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Probability Concepts 

In Geomorphology

GEOLOGICAL SURVEY PROFESSIONAL PAPER 500-C

o 

CQ
Probability Concepts 

In Geomorphology

By A. E. SCHEIDEGGER and W. B. LANGBEIN

THEORETICAL PAPERS IN THE HYDROLOGIC AND 

GEOMORPHIC SCIENCES

GEOLOGICAL SURVEY PROFESSIONAL PAPER 500-C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Page

Abstract ______-____--_-___--_________--_----____-____.._----_----_________-_--_-__________ Cl

Introduction ______________________________________________________________________________ 1

Primitive stochastic models, by A. E. Scheidegger____i_-.____-___-____-___-_.__-______._______ 3

Random-walk models, by A. E. Scheidegger___________________..________.________-___-__.______- 3

Markov processes, by A. E. Scheidegger.._____________________..______------_----_---___________ 5

River geometry: minimum variance adjustment, by W. B. Langbein._____________________________ 6

General statistical methods, by A. E. Scheidegger.___________ _..________________________________ 11

Summary, by W. B. Langbein_-___-_____--------_--__--_-___---.-_-_-____----_-_---_--------- 13

References-_________________________-___________-__--____----__----------------------_-_--- 13

ILLUSTRATIONS

Page

FIGURE 1. Lotka's pendulums histogram showing the frequency distribution of the number of pendulums on one side of the central position________________________________________ C2

2-4. Graphs of slope development:

2. Basic slope development-________________._.._______________-_--_----_---_-- 3

3. Error function, the slope profile resulting from a random walk ____________._---- 4

4. Decay of slope bank if autocorrelation is present..__ ___________________------- 5

5. Diagram of a stream junction-_--_____________-__.._____.__________-_-____-_-__---- 6

6-10. Graphs:

6. Variation of velocity and depth of Rio Grande near Bernalillo, N. Mex____________ 8

7. Relations of width, velocity, and depth to discharge, Rio Galisteo at Domingo,

N. Mex___-._._.._._._._......._._______.______.___._.___-__----------- 9

8. Sum of squares of hydraulic exponents for velocity, depth, width, shear, and friction

factor for constant slope-_--_---_________..__________-__-____-__--------- 9

9. Total of 22+(m + z)2-Hm+/+s) 2 4-(l + z) 2 +(l + m+z) 2 --- ____________________ 11

10. Linear decay of a mountain range.______--__..--____-___-_-__---_---_------- 12

in
THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES

PROBABILITY CONCEPTS IN GEOMORPHOLOGY

By A. E. SCHEIDEGGER and W. B. LANGBEIN

ABSTRACT

Rivers and landforms that are produced by the action of 

flowing water may be dominated by random processes. The 

complexity of the phenomena seems to be beyond complete 

resolution of the forces and resistances involved. An assumption 

of randomness appears to offer a direct approach to a study of 

landforms and of the hydraulic geometry of rivers.

INTRODUCTION

To understand the development of a landscape in 

terms of fundamental mechanical principles is a problem of extreme complexity. Landforms are numerous, 

and there are many aspects of otherwise similar landforms. The processes that are operative represent 

the cumulative effect of many small-scale events which 

are impossible to follow in detail. This is a condition 

that is not unusual in geomorphology and other natural 

sciences; it is so common that methods of statistical 

mechanics have been developed to cope with it. Bakhemeteff (1941, p. 99) argued for the use of statistical 

mechanics in hydraulics.

Throughout the development of geological science, 

many direct approaches at an explanation of landscape 

evolution have been tried. However, in view of the 

great complexity of the phenomena involved, these 

approaches may lead to as many explanations as there 

are cases, meaning that there is no general theory. 

A hypothesis based upon the concepts of statistical 

mechanics (that is, a probabilistic rather than a direct 

approach), therefore, may yield a satisfactory and 

practical theory.

One of the common assumptions that are introduced 

in statistical mechanics is the randomness of some of 

the processes. Under certain circumstances, this 

randomness can lead to logistic difficulties because in 

geomorphology nothing is really random; when it 

rains, the fall of each raindrop is governed by Newton's 

law of motion and by atmospheric friction. When 

the drop hits the ground and erodes some soil particles, 

again the appropriate laws of mechanics determine 

the initiation of events. The same is true for the 

formation and deepening of small gullies that form,

say, on a slope and ultimately cause its decay. Yet, 

it would be quite hopeless to try to account for these 

processes in detail. As stated by Melton (1958), 

"The variability in any natural environment is the product of the happenings in many geologic periods * * *. 

To argue that this variability could ever be entirely 

explained is absurd."

Our knowledge of the individual events can never be 

complete enough to deduce therefrom, say, the recession 

of a slope, the juncture of channels, or the velocity of 

the flow that caused them. Nevertheless, certain 

average relationships can be deduced from our incomplete knowledge of the individual processes, simply 

because the net effect of the many individual events is 

the same as if the individual events were to occur at 

random, although the events are, strictly speaking, 

entirely predetermined. This assumption of randomness, for example, has been used in the theory of flow 

of fluids through porous media (Scheidegger, 1964a). 

The flow channels (pores) are entirely rigid and given 

in any flow process, but then* complexities are so great 

that the net effect of the superposition of individual 

flow effects is the same as if these flow effects were to 

occur at random. The random-walk generation of a 

drainage network (Leopold and Langbein, 1962), 

verified as realistic (Schenk, 1963), does not belie the 

fact that rivers join at specific places for specific 

reasons. But, the whole result is as if the process were 

random.

Each process is deterministic; however, the rates and 

periods differ, and the result may be indistinguishable 

from the random. Lotka (1925), for example, showed 

how a set of pendulums, each oscillating in a perfectly 

deterministic way, produced a net effect that is random 

in its characteristics. Lotka described a set of 26 

pendulums having periods of 0.5, 0.6 * * * 2.9, 3.0 

seconds; these were set in motion simultaneously and 

then allowed to oscillate undisturbed. As the pendulums oscillated, the number that were on one side of 

the plumb position changed continuously about a mean 

value of 13. A count was made of the number of

Cl
C2 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPH3C SCIENCES

pendulums on one side of the central or plumb position 

at the end of each tenth of a second (this being the 

difference in the periods) over a 25-second period after 

the pendulums had been in motion for some time (250 

observations). The resulting histogram shown in 

figure 1 conforms to a binomial distribution with a 

mean of 13 and a 0.5 probability that each of the 26 

pendulums will be on one side of the plumb position. The theoretical standard deviation equals 

V26X0.5X0.5=2.55, compared with 2.50 for the 

sample.

