a new view of the landscape

A New View of the LandscapeAuthor(s): W. E. H. CullingSource: Transactions of the Institute of British Geographers, New Series, Vol. 13, No. 3 (1988),pp. 345-360Published by: The Royal Geographical Society (with the Institute of British Geographers)Stable URL: http://www.jstor.org/stable/622996 .

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345

A new view of the landscape W. E. H. CULLING 28 Under Ffrydd Wood, Knighton, Powys

Revised MS received 20 December, 1987

ABSTRACT The customary view of the landscape and of landscape processes emphasizes regularity. Macroscopic landforms are represented as contoured surfaces and mean values are held to provide a satisfactory knowledge of structures and processes at the micro level. However, it is known that geomorphic structures can exhibit a spatial variability that renders them unmappable and non-laminar flows are so complex as to defy regular analysis.

Irregularity in the landscape may be probabalistic in origin or the outcome of non-linearity where, despite being completely determined, systems can exhibit stochasticity. Two examples of probabilistic irregularity are presented in this paper: (i) the soil-covered landscape as a sample function of a Gaussian field; (ii) variable thresholds as the subtraction of random variables.

Non-linear stochasticity enters geomorphology mainly via dissipative structures in turbulent flow. For slightly perturbed, mildy non-linear, conservative systems, stability is provided for by KAM theory, while 'Hamiltonian chaos' guarantees ergodicity and the possibility of equilibrium.

KEY WORDS: Regularity, Irregularity, Dissipative chaos, Conservative stochasticity, Gaussian landscape, Random thresholds

INTRODUCTION Advances in science often follow from the adoption of a fresh point of view. That which was formerly disparate, confused or contradictory falls into place and is seen as an integrated and coherent whole. Hav- ing chosen a new perspective, the next task is to assemble the weaponry with which to tackle the new problems that a fresh viewpoint will bring in its train. For the physical sciences this is, in almost all cases, a mathematical armoury. It is these two aspects, a new viewpoint and the appropriate mathematics that forms the present subject matter.

The standard, orthodox, almost intuitive view of the landscape is one of regularity. In this it is part of our normal view of the outside world. In a well defined form this goes back to Galileo and the bring- ing of mathematics and physics together for the first time in any systematic form (Drake, 1970, p. 99). We know, however, that the universe is discrete. Solids are made up of particles, liquids are full of holes and gases embody molecular chaos. This much was glimpsed by the Pre-Socratics. Nevertheless regularity reigned supreme for almost three centuries after Galileo and despite Dalton and Mendeleev molecular

reality was not conceded until Perrin and Brownian motion.

The standard conception of a landscape such as that found in southern England is of a smooth regular surface admitting of representation as a contour map. Those parts of the landscape for which the cartogra- pher fears to place a contour form a very small proportion. Yet it is a commonplace that upon looking down at our feet the ground surface is highly irregular. The same applies to the view of the earth's surface from a satellite. In geomorphology, as in the physical sciences as a whole, a regular viewpoint is often assumed without question; or where recognized, great effort is expended to bring irregular phenomena back into the fold by a variety of mathematical and psychological tricks, so allowing full play to the immense weight of classical analysis, while assuaging an acquired mental set.

Without doubt there are great advantages in taking the regular viewpoint both for the scientist and the layman. Most phenomena encountered on the human scale in everyday life fit easily into the regular framework, at least to a first approximation. Conse- quently, a great deal of work has been directed

Trans. Inst. Br. Geogr. N.S. 13: 345-360 (1988) ISSN: 0020-2754 Printed in Great Britain



346 W. E. H. CULLING

towards its study and there is a wealth of results available. Nevertheless certain phenomena refuse to fit into a regular perspective. A prime example is turbulence. Despite many attempts turbulence is not to be rendered measurable by averaging or other established statistical techniques, not because of the technical difficulties, great though these may be, but because the frame of reference is inadequate.

Other examples are suited to a regular idealization at some scales but not at others. The physical landscape surface is one of these. In Kuhnian terms, the prevailing paradigm neglects certain aspects, sits ill upon others and, for some, tends to paradox. The time is ripe for a wider perspective. One that will encompass irregularity as well as regularity. The aim therefore is to present a fresh alternative or complementary view of the landscape, to cite examples of low and high irregularity and consider the implications for geomorphology.

In what follows, there will be a tendency to focus upon macroscopic landforms and to give the impression that irregularity in the landscape is to be explained exclusively in probabilistic terms. How- ever, the choice between a regular and an irregular viewpoint is present when contemplating processes as when dealing with landforms. In order to partially redress the balance a section is therefore included on non-linear stochasticity in the landscape. This topic is returned to in a final section dealing in more general terms with the implications for geomorphology of modern developments in non-linear theory. Prior to this, two examples are treated in some detail. The first, a special case, is that of the soil-covered landscape where both regular and irregular aspects can be brought within one mathematical model. The second example is of a more general application of random variable theory and is exercised upon the concept of thresholds.

The proposed view is new only in a relative sense. Inadequacies of the customary viewpoint have been noticed for many years, some for over a century, but in the last few decades the pace has quickened. Irregu- larity is no longer the exclusive province of random theory. Non-linear systems can trace orbits as being irregular, and in some respects more intricate than random processes, and yet be completely determined.

As usual great events have cast their shadows before. Non-linear dynamic theory had pre-cursors in Poincare and Birkhoff and more recently, at least two false starts. In probability theory it is germane to our purpose to instance the pioneer work of Norbert Wiener and Paul Levy in Brownian motion, well

before the 'official inauguration' of modem probability theory. In this we refer to the axiomatization of probability theory and the identification of random variables with measurable functions by Kolmogorov (1933). This view of probability theory holds in all that follows. We are not concerned with subjective, inductive or any other statistical variant; nor indeed with quantum mechanically inspired, negative or otherwise extended views of probability (Muckenheim, 1986).

Measure theory enters the discussion in a second fundamental way. The appropriate mathematical theory for dealing with highly erratic random phenomena is that of Hausdorff measures and dimensions. They also play an important role in non-linear chaotic theory and provide a bridge between the deterministic and indeterministic categories of extreme irregularity. We start, therefore, with some account of Hausdorff measures before looking at irregularity in the landscape.

