1967): Bodenuntersuchungen fur Ingenieurbauten. Springer, Heidelberg, 631 pp. SIEDECK, P. und VOSS, R. (1960): Die BodenprUfverfahren bei StraBenbauten, DUsseldorf, 124 pp. TERZAGHI, K. und PECK, R.B. (1961): Die Bodenmechanik in der Baupraxis. Springer, Heidelberg, 431 pp. TRAUZETTEL, G. (1962): Die Rutschungen im wUrttembergischen Knollenmergel. Arbeiten aus dem Geol.-Pal~ontol. I n s t i t u t der TH Stuttgart, Neue Folge (seit 1953) Nr. 3__22,143 pp. OBERLA, K. (1968): Faktorenanalyse. Springer, Heidelberg, 399 pp. WATZNAUER, A. (1965): Die Rutschung von Nieder-Tenzel (1941) - eine Korrektur. Zeitschr. f. angew. Geol., Bd. I i , 12, 667. WEIDENBACH, F. (1965): Die geologischen Voraussetzungen fur die Entstehung von Rutschungen. StraBen- und Tiefbau, 12, 1 - 8. ZARUBA, Q. u. MENCL, V. (1969): Landslides and Their Control. Elsevier Verlag, Amsterdam, 214 pp.
230