Among diverse landforms and among processes of 

land sculpture there are large numbers of examples that 

differ among themselves, not merely because of inhomogeneities of structure or of erosional resistance, 

but because readjustments to each action may not 

proceed at the same rates or time periods.

For an example of such readjustment consider the 

effect of a local depositional feature in a river channel. 

The chance emplacement of a bar, which slightly dams 

the stream, flattens the water-surface profile upstream 

from the obstruction and steepens the profile over the 

bar that is, the velocity will slacken above the barrier 

with the net result that deposition is encouraged and 

less material reaches the bar. Over the bar itself the 

steepened profile increases velocity and tends to erode 

the obstruction. This increased erosion in the oversteepened reach and deposition in the flat area will, 

over a period of time, tend to eliminate the original bar 

and thus restore the channel to its earlier condition 

(Leopold and Langbein, 1962).

The reaction to the initial perturbation takes time, 

and therefore restorative adjustment may not be com7 8 9 10 11 12 

NUMBER OF PENDULUMS ON

13 14 15 16 17 18 19 

ONE SIDE OF CENTRAL POSITION

FIGURE 1. Lotka's pendulums histogram showing the frequency distribution of 

the number of pendulums on one side of the central position.

plete before another chance event disrupts the adjustment, establishing a new series of events that inherits 

the characteristics of the old. If such readjustments 

are taking place, sporadically in time, not only at this 

one bar but at many, a variety of conditions will exist 

along the channel.

Many of these ideas are expressed in Hack's (1960) 

theory of dynamic equilibrium or in the steady-state 

theory of Chorley (1962). According to Hack, landforms can be explained on the basis of processes acting 

today through study of the relations between phenomena as they are distributed in space. This concept 

is close to the ergodic principle of statistical mechanics 

the replacement of time averages by space averages. 

The theory is independent of time and is concerned 

only with the relations between rocks and processes as 

they exist in space.

A homely example of Chorley;s (1962) illustrates independence of the steady state of history. A stream 

of water flows into a bucket that has a hole in the 

bottom. The discharge through the hole is proportional to the depth of water. The level in the bucket 

reaches an equilibrium such that average inflow equals 

average outflow this is a state of dynamic equilibrium. 

And then Chorley asks, was the bucket full or empty at 

the beginning?

The theory of dynamic equilibrium or the steadystate theory interpreted broadly stands in contrast to 

the formal stage-by-stage evolution of the landscape 

postulated by the Davisian geographic cycle. The 

dynamic equilibrium theory lends itself to quantitative 

analysis. The probability concepts outlined in this 

paper are of this school of thought. Both are uniformitarian in their approach.

The idea of using some type of statistical mechanics 

in geomorphology is, in principle, very old. When 

Davis (1924) spoke implicitly of the "average" geomorphic cycle, there was an element of probability 

implied hi the use of the very word "average." Whenever statistics are used, one ought to specify the "population" in which the " average" is taken. A consistent 

way of introducing statistical mechanics in geomorphology was suggested by Leopold and Langbein (1962). 

The method of these authors was later analyzed with 

regard to its stochastic basis by Scheidegger (1964b). 

Recently, Culling (1963) also introduced a statistical 

model which he based on the idea of a random-walk 

process.

In describing the steady state of rivers, a theorem of 

minimum variance adjustment among pertinent hydraulic factors is advanced. Its essence is that the 

increase in river discharge as it flows downstream is 

accommodated with minimum effect among the hydraulic factors, such as depth, width, and velocity.
PROBABILITY CONCEPTS IN GEOMORPHOI/OGY C3

The principle appears in this light to be an extension 

of Le Chatelier's principle a change in one element of 

a system in equilibrium is met by a reaction that tends 

to minimize the effect of the change.

Statistical mechanics may be adapted to the solution 

of geomorphic problems at three levels of sophistication. 

First, one can introduce primitive stochastic models in 

which semiempirical conjectures are made regarding 

the "average" behavior of a geomorphic system; second, 

one can postulate specific models, such as random-walk 

processes, and third, one can try to apply methods of 

ensemble theory (as originally introduced by Gibbs into 

the statistical theory of gases) into geomorphology. 

These various possibilities will be discussed, one by one, 

below.

Since these theories are of a very general nature, and, 

in fact, represent more of a philosophy of treatment 

than a closed and final scheme, their potentialities will 

be illustrated by a few simple examples taken from 

earlier papers by us that are listed among the references.

PRIMITIVE STOCHASTIC MODELS

By A. E. SCHEIDEQGER

The simplest way in which statistical (or quasistatistical) considerations can be introduced into geomorphology is by trying to set up a theory for the 

"average" pattern of evolution of a landscape. This 

theory is essentially the scheme which, albeit unconsciously, has been traditionally applied to describe 

geomorphology when one was speaking of "standard" 

or "characteristic" cycles and slope evolution. This 

type of reasoning can also be put into mathematical 

terms by setting up a theoretical model for the "average" 

evolution of a geomorphic element, neglecting the detailed "microscopic" phenomena.

Scheidegger (1961a), Bakker and Strahler (1956), and 

Souchez (1963) have tried setting up a theoretical 

model for evolution of slopes. Thus, Scheidegger 

(1961 a) reasoned that the denudation of a slope should 

proceed with a speed that is proportional to the declivity 

of this slope and that the denudation action would be 

normal to the slope. Denoting the height of a slopepoint in a cross section above some base line by h, 

the horizontal coordinate by x, and time by t, the above 

assumptions lead to the differential equation

where a is some constant of proportionality.

This differential equation is nonlinear, and hence 

solutions must be obtained by numerical computation. 

The evolution of an originally straight slope bank is 

shown in figure 2. The denudation of slopes having

varying lithology can easily be calculated (Scheidegger, 

1964c).