HAUSDORFF MEASURES AND DIMENSIONS

In order to obtain a mathematical grip on phenomena it is necessary to be able to determine their magnitude. If, as in everyday life we remain within the regular viewpoint, this is purely a practical problem. However, towards the end of the last century mathematicians were aware of the existence of sets that presented more than just practical problems. For a long time they were considered pathological and of no application to the physical world but this attitude has recently been reversed. The estimation of the magnitude of sets is the province of measure theory. In its inception this had two tasks. First, to provide a firmer foundation for the theory of the integral and, secondly, to make precise what had previously been intuitive ideas of the size or content of sets of points. On both counts a solution was proposed by Emile Borel and his student Henri Lebesgue. Lebesgue measure theory is now standard and clarifies elementary ideas of length, area and volume. It is the ground- plan of the regular view of the universe.

If it is to be of any use a measure must be able to make distinctions between sets of interest. Thus, a continuous line segment possesses a definite length which may range over the set of real numbers. What- ever the practical problems of measurement, the ideas of Lebesque theory are adequate to the task. But there exist certain sets, and these were known before



A new view of the landscape 347

Lebesgue, for which his measure theory proves inadequate. For example, certain sets that are certainly distinct, possess the same Lebesgue measure, e.g., all sets of discrete points have the same Lebesgue measure, namely zero.

In 1875, Weierstrass constructed a function that was continuous yet non-differentiable everywhere over an interval. That is to say, it is not possible to construct a tangent to the curve of the function at any point. Such a function is termed non-rectifiable; it has no unique length and all such curves have the same measure, namely + co. Since Norbert Wiener's (1923) study of Brownian motion it has been known that physical phenomena can approximate extremely closely to the mathematical conception of non- rectifiability. Closer to home, it has long been known to cartographers as a problem in the representation of the landscape. Steinhaus questioned the length of the left bank of the Vistula and Lewis Fry Richardson discovered discrepancies in the measured length of frontiers and went on to relate the apparent length of a mapped coastline to the unit length of measuring interval. The slope of a plotted log/log relationship between measured length and unit length gave a (similarity) measure for the irregularity of a coastline; low for the smooth coastline of Africa, greater for the irregular coastline of Wales and greater still for Aegean coastlines. Many more examples of systematic irregularity in nature have been given by Mandelbrot (1977, 1982) who coined the term fractal for highly erratic self-similar curves. The term fractal, though apposite, is not well defined and we use it solely as a general term for highly erratic phenomena without any further connotation.

Clearly the rich structure of the coastline of Wales is not to be captured by Lebesgue measure. Like every other coastline it would register as + co. At the other extreme, Lebesgue theory is no less inadequate. Since the days of Georg Cantor it has been known that certain point sets possess a surprisingly rich structure. His celebrated ternary set has the cardinality of the continuum. That is to say it has equivalence with the set of real numbers and is therefore non-denumerable. Yet as far as Lebesgue notions are concerned it is of measure zero and indistinguishable from any other set of discrete points. Coastlines, like Brownian motion and Cantor sets, slip through the mesh of Lebesgue measure. In order to study coastlines and other irregular phenomena in the landscape we need a finer conception of measure.

Such a conception has been available since 1919. Following upon the notions of an exterior measure of

Constantin Caretheodory, Felix Hausdorff (1919) introduced what is now known as Hausdorff measure. Instead of building up the idea of a measure from the sum of lengths of elementary rectangles, Hausdorff closed down upon a set of points with a series of spheres, but also enlarged the idea by taking a function of the diameters, of which the most useful is the power. This has the properties of a dimension, the Hausdorff dimension, differing from the intuitive idea of a dimension in that it can take fractional values, yet for well behaved sets gives the expected integral value.

The Richardson similarity dimension, already men- tioned in connection with coastlines, has recently been shown to be equivalent to the Hausdorff dimension provided the sets concerned satisfy a non-overlapping criterion (Hutchinson, 1981). Happily, sets taken from the landscape will almost certainly satisfy such a condition. For certain sets the power function version of the Hausdorff measure is not sufficient and recourse has to be made to more complicated functions such as an integrated logarithm. Mercifully this does not appear to be the case for any application to the landscape.

As far as can be foreseen, in the ideas of the Hausdorff measure and dimension, geomorpholo- gists are in possession of a concept adequate to measure and distinguish any irregular set to be found associated with any landform or landscape process.

IRREGULARITY IN THE LANDSCAPE

In any application to the landscape the Hausdorff dimension may be regarded as made up of two parts; an integral number and a fractional component, of which the latter takes values between zero and one.

Under certain conditions, the dimension of the Cartesian product of two component sets is the sum of their separate dimensions. The idea will be familiar from simple arithmetic. The product of two line sets is a plane of dimension two. If the plane set is multiplied by a further line set giving a volume of dimension three, this value is seen to be the sum of the dimension of the plane (2) and of the line (1). What will be unfamiliar is that the rule can be extended to cases where one of the components is of fractional dimension.

If we denote the fractional dimension as DH, where 0< DH < 1, then a fractal surface has a Hausdorff dimension of 2 + DH, a fractal line graph of 1 + DH and for a discrete fractal set of points, O+DH. This is




strictly true in a fairly general sense only as an in- equality. However, for a special case, of importance geomorphologically, namely highly erratic Gaussian random fields, it is true as the equality. In which case, having determined the Hausdorff dimension for one integral valued 'topological' dimension we immedi- ately have all the others. Thus on successive division by a line set of dimension one, we can reduce any problem relating to a surface of dimension 2 + DH, to one relating to a line set of dimension 1+ DH i.e., to a profile or a contour curve, and then in turn to a set of points of dimension DH. This relationship makes possible the reduction of all dimensional problems relating to the soil-covered landscape to equivalent problems on the unit interval. Thus the determination of the dimension of the landscape surface can be made from the set of intersection points between an appropriate contour and a chosen vertical plane. This problem is illustrated in Culling and Datko (1987), with a fuller treatment of the calculations in Culling (1988).