Although the general picture of the evolution of an 

average slope profile obtained in the present theory is 

not unreasonable, it would be desirable to specify more 

accurately the statistical concepts that are involved 

than can be done by the simple models that have been 

given.

RANDOM-WALK MODELS

By A. E. SCHEIDEGGER

The evolution of a landscape is primarily accomplished by a transport process: what is transported is 

simply the material that makes up the landscape. 

This material is in a relation to the height above base 

level of the points under consideration. The transport 

depends on the height above base level and involves a. 

constant which depends on the density decrease that 

occurs where the ground is broken up.

Taking the grand view, it is obviously possible to 

regard the evolution of a landscape itself as a transport 

process: what is transported are the materials that 

constitute the heights above base level. The transport 

process has a stochastic element in it, that is, there is a 

random element determining which "particle" moves 

at what time so that the transport process can be 

regarded as the result of a "random walk" of the individual "particles" making up the landscape. As noted 

earlier, the process is not truly random, since the 

small-scale effects involved are all mechanically completely determined. However, the mechanics of these 

small-scale effects is so complicated that the details 

concerning most individual particles will remain forever 

unknown. Because of this, the process of landscape 

evolution can be treated as if it were random. As 

previously explained by Scheidegger (1964a), the following three types of specifications have to be made to

1.0

0.8

0.6

0.4

0.2

x = horizontal coordinate 

y= vertical coordinate 

t= Total time 

(arbitrary units)

0.2 0.4 0.6 0.8

FIGURE 2. Basic slope development.
C4 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES

arrive at a valid transport model for such random-walk 

processes :

1. One must specify the ensemble. Such ensembles 

may consist of all systems that can be considered 

as identical to the sought-after relationships. 

To give some specific examples : it one considers 

a slope profile, then the ensemble includes all 

normal cross sections of a slope bank having 

certain properties, and the same would apply if 

one were to consider a mountain range; if one 

considers rivers having a given distribution of 

discharge, the ensemble includes all rivers having that distribution of discharge. It is for 

averages over such statistical ensembles that 

relationships are sought.

2. One must specify a priori the statistics within the 

ensemble. This includes questions as to 

whether the events are independent or correlated. The evolving patterns depend very 

much on the choice of statistics.

3. Finally, one must specify the microdynamic 

transport law.

Let us again illustrate the general theory of the 

evolution of a slope regarded as the result of a randomwalk process.

Case a. Assume that the phenomena occurring in 

the individual time steps are completely independent of 

each other. Several subcases can be distinguished.

Case a i. There are no microdynamic forces; that is, 

all changes are determined by stochastic processes. In 

a geomorphological context, this means that the 

"particles" on a slope are simply performing a random 

walk. Every "particle" on a slope has a certain chance 

to move forward; this chance is not dependent on conditions along the slope. Because of the central-limit 

theorem of probability theory (see, for example, Mises, 

1931), the distribution function of the "particles" (and 

therewith of geomorphic height h) is subject to a diffusivity equation

where t is time, and D is a diffusivity constant. The 

pattern of the decay of a slope profile corresponds to 

the solution of the diffusivity equation

where erf denotes the "error" function. This equation 

states that when the particles are subject to movement 

as described, they become distributed along the slope 

length, such that the height h, at any point x, takes the 

form of the "error" curve. The pattern is shown in 

figure 3.

-cO.5

-3 -2 -1

FIGURE 3. Error function, the slope profile resulting from a random walk.

Case a n. Assume now that there are microdynamic 

forces, that is, forces in addition to those causing the 

random motion of the "particles." As previously explained in regard to the hydrodynamics of porous media 

(Scheidegger, 1964a), this introduces a mass-transport 

term in the diffusivity equation so that it reads

where c is the average mass-transport velocity, that is, 

in a frame of reference moving with velocity c, one 

would again have the diffusivity equation. In fact, 

Culling (1963) has already investigated problems of 

this type where h, instead of denoting the height of a 

landscape, refers only to the thickness of an active layer 

(say, "soil") lying upon a slope; then any "accumulation" will gradually move downslope and spread out.

A possible generalization of the diffusivity equation 

can be arrived at if D is no longer considered as a scalar, 

but as a tensor. Then, in customary tensor notation

where the summation convention is assumed to be applied (see, for example, Jeffreys, 1932). Culling (1963) 

calculated several cases corresponding to the preceding 

two equations, which refer, as noted above, to the creep 

of a soil layer of thickness h over a slope.

The present context concerns an interpretation of 

landscape height. Then, the mass- transport coefficient 

c, if not taken as zero as in case a i, will presumably depend on the declivity of the slope. Since this coefficient 

describes the mean motion of "particles," one will presumably have to refer to the expression corresponding 

to that introduced earlier. There it was assumed that 

the vertical speed of lowering of a slope with declivity 

was given by

 >k 7 \ 9

ohY -)
PROBABILITY CONCEPTS IN1 GEOMQKPEDOI/OGY C5

where a is some proportionality constant, 

horizontal speed c will be

Hence, the

C='t

Thus, the differential equation (in one dimension) becomes _____ 

dA_ n d2h fdh\ I-i ./^V

which is nonlinear. It is, in fact, the same differential 

equation as in the section on primitive stochastic 

models, except that now a diffusivity term is added on 

its right-hand side.

Case b. Assume that there is autocorrelation between "particle motions" in succeeding time steps.

This case is again analogous to one occurring in the 

physics of flow through porous media (Scheidegger, 

1958), where it has been shown that one now arrives at 

a "telegraph" equation (so called because it describes 

the electric behavior of a long telegraphic cable) rather 

than at a diffusivity equation. In one dimension this 

telegraph equation is

~W~Ato?*Ato =()

where A and D are two constants. The chief difference is the occurrence of the second time derivative 

which gives the equation some of the character of a 

wave equation. The solution of an originally vertical 

slope bank (at t=0,.H=l for z>0, zero otherwise; 

with u.= (D/Ay/2 is (Goldstein, 1951)

h(x t}-- fi(x, t)-,

, In 2Au 

=0 (otherwise).

The situation is shown in figure 4.

1.0

0.5

FIGURE 4. Decay of slope bank if autocorrelation is present. 

223-837 O 66 2

MARKOV PROCESSES

By A. E. SCHEIDEQGER

The random-walk model without autocorrelation, 

discussed earlier, represents a special case of a rather 

general type of stochastic processes, namely, of Markov 

processes. Such processes are characterized by the property that the probability distribution of whatever 

stochastic variable is under consideration is completely 

determined for all C>£o by the value of the variable at t0 .

Let us consider the "average" profile of a slope bank. 

Its state, at a particular tune t, is given by the probability distribution (not normalized) h(x, t) of a 

continuous stochastic variable. We identify this distribution with the "average" profile of our slope bank 

(h denoting the height above a certain base level). 

We then assume that the evolution of the slope bank 

during the time interval t, t-\-dt is determined solely 

by the state at time t. As noted, processes where 

this property obtains are called Markov processes.

One should again remark, of course, that there is 

nothing probabilistic about the actual development of 

the slope bank. The laws of mechanics determine 

what happens to each soil particle. However, we do 

not know the exact configuration of a particular slope 

profile; we know only its probable configuration within 

a probable ensemble of slope profiles. This probable 

configuration, in spite of the determinism of the laws 

of mechanics, changes with time, and hence transition 

probabilities from one state to the next can be set up.

We have introduced h(x] as a probability distribution 

which changes with tune. This distribution, resulting 

from a Markov process which is continuous in space 

and time, can be described by the corresponding 

Fokker-Planck (sometimes also called Forward-Kolmogorov) equation, which has the form of a general 

diffusivity equation (see Bharucha-Reid, 1960)

x) , x}\ D[b'(t,x)-h(t,x)]

Here., a! and b' are "coefficients" equal to the time 

derivative of the variance and mean of the stochastic 

variable.

The mere assumption of the evolution of the state 

of a landscape corresponding to a Markov process, 

therefore, entails a description of this evolution by 

means of a diffusivity-type (Fokker-Planck) differential 

equation. The specification of the coefficients a and b 

allows one enough latitude to describe a variety of 

cases. A particular choice for a and b leads to the 

random-walk theory explained in the preceding paper; 

however, there is the prospect that full generality of 

the cases that can be adduced is not yet clear.
C6 THEORETICAL PAPER'S IN THE HYDBOLOGIC AND GE'OMORPHIC SCIE!N E,S

RIVER GEOMETRY: MINIMUM VARIANCE 

ADJUSTMENT

By W. B. LANGBEIN

Consider a simple network, as shown in figure 5, 

of the channels in a homogeneous noncoherent alluvium. 

The problem concerns how the channels represented 

accommodate the given discharges. The increased 

discharge below the junction might be taken up in 

increased width, depth, or velocity, or any combination with corresponding variations in slope, shear, and 

friction. The problem is examined on an assumption 

that the increased discharge is accommodated with 

least variance in these several properties.

Use is made of the Leopold-Maddock (1953) model 

of the hydraulic geometry of rivers in which the dependent variables among the several river properties vary 

in terms of discharge, Q, as the independent variable

velocity ~Qm, 

depth ~Q', 

width ~Q6 ,

since log v= constant -f-ra log Q, etc.;

J Vlog Dialog Q> 

0 0" log lo/O'log QThus, the hydraulic exponents of the dependent 

variables equal the ratios of the standard deviation of 

the logarithms of a dependent variable to the standard 

deviation of the logarithms of the independent variables. 

The squares of these ratios are called variance.

The model in this treatment is linear with regard to 

the hydraulic exponents or the logarithmic standard 

deviations. Thus, those standard deviations are related 

to each other by simple linear equations. For example,

which, according to the Leopold-Maddock model 

becomes

or- <? '<?*<?

or

m-h/+6=l.

FIGURE 5. Stream junction.

This follows from the relationship involving the 

hydraulic indices which are exponents. However, 

the same results are obtained from the statistical 

statements.

Given

therefore

and

If, as in the model, the several coefficients of correlation, r, are each equal to unity, then,

Q=vXDXw; 

log #=log w-flog D+log w

+ 0Io, «

Q »~T 0'logZ)~r 0"lOg w)

°" log Q = °" log v ~r 0" log D ~T ff log w

1 og " I fflog D i °" log 'to 

"

°"log Q fflog Q

l=ra+jH-&, as before,

 >=<TlOg1>!<r\OgQ', /=0'lOg£/0'lOg QJ b =

and

or 

and

With actual data and a finite standard error, the 

coefficient of correlation can be as close to unity as 

desired by extending the range of the data.

A river is described by its several hydraulic properties, which are presumed to be distributed at random. 

Let each property be represented by a standardized 

variate (x' x')/e, where x' is the logarithm of a given 

property (such as velocity, depth, and width); xr is the 

logarithmic mean of the property at N points of observation, each associated with a value of the independent 

variable; and e is the standard deviation of the logarithms of the hydraulic properties. Similar standardized variates exist for x", x'", etc. Moreover, the 

standardized variates of the several properties are 

initially assumed to have the same value of c and to

constitute a homogeneous population. If all properties 

are independent, then the most probable combination 

is that where

[(x"-x")/e]2+etc.=A minimum.

Letting

JV 

1

JV

E
PROBABILITY CONCEPTS IN GEOMORPHOIX)GY 07

where N is equal to the number of points where observations are made of x', x", etc., and c is a factor common 

to all variables. The above statement is equivalent to 

the statement that

=a mnmum.

If the ff's are each divided by standard deviation of 

the sum (x'+x"-|-etc.) at the points of observation (all 

in logarithms), then the ratios correspond to the hydraulic exponents.

The independence among values of the hydraulic 

properties, referred to in the above equation, does not 

preclude constraints that specify relationships among 

the several factors. An illustration will show what is 

meant by constraints.

Assume that each river of a large ensemble of many 

identical rivers is characterized by several distinguishable factors, 'Fi, F2 , Fz, etc. These values have a 

mean and a standard deviation among the rivers which 

may be modeled by tossing a large number, N, of fair 

devices (such as coins, dice) each with F distinguishable 

faces. Each toss of the N devices represents a river. 