High, low and intermediate dimensioned structures can be identified in the landscape. The mature, well-developed soil-covered landscape of the rolling Chalk downlands possesses a surface of low Hausdorff dimension commensurate with its smooth- ness and regularity. Found values for the fractal component range up to DH - 0.3, but with most below DH = 0-25 and they can fall to DH - 0-05, for the very smooth. In marked contrast the surface micro-relief is highly irregular with values of DH - 0.8. This implies that processes tending to maintain irregularity predominate whereas the opposite is the case on soil-covered slopes.

Intermediate values (0-4-0-6) are found associated with the drainage network. It is upon this base pattern that the smoother structures of surface degradation are superimposed and it is usual to find evidence of both in samples from southern England. There may be more than one structure superimposed on that of the drainage, a result of geological or geomorpho- logical differentiation of surface and an example of the latter, from the Shelve area of south Shropshire, is given in Culling and Datko (1987).

Values about the mid-point are typical of branching processes, where at each node there is no significant predilection for one way or the other. Examples outside geomorphology abound; Hele-Shaw cells, where a jet of viscous fluid penetrates one of lower viscosity, electric discharge, electro-deposition, diffusion aggradation, neurological growth and, possibly, unrestricted settlement growth out from a central point; while within geomorphology, gully

growth into an homogenous plateau. Where the precise origin of the drainage net is unknown the mere pattern can be regarded as a branching process some- way between diffusion aggradation, where the aggrading particle executes a random walk like rain drops on a gently sloping pane, and ballistic aggradation, where the aggrading particle is shot at the ac- cumulation network in randomly selected straight lines, like rain drops on a vertical pane. In such cases the dimension of the pattern will fall between 0-4 and 0-6.

An example of low to intermediate values is discussed in a later section on the Gaussian landscape. For the remainder of this section we concentrate on high fractal dimensioned structures and particularly on the implications for geomorphology. Some time is spent on this topic as it is apparent from a recent discussion (Brinkman, 1987; Culling, 1987b) that the implications for geomorphology arising from a study of highly erratic phenomena on Iping Common (Culling, 1986a), remain incompletely appreciated.

These implications are as follows. (1) In all measurements of properties of the soil or

soil surface or any other geomorphic property liable to local spatial variability the degree of erraticism should be determined. If the value of the Hausdorff dimension records over 0-7, or possibly even lower, then points (2) to (5) below apply.

(2) The existence of high spatial variability precludes the use of mean values for both measurement and interpretation and this will include statistical techniques such as linear regression analysis.

(3) The maintenance of high spatial variability in the face of the many processes that tend to smooth differences requires the existence of high temporal variation.

(4) The measurement of temporal variation using destructive sampling techniques is vitiated by the existence of high spatial variability.

(5) The appropriate mathematical apparatus for the study of the properties and relationships of high spatial/temporal variable parameters is that of probability theory, where the parameters are regarded as sample functions of random variables.

There is a further practical implication. The variability of many, if not all, soil based properties is of fine resolution. So much so that the determination of the spatial variability is in many cases beyond the resolv- ing power of existing measurement techniques. This was certainly found to be the case in the measurement




of soil-pH on Iping Common, West Sussex (Culling, 1986a). The first series of measurements had to be rejected and a special integrated and laborious tech- nique devised to measure soil-pH to the precision required to demonstrate the intrinsic variability (Brinkman, 1987; Culling, 1987b).

The highly erratic nature of the surface micro-relief on soil-covered slopes has been the subject of a study by Armstrong (1986), who finds for a fractal component of DH -, 0-8. A similar value was found for the spatial variability of soil-pH (Culling, 1986a), and there can be little doubt that the same applies to other soil variables. A fractal component of DH -- 0-8 is too irregular to be mappable for practical usefulness (Culling, 1986a, Fig. 9). High variability of soil properties is not a new discovery. A review is given in Beckett and Webster (1971) and of more recent work in Burrough (1981) and Culling (1986a).

Irrespective of the interpretation, the highly erratic nature of soil properties poses a problem for the geomorphologist. Most geomorphic processes find their basic unit at the particle level and as such operate within an irregular fractal environment. Faced with such diversity the customary practice is to take refuge in the mean value of a series of observations, either explicitly or disguised in some form of statistical pro- cedure. This course of action rests upon the pre- supposition that there exists a 'true' variable hidden beneath the data, that is well behaved and of a simple deterministic nature. The distribution of observed values is regarded purely statistically as a signal plus noise. The task of the geomorphologist is merely to disinter the structural variable from the accumulated noise. If any intrinsic variability is recognized then it is assumed to be strongly peaked, like temperature, and can thus be safely neglected.

This is not the case with soil properties and we have already noted how great pains were taken in the measurement of soil-pH on Iping Common to make sure that it was structural variation that was being recorded and not observational or experimental noise. The understanding of many geomorphic processes depends upon the establishment of relationships between soil properties and also of processes that are intimately related to the soil structure or surface. In the light of their highly erratic spatial and temporal variation the couching of relationships in terms of mean values will, at best, lead to crude first approximations that in the long term are more likely to obscure rather than promote understanding.

The apparatus is available to undertake regression analysis between two random variables. The out-

come will be expressed in terms of a conditional probability distribution. That is, for any chosen value of one of the random variables the other is expressed as a probability distribution. All the usual statistical quantities-means, variances, covariances, correlations and quartiles--can then be expressed in terms of the conditional distribution. If the joint distribution of the two random variables is of multi-variate Gaus- sian form then the conditional formulae are relatively simple. In practice the joint distribution will be experi- enced as a series of sample functions and the Gaussian nature, if present, will need prior establishment. The procedures have been extended to the non-Gaussian case and, in fact, to the non-parametric case (Stone, 1977).

Faced with irregular surfaces, experimental scientists have often turned to spectral methods as a measure of roughness; for example, of road surfaces (Pevzner and Tikhonov, 1964), of ocean waves (Pierson and Marks, 1952), of lunar terrains (Jaeger and Schuring, 1966) and, in three dimensions, of soil properties (Dexter, 1977). However, for the understanding of phenomena as distinct from description or prediction, Fourier methods suffer from the disad- vantage that the coefficients bear no relationship to the underlying structure.