The relative frequency, k1} k2 , etc., among N devices 

represents the values of Fi, F2 for that toss or, as it 

were, for that "river." The question concerns their 

most probable values. It is asserted here without 

proof that the most probable frequencies occur when

4- (&2/A0 2, etc. = a minimum,

where k\, k2 , etc., is the frequency of each distinguishable 

factor and where ki-{-k2 , etc. = TV. This statement, 

without constraint, is obviously satisfied when ki=k2 , 

etc.

To illustrate the effect of a constraint let us now 

consider only those tosses in which k2=2ki. This 

means that even though all tosses and all devices 

remain independent of the others, only those tosses 

of the N devices in which k2=2ki qualify. What are 

now the most probable values for k\, k2, kz, etc.? 

Introducing this constraint in the sum of squares

+ (k*INY+ ... a minimum.

The general case has not been worked out but consider the case where there are three factors, FI, F2 , and 

F* Then

or

*+ (1 -3fc!/A02=a minimum, 

Ar,A/V=0.215,

These values can be verified for large values of N by 

the ordinary statistics of combinations. The product 

p(k^'p(2k1 )-p(N 3^) is a maximum when C^-C^ 

-CN- 3kl is a maximum. This product is a maximum 

whesn ki=0.21 5N for 1 arge values oi N. '. , Although in the model the factors FI, F2 , etc., are 

drawn from a population having identical standard 

deviations, by stating a constraint kz=2ki, the standard deviations of the factors FI, F2 , etc., as well as their 

most probable values, become distinctive. Note, too, 

the sum of squares gives the most probable values 

without need to develop the separate frequency 

distributions.

One may also observe that the introduction of 

constraints increases the sum of squares. The fewer 

the constraints, that is, the more relaxed, the lower 

the sum of squares. For example, the sum of squares 

without constraint is 0.33; the introduction of the 

constraint &2 =2&i increases the sum of squares to 0.36. 

Effects of constraints and relaxations on hydraulic examples will also be shown.

The statement that the sum of squares of the variables is a minimum is very close to the principle of 

total least work in the relaxation adjustment of linear 

networks, such as structural frames. Southwell (1940, 

sec. 98) stated, "In a framework we have to determine 

that distribution of joint displacements which, subject 

to the overriding condition of continuity, entails a 

minimum value of the total energy this being (by 

Hooke's law) a quadratic function of the displacements." In a network of pipes, least work occurs 

when the product discharge (Q) by head loss (H) is a 

minimum, that is, when

=& minimum.

In a linear network H=aQ, where a is a constant of 

proportionality of head loss to discharge; therefore,

and

s a mnmum

zH2/a is a minimum.

If the diversity among the several branches is not 

extreme, then with little loss of accuracy one may use 

average values; thus,

s a mnmum.

In a set of trial solutions for a given network, it will 

be found that one quantity increases as the other decreases. Thus, decreasing variability in one quantity 

is purchased at the expense of increasing variability of 

the other. The relaxed solution occurs when the sum 

of squares is least.
C8 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES

Le Chatelier's principle as defined in the American 

Geological Institute Glossary (1957, p. 166) can lead 

directly and simply to a statement of minimum variance adjustment. For example, consider an increase 

in discharge in a flume in which flow had produced a 

stable regimen. Or, consider the addition of flow at a 

junction with a stream that has a stable regimen above 

the junction. The increase produces changes in velocity, depth, width, shear, and possibly other things. 

Since the regimen is stable, according to the Le Chatelier 

principle, each of these changes produces effects in the 

others that react against the change. The effect is the 

same as stating that each factor resists change, or in 

other words, the variance of each tends toward zero. 

Since all of them cannot be zero, it appears that the 

adjustment is such that the total variance is minimal.

The concept of the probability of a state of nature 

raises philosophical difficulties that are at present 

controversial. One difficulty is that the concept is 

not operational. It does not give rules for deciding 

which data are relevant and therefore to be included 

in the "state of nature."

Such knowledge must come from the mechanics of 

the processes. Hydraulic geometry is described by 

such properties as width, depth, hydraulic radius, velocity, shearing force, friction factor, sediment transport, intensity of power expenditure, and possibly 

others. There is therefore a complex mix of properties 

that might be involved in a minimum variance adjustment. However, not all have equal weight. Their 

relative weights could be resolved by data. However, 

initial inquiry indicates that a large set of problems 

can be explained by giving the dominant factors; 

width, hydraulic radius, velocity, shear, and friction 

factors a weight of unity, and the others a weight of 

zero. The matter is simplified if the channels are 

sufficiently wide that mean depth equals the hydraulic 

radius.

Five cases are considered, as successive factors are 

relaxed or constrained, and these cases are compared 

with laboratory and field data.

Case 1. If width and slope are constrained, then 

all the increase in discharge below the junction must 

be accommodated in depth and velocity and accompanying changes in shear and friction

Since discharge is the independent variable, it may 

be assigned a variance of unity. Width and slope 

have zero variance. If the variance of depth is represented by f 2 , then those of all dependent variances are

velocity_______________ (1 f) 2

Dependent variable 

Velocity.. ________ _

Depth _ _________ _

Shear. _ _________ _

Friction. _________ _

Sum. ___________

exponent 

_-_--__.- 0.42

-_--_____ .58

-_--.____ .58

___.____. -.25

Square 

0. 17

O A

.34

(\C

. 94

(3f 2) 2 .

. The variance of velocity follows because velocity equals 

discharge divided by depth and width, and width is 

constant. Shear equals the product of unit weight of 

water, depth, and slope. Since unit weight and slope 

are constant, shear varies only with the mean depth. 

Friction (the Darcy-Weisbach friction factor) with 

slope constant is proportional to the mean depth 

divided by the square of the velocity. It is assumed 

that the mean depth equals the hydraulic radius.

If the change in discharge is accommodated equally 

as between changes in depth and velocity, then f=0.50. 

If, in addition, the variances in shear and friction are 

also minimized, then f=0.58. Actual data give results within this range. These results indicate that 

when width and slope are constant, then 50-58 percent 

of a change in discharge is accommodated by change 

in depth, and the remainder by change in velocity.

The sum of squares of the dependent variables is 0.94, 

computed as follows:

Hydraulic

A multiple regression of the 42 runs for which bedforming processes were active in a set of constant-width 

flume experiments reported by Simons, Richardson, 

and Albertson (1961) indicated that depth varied as 

tbe 0.52 power of the discharge for constant slope. On 

the Rio Grande at Bernalillo, N. Mex., where the riverbed is sand and width and slope are constant over a 

wide range of discharges, depth varies as the 0.55 power 

of the discharge. (See fig. 6.)

10

8

Velocity

10

LU 

U_

z

I

ui 2'Q

500 1000 2000 5000 10,000 20,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 6. Variation of velocity and depth of Rio Qrande near Bernalillo, N. Mex., 

1948; width between 270 and 280 feet.
PROBABILITY CONCEPTS IN1 GEOMOEPHXXLOGY C9

Case 2. Width is relaxed and becomes a dependent 

variable; slope is still coastant. Again taking discharge 

as the independent variable, it may be assigned a 

variance of unity. If the variances of the dependent 

variables, velocity and depth, are represented by

velocity._______________________ m2

depth._________________________ /2,

then those of the remaining dependent variables are

width.._______________ (1-m-/) 2

bed shear______________ /2

friction______________ (/ 2m) 2.

Summing these five terms, one finds that a minimum 

is reached when ?M=5/23 and /= 7/23. These results 

signify that 22 percent of a change in discharge would 

be accommodated by a change in velocity, 30 percent 

by a change in depth, and consequently, the remaining 

48 percent by a change in width. The sum of squares 

is now reduced to 0.478.

Figure 7 shows the way in which the Rio Galisteo at 

Domingo, N. Mex., accommodates changes in discharge. 

The river at this gaging station is at virtually constant 

slope, and the bed and banks are of medium sand. 

The slopes of the graphs shown correspond to values 

of m=0.24 and/=0.27. Results obtained for stream 

channels in noncoherent materials that are reformed 

with each change in discharge as, for example, the 

continuously reforming channels below Emmons Glacier, vividly described by Fahnestock (1963) gave values 

of m=0.27 and/=0.33. However, these results apply 

to different channels having different slopes. Graphical 

analysis of a set of flume experiments with channels 

developed in noncohesive 0.67 millimeter sand by Wol200

I III III

1 2 5 10 20 50 100 200 500 1000 2000 5000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 7. Relations of width, velocity, and depth to discharge,. Rio Qalisteo at 

Domingo, N. Mex., 1957-58.

man and Brush (1961) gave values of m=0.22 and 

/=0.29, at constant slope.

Four "at-a-station" cases (slope constant) are considered. The sum of squares is shown by the family 

of curves on figure 8. In case 1 width, as noted, is 

fixed. This solution lies along the line 6=0. In case 2 

width, as noted, is free to adjust to each flow, and there 

is no constraint. The solution occupies the "eye" of 

the family of curves. In case 3, which has not yet 

been discussed, there is the constraint that depth varies 

as the square of the velocity. The solution lies at a 

point of tangency to the line/= 2m.

Case 2A. This case is shown in figure 8 and describes 

the usual "at-a-station" case on a river in a humid 

region. The channel is developed at a bankfull or 

channel-forming discharge, and the banks are sufficiently cohesive that the channel is not reformed at 

lower flows. This relation of channel to discharge is 

described by the constraint 6=0.55/, where the value 

of 0.55 is the slope of the graph between mean depth 

and width on logarithmic chart as given by the sine 

cross section cited by Nizery and Braudeau (1955).

Case 2B. It is of interest to relax m2+/2 +62 

+ (/+ 2) 2 +(/+2 2m) 2 , these terms again representing 

velocity, depth, width, shear, and the friction factor, 

without constraint, giving m=l/7; /=3/7; and 2 

 2/7, where z is the hydraulic exponent for slope. 

The sum of squares is 0.428, lowest for cases 1-3. 

This case and the figures obtained correspond with the 

flume experiments reported by Ackers (1964), who 

obtained the hydraulic gradients: m=0.15; /=0.42;

0.7 r

^ ^ Constant 

concentration 

(Case 3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

FIGURE 8. Sum of squares of hydraulic exponents for velocity, depth, width, shear, 

and friction factor for constant slope.
CIO THEORETICAL (PAPERS IN THE HYDROLOGIC AND GEOMORPHIC SCIENCES

6= 0.43 ; and z uncertain. Because in these experiments 

there is no continuity of sediment, this case is not to 

be confused with a canal network, case 4, that will be 

discussed.

Case 3. It is specified that sediment transport varies 

in proportion to the rate of discharge and that slope is 

constant. Load per unit width is proportional to mean 

velocity cubed for mixtures of sediment of different 

sizes (Thomas Maddock, Jr., written commun., 1965). 

Then, because load per unit volume is constant, depth 

must vary as the square of the velocity. In this case, 

the respective variances are

width_____________.___ (1-1.5/) 2

shear_____ ______ _______ (/) 2

friction. _______________ 0.

Summing these terms, one finds that a minimum is 

reached when /= 1/3 and therefore m=l/6 and 6=1/2. 

Because of the addition of a constraint, the sum of 

squares is increased to 0.50. There are no flume experiments that apply to this case.

Case 4 (straight canals). If slope is relaxed, or in 

other terms, slope also is a dependent variable, and 

there is continuity of sediment as well as of water, then 

a longitudinal profile is developed and a system of 

canals is approximated. In this case, the profile can 

develop so that the variation in shear and friction is 

equally divided, for the above sum of squares is a 

minimum only when this is true. The variance of 

shear is (J-\-z) 2 and that of friction (j-\-z 2m) 2 =z2. 

Since uniform charge requires that m=0.5/, these two 

variances are equal when z= m. However, the profile 

developed also is free to adjust so that the several 

aspects of stream power are as equably accommodated 

as is possible. Various aspects of stream power and 

their variances are

Power per unit 

Length per unit discharge.

Volume. _________________

Variance

QAH 

QAL

QAH

Bed area 

Length. -

QAH

wAL

~vDs

Time of stream travel(Rubey, 1938).

AL

~Qs 2) 2

(1 + m+z)2 .