Theoretical scientists, when presented with such a problem, have often turned to a stochastic approach. Representations of irregular surfaces as random fields include: ocean surfaces (Longuet-Higgins, 1952, 1957a and b; Cartwright and Longuet-Higgins, 1956); metallic surfaces in contact (Greenwood and Williamson, 1966; Whitehouse and Achard, 1970; Nayak, 1971); earthquake fault planes (Andrews, 1981); contoured surfaces in general (Swerling, 1962) and, in three dimensions, soil structure and soil creep (Culling, 1963, 1965).

Random variables are measurable functions map- ping events from a probability measure space onto an appropriate Euclidean space. Random functions (fields if the dimension is greater than one) share all the usual properties of ordinary non-random functions. Equivalents exist, for the rules of simple arithmetic, for convergence properties, and the differential and integral calculus, so that random variables can take their place in stochastic equations. In the simplest case two random variables can be added or subtracted, taking into account the whole distribution and not just the mean value. Each value for the one variate is added (or subtracted) to all the values of the second variate, weighted in both cases according to the probability of occurrence as expressed by the




probability density function or frequency distribution.

As is to be expected, operations with random variables are more complicated than if practised upon ordinary variables. However, we are not concerned with a general theory of random functions here but merely with those aspects of application to geomorphology. In this we are fortunate in that it is the relatively simple aspects that are of particular use and this can be seen clearly in our first example in which the theory of Gaussian random fields is applied to the surface of the soil-covered landscape. Gaussian var- iates obey a simple additive law and are otherwise relatively easy to work with. They are of great practical importance and as a consequence have been much studied. Our second example, in contrast, is more straightforward and mechanical and exemplifies the application of simple arithmetical operations with random variables.

But, before moving on to these two examples, consideration needs to be given to non-linear stochasticity.

NON-LINEAR CHAOTIC IRREGULARITY

The main emphasis throughout this paper is on the probabilistic approach to irregularity. However, some consideration must also be given to chaotic irregularity, wherein non-linear systems can exhibit behaviour that mimics that of random systems yet is completely determined. The background to deterministic irregularity is sketched in an earlier paper (Culling, 1987a). Here we can only trace some of the implications of the most important aspect of non- linear dynamics for geomorphology, that of turbulence. Turbulence has been described by David Ruelle (1980) as the greatest unsolved problem in classical physics and a recent reviewer (Lumley, 1987) regards present thinking on the subject as a morass. It is not within the gift of geomorphology to do much about a solution, however progress in certain purely geomorphic matters will not be forthcoming unless due allowance is made for the difficulties turbulence brings in its train. Even the slightest perusal of the literature on fluvial geomorphology will show that it is heavily dependent upon averaging and the very imagery and conceptual frame of reference, assumed almost sub-consciously, tends to stand in the way of any refined understanding.

Earlier the variability of soil structure was treated as if static, whereas we know that it can partake of a slow creep flow. This can be expressed in terms of

random theory as a diffusive type flow (Culling, 1963, 1965). If a particulate medium is subject to greater velocity in relation to the viscosity, it can assume liquid-like properties and exhibit laminar type flow. At higher velocities, the flow will transform, first to transitional, and then to fully developed turbulent flow. Laminar flow, where layers of fluid slide past one another, is simple to conceptualize and easy to study. It is the type regular flow. There is an ever present tendency to regard all flow according to the norms of laminar flow; they pervade the language and threaten to cloud geomorphic thought.

From this standpoint turbulent flow is visualized as laminar flow that has somehow gone wrong. Under- neath the swirls and eddies, the bursts and bubbles, lies the 'true flow', which by dint of statistical artistry can be disinterred. The use of mean velocities in this case parallels the use of mean values in the earlier static examples. Here, as there, this is a totally mis- leading conception and contrives to 'throw the baby away with the bath water'. Turbulent flow is the confused hierarchy of eddies. In detail, the flow pattern is unmanageably complex and refuses categorically to sit in a regular straight jacket.

Fully turbulent flow approaches very closely to a strongly mixing flow in which a flow unit at any one point in a container can eventually occupy any other interior point and not merely in an ergodic manner. Thus, if a proportion of ink is stirred in a beaker of water we expect to find that proportion in any sample taken and not just as a limiting average, as is the case for an ergodic system. Mixing implies the tendency to equilibrium and we expect this to take place in short order which implies that the system is not only mixing but is a K-flow (Arnold and Avez, 1968, pp. 19 and 32). This applies to position only. In velocity terms turbulent systems do not occupy the whole of available phase space, though they may occupy stochastic bands within which it is assumed, for a rigorous argument is rarely possible, that they are strongly mixing. In view of the importance of the concept of equilibrium in geomorphology it is highly desirable that particulate transport systems, to name just one example, possess the high degree of stochasticity commensurate with mixing and K-flows.

Until the advent of modem non-linear theory it was natural to study a system as complex as turbulence as a problem in random theory. In modem form this approach can be traced back to Norbert Wiener (1939) and the programme for 'the use of statistical theory in the study of turbulence' (see Wiener, 1977, pp. 435-758 and note that, today, we would prefer to




use the term probabilistic instead of statistical). The question, 'how is it that a deterministic system can give rise to such complexity that it demands probabilistic treatment?', tended to be half forgotten.

This programme arose out of Wiener's work on pure or homogenous 'Chaos' (Wiener, 1938). Today, this would be called a stationary or homogenous random field and the use of the term 'Chaos' by Wiener needs to be distinguished from its later use by non- linear dynamicists. To this end we denote it by a capital C. At the time, the only kind of systems studied with anything approaching adequacy for the intended task were those of Wiener himself on Brownian motion and, its multi-dimensioned version, the Wiener process or Brownian sheet. Thus Brownian motion was the type homogenous Chaos. Therein, the mass, or density distribution of a relevant parameter, such as velocity, over the whole space occupied by the ensemble was represented so that: (i) the mass in a given region has a Gaussian distri-

bution dependent only on the volume of the region;

(ii) the mass distribution in non-overlapping regions is independent (Wiener, 1939, p. 4).

The central problem of turbulence, to be solved by Chaos theory was that given

the statistical configuration and velocity distribution of the medium at a given initial time and the dynamical laws to which it is subject, to determine the configuration at any future time with respect to its statistical parameters (Wiener, 1939, p. 6).

Although Chaos theory was held to be the key to progress it was recognized that

the demands of chaos theory go considerably beyond the best knowledge of the present day (Wiener, 1939, p. 9).

The carrying out of the programme was left to others (McMillan and Deems, 1977). The contemporary but largely independent programme of physicists and applied mathematicians, stemming like Wiener's from the work of Taylor (1921, 1935), can be followed in the early pages of Batchelor (1953, pp. 7-13).

Fully developed turbulence, according to Stuart (1977), is such as to defy rational understanding and the more restricted objective of tracing the early stages of the transition to turbulence was felt to be a more realistic target, though still of great difficulty. It

is profitable to spend some time on this topic as modem chaos theory has had its greatest impact on the roads to turbulence rather than on fully developed turbulence (Eckmann, 1981). On the practical side, the fluvial entrainment of, at least, the finer grades of sediment is most likely to take place well before fully developed turbulence sets in.

A theory of non-linear hydrodynamic stability was laid down by Landau (1944) and (belatedly) became the orthodox physical view. Increase of the Reynolds number above the critical value, beyond which the flow is unstable to infinitesimal perturbation, gives rise to a periodic flow. As the Reynolds number is increased further, Landau suggests that the periodic flow itself becomes unstable leading to a flow with two different periods. Continued increase in the Reynolds number witnesses further periodic modes, the intervening 'stable' intervals diminishing rapidly, giving rise at the limit to a complicated and confused flow represented by an infinite summation of periodic terms (Drazin and Reid, 1981, pp. 370-76). Indepen- dently Eberhard Hopf (1948) had come up with much the same picture with turbulence arising from a succession of (Hopf) bifurcations.

The Landau-Hopf scenario is now largely dis- counted as a route to turbulence. If the coefficients in the summation are independent the sum is an almost periodic function and if they are distributed in a Gaussian manner the resulting expression is Brownian motion, so the connection with Wiener is close. However, 'almost periodicity' implies persist- ence of velocity correlation, whereas it is characteristic of turbulence that it decays exponentially. The assumption of independence between the modes is not borne out in practice, where 'frequency locking' is found and this makes a Gaussian outcome remote. The set of bifurcations requires the successive appearance of fresh incommensurate frequencies in the power spectrum, which remains discrete over the whole range of Reynold's numbers. Experimental spectra do show a small number of isolated peaks but they soon dissolve into broad noisy bands. A more serious difficulty, that is also shared by Chaos theories, concerns the relationship between the flow and the initial conditions.

In both theories the velocity distribution follows the familiar linear pattern in that with time the distribution rapidly assumes independence of the initial conditions. It is now known that this is not the case for dissipative systems exhibiting strange attractors, and there is good reason to believe that the Navier- Stokes equations governing tubulent flow harbour a




strange attractor on which all orbits are highly sensi- tive to the initial conditions. This is not to suggest that random theory has no further role to play, particularly with fully developed turbulence, and this is certainly the case in geomorphic contexts; in the study of the turbulent diffusion of sediment particles or, indeed, of particulate flow phenomena in general (Culling, 1985b). But it is apparent that things will never be the same after the discovery by Lorenz (1963) of the intricate structures, named 'strange attractors' by Ruelle and Takens (1971); for further details see Culling (1987a and 1985a).

Despite the overall structural stability, because of the existence of exponential divergence in at least one dimension of a strange attractor, points which are arbitrarily close to start with often end up widely spaced along the attractor orbit. However, because of its boundedness and intricate folding, they remain within a small volume. This has two important implications: (i) despite immediate determinacy there exists long

term unpredictability both in retrospect and prospect;

(ii) conversely there is extreme sensitivity to initial conditions and, apart from a very few exceptional cases, these conditions cannot be replicated.

This means that knowledge of the system is localized about the present. No observation or experiment can be repeated; no eventual outcome can be foreseen with precision. There is no role for Lapace's 'superior intelligence', that given the present state of the universe, could predict its entire future.

One important implication for geomorphology, the non-repeatability of observations of strange dissipative phenomena, is well known as a fact of life for students of turbulent phenomena (Yalin, 1977, p. 25 7) and, much earlier, Heraclitus claimed that 'you cannot step into the same river twice'. The normal reaction would be to regard the observation as a sample function of a random field and to proceed along the lines of homogenous Chaos theory. But, as we have already discussed, this cannot be the complete answer. The irregularity of strange attractors is not well catered for by standard random theory. They exhibit non-stationarity with a vengeance. This is well shown by the Malkus wheel (Malkus, 1972; Lorenz, 1963; Martin, 1979), which can start, stop, speed up, slow down and reverse direction both periodically and aperiodically without any change in the external parameters. It is as if the rules of the game are changed as it goes along.

It is the system itself that changes the rules. Expo-

nential divergence along the unstable dimension of a strange attractor is measured by the K-entropy giving the rate of information flow. In entropy terms, turbulence creates information and at an alarming rate. The stirring of a cup of coffee produces _ 1012 bits of information per second (Shaw, 1981, p. 106). The K-entropy (the K stands for Kolmogorov) is closely related to the Hausdorff dimension (Billingsley, 1965; Culling, 1987c). They both measure irregularity or unexpectedness and it is natural to turn to the Hausdorff dimension as a measure of chaos (Shimada and Nagashima, 1979). Care needs to be taken as the idea of dimension can be applied to strange attractors in more than one way. Cross sections exhibit a Cantor-like structure but the dimension is usually low, often less than 0-1. Despite their visual complexity strange attractors occupy very little volume. On the other hand the artifacts of turbulence, such as clouds, register values of 1-35 + 0-05 (Lovejoy, 1982). This has been supported by a theoretical derivation from Richardson's (1926) idea of relative diffusion in turbulent flow by Hentschel and Procaccia (1984). The same authors estimate the fractal dimension of fully developed turbulence as 2-50 < D < 2-75, while Mori (1980) quotes a value of 2-659, indicating a stochasticity greater than standard Brownian motion. The most promising approach appears to be that of the conjectures which link exponential divergence, Liapounov exponents and K-entropy with dimension (Culling, 1985a, p. 31). Whatever the outcome, the Hausdorff dimension as a fundamental measure of the irregularity or erraticism of a set of points appears to be the natural connection between probabilistic and deterministic irregularity.