The total variability is therefore the sum of the 

squares

Since m=0.5/ and z=m, this sum of squares is a

minimum when z= K, and ra=%, /=0.33, and 

6=0.50. These values correspond with those that have 

long ago been derived empirically (Lacey, 1930) for 

"regime" canals in homogeneous noncoherent materials. 

This analysis contains as implicit or explicit conditions, 

steady discharges in each reach, uniform sediment 

concentration along the channel, and active sediment 

movement in the range satisfied by Maddock's statement of transport.

The theoretical basis of regime equations for stable 

channels rests, it is suggested, not so much on considerations of dynamics as on statistical mechanics.

Case 5 (rivers, downstream). Rivers have additional liberties and are even further relaxed than canals. 

Rivers, for example, have no lateral constraint and are 

free to meander. For the river case there are no 

constraints save the condition regarding sediment concentration or charge. In rivers, as is well known, 

sediment concentration decreases downstreamward 

because of flatter land gradients. Although this reduction in sediment concentration has been in the past 

associated with discharge (Brune, 1948), slope seems a 

better parameter because it is more directly related to 

the factor directly responsible. An analysis of existing 

data shows that concentration varies about as the 0.25 

power of the slope. Since, as before, load per unit 

width varies as the velocity cubed, the condition that 

would apply to rivers is m=K/-j-0.122.

The total sum of squares is then a minimum when 

w=0.12-|- and z= 0.55. Using these figures, it 

follows from ra=0.5/4-0.122 and m+/+6 = 1.0 that 

y=0.38 and 6=0.50, results that have close resemblance to nature (Leopold and Maddock, 1953).

If concentration remains constant (that is, ra=K/), 

then the sum of squares, without other constraint, is a 

minimum when 2= 0.56, m=%, /=K, and 6=0.50. 

The ra, /, and 6 values are the same as the canal case, 

whereas the hydraulic exponent for slope is the same as 

the river case. Evidently the slight downstream decrease in concentration does not affect either the profile 

or width. The decrease in concentration is accommodated entirely by changes in velocity and depth.

These relationships are clearer on figure 9, which 

shows a family of curves defining the sum of squares 

22+(m+z) 2+(m+/+2) 2 -f-(l + 2) 2+(l+m+2) 2 in terms 

of m and z. The diagram shows the curves for the 

condition that ra=}_/ (constant concentration) and 

that m=K/+0.122 (downstream decrease in concentration). The two sets of curves differ only for large 

values of 2. This diagram may help to show how the 

"profile" cases are related to each other:

1. The canal case is subject to the constraint as 

derived above for case 4 that variance in 

shear and friction are equally divided, and so
Cll

/Solid lines: m = 

Dashed lines:m =Vz f+.122

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 

VALUE OF z

FIGURE 9. Total of zi+(m+z)i+(m+f+z)*+(l+z)*+(l+m+z)i.

z= m. A line corresponding to this condition 

is shown on figure 9. This line is tangent to 

one of the family of curves at a point defined 

by m z=% and therefore /=0.33, and 

6=0.50, as before.

2. There are two minimum sum of squares, without

constraint. That minimum defined by the

curves for which m ^j and which occurs at

2= 0.56 and m=0.167 corresponds to the

relaxed canal case just dscribed above. The

minimum sum of squares defined by the curves

for which ra=K/-|-0.12s and which occurs at

z= 0.55 and ra=0.12 described the river case.

The agreement between computed and observed

hydraulic exponents appears to suggest that the

mutual adjustment among them is indeed toward

minimum sum of their squares.

One notes that the mutual adjustment among the 

dependent or adjustable parameters need not be 

unique but may consist of a set of values distributed 

about some modal value which can be ascertained 

from a number of examples as a statistical mode or 

mean. Average values exist; averages for flume 

experiments are replicable. Any one example displays 

equilibrium even when it varies to some degree from 

this mode or most probable case. The total variance 

as indicated by the sum of squares of the several 

examples will be relatively conservative. Wide departures that would contribute greatly to the sum 

of squares rarely occur, and if they do, they represent 

temporary conditions in progress toward a more stable 

form, as indicated by a lower sum of squares of the 

pertinent hydraulic exponents.

GENERAL STATISTICAL METHODS

By A. E. SCHEIDEGGER

Recently, Leopold and Langbein (1962) postulated 

empirically an analogy between the statistical evolution 

of g;eomorphic features and thermodynamics. 

Scheidegger (1964b) then investigated the underlying 

physical conditions for the validity of this analogy and 

showed that, indeed, a reasonable microdynamic model 

could be set up which justified the Leopold-Langbein 

hypothesis.

A general statistical model of landscape evolution 

can lead in a broad context to theories of the nonequilibrium and nonstationary state. The basis of 

this development lies in the idea of considering a 

particular landscape as a sample of a Gibbsian ensemble. Only statistically valid statements can be 

deduced for any particular landscape rather than 

unique solutions. The fact that there is a conservation 

law of the mass contained in a landscape permits one 

to introduce a Hamiltonian function and canonical 

equations, terms that are defined in any textbook on 

analytical mechanics. The further development of the 

theory is then automatically the same as in statistical 

thermodynamics (for this theory see Sommerfeld, 1964).

The general statistical theory in geomorphology of 

concern here is based upon the concept of conservation 

of mass. In a one-dimensional section of a landscape 

(this may be an averaged, representative section) the 

distribution of heights h can be treated as a statistical 

function. If we assume that mass transport can occur 

only within this longitudinal section, then the sum 

total (that is, the integral) of all the heights taken over 

the section must be a constant of the motion. In 

order to have a system with a finite number of degrees 

of freedom, one may split the length L of the section 

under consideration into N equal segments. For the 

iih segment, the (mean) height is ht. Thus,

i=N

must be a constant of the motion and can be treated 

as Hamiltonian. It has been shown (Scheidegger, 

1964b) that canonical variables can be introduced by 

setting

so that

The function U is the interaction function between the 

various positions indicated by the index i along the 

section.
C12 THEORETICAL PAPERS IN THE HYDROLOGIC AND GEOMGRPHIC SCIENCES

One can immediately define a partition function at 

point" i by

= f+m f+

J CO J 00

If the interaction function U is small, this yields

Vis

with

(7s

Thus, the average height comes out as

Using the usual thermodynamic functions, we see that 

the canonical temperature is

I 2h -

if we set k=2. There is, therefore, a complete analogy 

between temperature and average height.