This is not the place to go any further into the problems of turbulence. Enough has been covered to demonstrate that if the geomorphology of the landscape is to get the measurement, let alone the interpretation and comprehension, of those processes subject to turbulence, into coherence, then the present vague statistical motivation will need to be replaced by some new thinking. New attitudes are required, and a frame of reference more in tune with the intrinsic irregularity of the subject matter.

EXAMPLE 1: GAUSSIAN LANDSCAPE

Wiener's programme has been carried out for two geomorphic cases, albeit of a degenerate nature. The first case relates to the idea that soil particles undergo random translations which, in turn, leads to a diffusive flow on the part of the soil cover as a whole




(Culling, 1963, 1965). It is usual to transform the reversible behaviour on the part of the Brownian particles into a differential equation in terms of the concentration. The flow itself is expressible as a random field but despite the complication of a variety of directed and cyclic movements the fact that it is irrota- tional makes this type of flow much simpler than turbulent diffusion. Various predictions of the stochastic theory of geomorphic soil creep are now begin- ning to be verified; the concentration distribution about a circular cylindrical obstacle (Flavel, 1986a, 1986b), the diffusive blurring of soil boundaries both in the field (Sibbesen, 1986) and in the laboratory (Moira Hansen, personal communication), and the application to the dating of sea cliffs (Nash, 1980) and earthquake fault scarps (Hanks et al., 1984).

The second case, which we will dwell upon, is the application of random fields to the geometry of the surface of the soil covered landscape. As the real time dimension is trivial in flow terms, we deal with a degenerate static case of random flow. The theory is outlined in a series of papers (Culling, 1986b, 1987c, 1988; Culling and Datko, 1987) from which we abstract the essential ideas.

In order to apply the notions of random theory, the given landscape surface is to be regarded as a sample function of an ensemble of a large number of surfaces. From the study of these, properties are to be deduced relating to the whole family of surfaces. There are difficulties over the precise definition of the compo- sition of the ensemble and as to the possibility of the recognition of more than one sample function (Culling, 1988). However, the given surface, if homogenous, can be used as the basis of an ensemble that comprises the set of all profiles of the surface (apart from non-generic examples such as those following a river valley). This is not the same as the given surface though many, but not all, properties are transferable. As a result, vector properties of a sample profile are not applicable to the surface; for example, the number of peaks and the mean gradient differ between the two. In general the extension from the linear to the planar case is not straightforward.

An ensemble is not just a collection of functions, it must also possess the apparatus associated with an appropriate probability measure. Further properties are also required to be present before any practical application can be made. Spatial homogeneity implies that the distribution of a parameter at any one point is the same as at all others. A translation conserves the measure properties; that is, the probability of occurrence of events remains unchanged. If such a measure-

preserving transformation of a set of values of a parameter into itself is associated with an integrable function of the parameter, then the time average of a sequence of such transformations of the function exists. This is the essence of the Birkhoff individual ergodic theorem; that under such conditions a time average exists. In the case of the landscape surface it will be a spatial average along a profile but, following convention, it will be termed a time average.

The immense practical importance of the ergodic theorem lies in its substantiation of the translation of averages over all points of the space available to a given system into averages over all systems of the ensemble. In this way, averages derived from observations on one individual sample function can be used to construct the distribution of the averages of the same parameter for all systems of the ensemble. On the other hand, results computed for the whole ensemble provide predictions of the averages to be expected in relation to any individual system. With- out such ergodic justification, measurements in the field cannot be held to apply to any other part of the landscape or, if the ensemble is that of a family of surfaces, measurements on one surface cannot be held to apply to any other surface. As measurements usually involve some form of averaging, without the existence of the average guaranteed by an ergodic ensemble, the very act of measurement in the field is an act of faith.

The two cases we have considered are connected. From the theory of soil creep expressed in terms of Fourier solutions to the diffusion equation, it can be predicted that a mature soil-covered surface will eventually tend to a form that is well represented by a Gaussian random field of Brownian form (Culling, 1965, 1986b). Subsequent observation has verified this prediction for certain terrains in southern Britain (Culling and Datko, 1987; Culling, 1988). Brownian surfaces possess the desirable ergodic properties. Time averages for important parameters exist and are equivalent to space averages over the whole of the surface. Hence the application of the theory of Gaussian random fields to the soil-covered landscape supplies a good reason for what most geomorpholo- gists do by instinct.

What does the idealization of a landscape surface as an homogenous random flow imply? To answer this question it is convenient to regard the flow as of a two-dimensional profile in the remaining dimension. Then, if we consider the area covered by a 1: 25000 O.S. map and take a profile along the western edge, we may derive certain properties from the trace of the




profile; the distributions of the elevation, gradient and curvature, of crossing points and excursion sets, and of the dimension and entropy. We now envisage the translation of the meridian of the profile east- wards across the sheet in what is taken as the time dimension. Each point on the initial profile will trace a profile in the time dimension. We do not possess a set of equations to govern the behaviour of each point as we would of each particle in the ensemble of an ideal gas. However, we do possess, from the theory of stochastic soil creep, certain properties relating to the elevation, gradient and especially the curvature. From these can be deduced the Brownian nature of the time profiles and the ergodic nature of the flow. Averages of properties along the time profiles will be equivalent to those along meridian profiles or taken from integrations over the whole area. When the eastern margin is reached, the properties of the profile will be equivalent to those of the initial profile. These properties will be expressed, in the first instance, as probability distributions and then, in virtue of the ergodic nature of the flow, in averages, correlations and integrals.

There is no particular virtue attaching to the N-S, E-W arrangement and the construction of the surface can be started off at any angle to the margin. All possible profiles across the surface will be included in one of these constructions. The ensemble of all possible profiles is then made up of an infinite set of rep- licas of the surface. Properties such as the probability measure, ergodicity, dimension and entropy inherent in the surface apply also to the ensemble.

It is a characteristic of Gaussian functions that they often obey a zero-one law with respect to a certain property. That is either all (or 'almost all' in the technical probability sense) sample functions possess the property or none do. Regularity is such a property. If a Gaussian random field is smooth, it is very very smooth, but if it is not, then it is highly irregular or fractal in the manner we have already discussed.

Regularity is a local property. It resides in the behaviour of a function over small neighbourhoods and is defined by limiting procedures. However, when placing such an idealization up against the real landscape in order to make its perception manageable we cannot follow the model to the limit, even if we wished to do so. Such limiting behaviour can have no counterpart in the real world. A stop is made some way short of the limit and, in effect, the local properties and their embarrassing consequences are neglected. In this way, we are able to apply either a rough or a smooth approximation almost at will.

A transparent, though not very precise, criterion to distinguish between the two modes of Gaussian surfaces is the existence or otherwise of contour lines. In the regular case, a Gaussian random field possesses continuous level sets for all elevations which guaran- tee the existence of connected contours. This is the conventional regular idealization epitomized by the contour map. In the irregular mode, this is not so. The surface is so irregular that, at the limit, a level (contour) set withers away as a set of disconnected points. However, as already noted, in neither case is the model taken to the limit.

Thus both the regular and irregular viewpoints can be encompassed within the theory of Gaussian random fields. On the one hand, the satisfaction of certain regularity conditions (which in practice cause no hardship) allows the deployment of the many results relating to crossing points, excursion sets, local maxima and extreme values. Crossing points, as the name implies, are those points at which a profile curve crosses a given level, and excursion sets are the associated intervals within which the profile curve takes values at or above the given level. Not only can the distribution of the mean number of up or down cross- ings be predicted, but also the distribution of excursion set length and the probability of clustering of crossing points. It is also possible to use the set of mean level crossing points to calculate the Hausdorff dimension of the set of points and, in turn, of the surface as a whole. Thus it is possible to deduce a great deal of information about a landscape surface upon a slender basis of empirical data, provided we remain within the confines of a regular Gaussian view of the landscape.

Alternatively, an irregular viewpoint may be adopted and the landscape represented by an index-f Gaussian field. These fields are highly erratic and the Hausdorff dimension of the fractal component is given by the complement of the index, (1-fl). The Hausdorff dimension gives a measure of the irregularity or roughness of the landscape surface. As degradation proceeds, and the surface is progressively smoothed, the value of the index rises towards unity, while the dimension of the fractal component falls towards zero. The Hausdorff dimension is directly related to the entropy of the surface provided it is ergodic (Culling, 1987c, 1988). Both are measures of irregularity or unexpectedness and, contrary to the impression given by some physical geography textbooks, the entropy of the surface falls towards zero with the dimension as degradation advances.




The most important advantage accruing to the geomorphologist from the adoption of an irregular idealization of the landscape in terms of index-fl Gaussian fields is that it enables the application of the results of the theory of Brownian motion and, in effect, the carrying out of Wiener's programme.

EXAMPLE 2: THRESHOLDS

The concept of thresholds in geomorphology pro- vides an opportunity to deploy those aspects of probability theory involved in the extension of simple arithmetical procedures to random variables. But the main reason for the choice, however, lies in the existence in the literature of a detailed worked example (Culling, 1983b). Devised as ancillary to the application of rate process ideas to geomorphic topics, its full potential has not yet been called upon.

Thresholds in geomorphology have their origin and most of their widespread currency in large scale phenomena (Schumm, 1973). However, it is most likely to be the case, in geomorphology as in the physical sciences in general, that the most important thresholds are on the small scale; at the particulate, molecular and atomic levels. In the application of rate process theory to geomorphic phenomena, the activation energy is a measure of the chemical bond energy, or the mechanical energy of friction between two particles, that has to be overcome before re- arrangement of matter can take place. Since some form of re-arrangement of matter lies at the heart of almost all geomorphic activity, the activation energy as a geomorphic threshold is arguably the most important parameter in the study of the landscape.

Before the discussion of variable thresholds a point of clarification is required. This might be called after Charles Holland (1986) the 'golden spike syndrome'; a picturesque way of describing the search for a precise boundary. In general, geomorphic as well as fau- nal changes are multidimensional. Apart from chance degeneracies, a threshold in a N-dimensional system will occupy N-1 dimensions. For instance, the boiling point of water as a single number requires elaborate precautions. A mixture of ethanol and water will boil over an interval. The addition of solutes and variation in pressure will add further dimensions to the threshold surface. The search for a single value, or as is more usual in geomorphology, a line on a graph, is to court disappointment in all but the simplest of situations. At best, a transition zone is plotted that is bounded by the projection of the threshold surface

+5 A

-5

+5

-sL

+5

-5-

FIGURE 1. Variable thresholds A. Variable parameter B. Variable threshold C. Threshold excursion set: the difference

between A and B

onto the plane of the graph. The treatment of thresholds as random variables will need to be able to cater for multi-dimensioned examples.

We are concerned with those aspects of threshold theory where a parameter of interest varies in time with respect to a threshold value, usually taken as a fixed constant. When the parameter records values above the threshold level or, in some cases, where the parameter value merely rises through the threshold level, some significant change is held to occur or to be set in motion (Fig. 1A). It will be recognized that this is a description of excursion sets and up-crossing points. If the variation in the parameter can be reason- ably approximated by a Gaussian function then the analysis referred to in the previous section will also apply here. Apart from extreme value estimations of the Gumbel (1958) variety, little attention has been paid to the probabilistic properties of threshold excursion sets.

A fixed value for the threshold cannot be more than a first approximation in any natural system. Most, if not all, components will possess an intrinsic variability and this will be compounded by the presence of noise. Where recognized, it is customary to reduce this variability by the taking of a mean value




or to clothe the mean value with a narrow transition band. This is almost a contradiction in terms; certainly the essence of the threshold concept is vitiated. This is a further example of the imposition of a regular model upon an inherently irregular natural system.

A fluctuating threshold value is illustrated in Figure lB. It is in one dimension for convenience but we later generalize to many dimensions. The idea of excursion sets is widened to comprise those points at which a parameter takes a value greater than the corresponding value of a variable threshold. The question then arises, is it possible to take into account the irregularity on the part of the threshold and yet make useful statements that can be used as predictors or in the analysis of the process? A detailed answer is given in Culling (1983b) but from Figure 1B the threshold sets can be seen to be given by the difference between two sample functions.