An equilibrium theory sketched above, which is 

discussed in detail by Scheidegger (1964a), justifies 

certain analogies that already have been made in 

geomorphology. However, of greatest interest are its 

extension to nonequilibrium processes.

A general way by which the theory can be extended 

to nonequilibrium processes has been indicated by the 

present author (Scheidegger, 1961b) in a different 

context. For all practical purposes, this can be 

accomplished by using Onsager's relations (DeGroot, 

1951) and making the assumption that the fluctuations 

from equilibrium regress linearly. The generalization 

has been carried out, for example, by DeGroot (1951, 

p. 41 ff.) for the conduction of heat in solids; the theory 

for the time dependence of the temperature distribution in solids using the canonicity of the variables 

introduced earlier can be taken over as valid in general 

for any constant of the motion of a large system that is 

made up of positive definite contributions from component systems. One thus has, quite generally, as 

shown earlier (Scheidegger, 196 Ib), the following 

theorem: Whenever a system is a linear combination 

of a large number of fluctuating component systems, 

and in the "large" systems a certain quantity is a 

constant of the motion to which the component systems 

contribute positive definite amounts, then, under 

equilibrium conditions, that quantity is canonically 

distributed in the component systems, and under

nonequilibrium conditions, assuming linear regression 

of the fluctuations and microscopic reversibility, the 

quantity in question is subject to a diffusivity equation 

with a symmetric diffusivity tensor.

The application of this theorem to the heights in a 

landscape immediately yields

dx

with i=l, 2 (referring for example to xt = "north" 

coordinate and £2 = "east" coordinate, respectively), 

A=the (statistical) height, £=time, and Di} = diffusivity tensor; furthermore, the summation convention has been used.

Two exemplary applications of the above diffusivity 

equation will be investigated. A very well known 

solution of the diffusivity equation in two dimensions 

(h, x) is given by a Gaussian curve

h=

This can, for example, be regarded as the time dependence of the decay of an (average) profile of a mountain 

range. As time proceeds, the mountain range gradually 

flattens out. The phenomenon is shown in figure 10.

Another well-known solution of the diffusivity equation in two dimensions (h, x) is

where erf is again the error function.

This solution can be regarded as representing the 

decay of the (average) profile of a slope bank. It is the

FIGURE 10. Linear decay of a mountain range.
PROBABILITY CONCEPTS UST GEOMQRPHODOGY C13

same as that obtained in the random-walk theory 

without autocorrelation (fig. 3).

The theory given above is completely analogous to 

the theory of transport processes with mass conservation. The latter encompass such processes as sediment 

transport hi rivers, the spread of a contaminant in a 

porous medium, and the turbulent mixing of poisonous 

gases (see Scheidegger and Chaudhari, 1964). In transport processes with mass conservation, the Hamiltonian 

refers to a quantity that is conserved in the "large" 

system.

All the above cases, in turn, are analogous to conditions where the Hamiltonian refers to the energy, 

that is, to what is commonly called thermodynamics. 

The discussion of the last paper referred to (Scheidegger 

and Chaudhari, 1964) is directly applicable to the 

geomorphic problem. Thus, the interaction between 

the points of a landscape can be taken into account by 

defining a suitable interaction function in the Hamiltonian. In the Liouville representation of statistical 

mechanics (see Prigogine, 1962), the equation describing 

the evolution of the probability distribution function 

for the geomorphic heights becomes then very similar 

to a quantum-dynamic Schrodinger equation. (See, 

for example, Sommerfeld, 1964.) Similar approximation procedures, as are well known in quantum theory, 

can therefore be applied.

The proper choice of the interaction function between 

the various points on a map is not yet clear. If the 

analogy between landscape evolution and temperature 

conduction is exploited, various types of diffusivity 

equations are arrived at for the geomorphic problem. 

These equations can describe the statistical evolution of 

a landscape.

SUMMARY

By W. B. LANGBEIN

There is today a considerable growth in geomorphology toward a common set of ideas implicit in Chorley's 

"General Systems" theory, Hack's "Dynamic Equilibrium," or in the statistical concepts described in this 

paper. In each theory the landscape is viewed in terms 

less rigid than in the Davisian geographic cycle, with 

its considerable emphasis upon stage or history.

Dominant, perhaps, is the idea of the landscape as an 

open system maintained by a continuous (though not 

constant) supply and removal of material. The steady 

state exists when there is an average equality between 

supply and removal. This equation of material is accomplished by means through adjustment of the geometry of the system itself (Chorley, 1962, p. 3). The 

steady state is believed to correspond with Hack's dynamic equilibrium. It is, however, tune independent;

that is, different initial conditions may yield similar 

results (Chorley, p. 8).

Because the steady state is rarely characterized by 

exact equilibrium, the landscape itself manifests a tendency toward a mean condition of forms, recognizable 

statistically, about which variations may occur 

(Chorley, 1962, p. 7).

These views that the open-system steady state of the 

landscape can be described statistically have led us to 

an attempt to introduce consistent methods of giving 

these concepts quantitative expression. The theories 

of diffusion processes, random walks, and Markov processes provide some of the techniques. Minimum variance theory offers a method for estimating the variation in the geometry of rivers and possibly of other 

features of the terrain.

The present analysis, instead of one specific landscape or river, treats of a whole ensemble of landscapes or rivers which are "macroscopically identical" 

in the face of one's ignorance about their details. Because of one's ignorance, only averages can be derived.

It is further assumed that each particle (= a small 

volume of solid or a volume of fluid small enough not to 

be separated in its journeys) will, under steady-state 

conditions, in time meet all the conditions that are present in the many landscapes or rivers composing the 

ensemble. In other words, tune averages and ensemble 

averages can be exchanged. This is called the ergo die 

hypothesis.

If, for example, the ensemble consists of rivers having 

a prescribed rate of change of discharges, then only 

average rates of changes of the associated river properties can be dervied, although inferences may also be 

derived about the variance among the individuals composing the ensemble. It is assumed that the ensemble 

of all rivers having the given rate of change of discharge 

at any time contains samples of the changes that will 

occur in any given river over time.

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