The addition, X + Y, of two mutually independent random variables X and Y, is given by the one- dimensional vector function of the two variables. This is a random variable in its own right with a distribution function given by the convolution of the original pair. The convolution is a complex integral expression but, if we transform into the Fourier do- main forming the corresponding characteristic functions, then the symbolic multiplication of the convolution is replaced with a straightforward product of the characteristic functions. This simplifies things considerably though it is still necessary to invert the resulting Fourier transform. In this way, random variables can be added or, with a change of sign, subtracted (Cramer, 1962, pp. 24-38; Kullback, 1934).

If the two random variables are Gaussian the outcome is particularly simple. The mean values are added or subtracted while, in both cases, the variances are added. This is sufficient to define uniquely the distribution function. In Culling (1983b) this pro- cedure was applied to the escape of activated particles over a potential barrier where both the energy of the particle and the activation energy of the barrier vary in a Gaussian fashion. The method was then extended to cover distributions of Gamma type. At the cost of considerable complication, this extension should give a reasonable approximation to most of the empirical distributions which are met in geomorphology. Pro- vided both the parameters and the threshold can be represented by well defined random variables, and that the final inversion can be completed, then the method can be extended to any number of dimensions. Multi-dimensioned Gaussian examples will obey the simple addition laws.

Returning to the time series aspect of thresholds (Fig. IC), if the parameter and threshold are represented by Gaussian functions then so will the threshold excursion set. The resultant mean value can be translated to zero about which the excursion set is symmetrically distributed. In effect, the variation of the threshold has been straightened out but this time it has not been neglected. It follows that all the results referred to previously, relating to excursion sets and crossing points, will now apply.

IRREGULARITY IN GEOMORPHOLOGY

The change of emphasis from the regular to the irregular is very much in the spirit of the age (Lichtenberg and Liebermann, 1983). In the physical sciences, attention has shifted from the computation of individual orbits, which in general is too difficult, to the discovery of qualitative properties of classes of systems. This has hitherto implied a probabilistic approach, in which one makes no attempt to follow the motion but is satisfied with macroscopic properties made available by long-term averages and with describing the behaviour of 'almost all' the systems for 'most' of the time. Even this proves difficult. In the original sense of occupying almost all of phase space, 'hardly any' natural systems are known to be ergodic. What is known is that between the extremes of integrable and ergodic there exists a whole range of stochastic behaviour (Berry, 1978, p. 18).

Dissipative systems of many degrees of freedom have been discussed in the section on turbulent flow. Here we need to say something about the other great source of deterministic irregularity, that of Hamiltonian systems of relatively few degrees of freedom. This is doubly necessary as Hamiltonian systems were omitted from an earlier paper (Culling, 1987a).

The motion of a classical (energy) conservative system of N degrees of freedom can be completely specified by 2N Hamiltonian equations. If these can be solved to give 2N constants or integrals of the motion the system is described as integrable. Examples include the simple pendulum, harmonic os- cillators, and Kepler planetary motion. Orbits of finite (bound) motion explore a compact surface (manifold) of N independent smooth vector fields and therefore the manifold is an N-torus. That is a completely connected surface with no edges which allows Ndistinct irreducible flow patterns. If N= 2 the torus is doughnut shaped or, more colloquially, a 'hairy doughnut'




that can be combed in two independent ways without singularity, unlike a sphere.

The tori are invariant. Once an orbit starts out on a torus then in the absence of external disturbance it stays on the torus. Orbits wind round the torus with N independent rates of rotation. If the ratio between the winding rates is rational, the orbit will eventually retrace itself and the motion is periodic. If the ratio is irrational, this does not happen and the orbit con- tinues to cover the torus densely to give quasi- periodic motion. Like an almost periodic function, there exists a recurrence period for any distance from a given point on the orbit, but it never retraces the point exactly. Even though it covers the torus densely it is of N dimensions and, for N> 1, this means it is always less than the 2N- I dimensions of the energy surface. Ergodicity on the phase space and, therefore statistical mechanics, are completely ruled out for integrable systems. Though they bulk large in physics textbooks, integrable systems are exceptional in nature. In geomorphology, as in the physical sciences in general, almost all Hamiltonian systems are non-integrable. What does this mean?

The stability of weakly perturbed, mildly non- linear, near integrable systems, is the province of KAM theory. The KAM theorem was outlined by Kolmogorov in 1954 and proved by Arnold (1963), Moser (1966) and Arnold and Avez (1968, p. 97). It states that, under certain non-resonance conditions and for specified sufficiently small perturbations, the orbit remains stable, in the sense that it remains on the N-torus (apart that is from a negligible set of initial conditions of measure zero that may wander over the whole energy surface). Near integrable orbits are con- fined to sectors or bands of the torus separated by continuous KAM curves. However, for N> 3 a slow (Arnold) diffusion of a small number of orbits can take place across the KAM surfaces. Although there is distortion in detail, the qualitative picture of KAM motion is much the same as for the unperturbed system; as is the case, for instance, with the earth's orbit perturbed by the presence of Jupiter. Consequently, near integrable KAM type motion cannot be recognized by observation alone. Nevertheless from categ- orical arguments the stability of geomorphic systems of relatively few degrees of freedom is a problem in KAM theory. Thus, it is possible to make qualitative statements about the behaviour of dynamical systems in the landscape, notwithstanding our inability, as yet, to specify any such system in Hamiltonian terms.

Probably the best introduction to KAM theory is the chapter in Bai-Lin (1984, pp. 7-14) and then Berry

(1978, pp. 18-57); Lichtenberg and Liebermann (1983) and Arnold and Avez (1968). Kolmogorov's address of 1954 is translated and reprinted as an appendix to Abraham and Marsden (1978, pp. 741- 57); a translation of Kolmogorov's original article announcing the KAM theorem is included in Bai-Lin (1984, pp. 81-6).

When the conditions of the KAM theorem are violated, the KAM curves between the orbit bands break down, the most irrational being the last to dis- appear. The space made available is gradually taken over by stochastic bands within which orbits trace highly intricate courses that effectively cover the band with strongly mixing flows. In fact, the normal case for Hamiltonian chaos is a K-flow, guaranteeing ergodicity over the relevant space and the possibility of equilibrium. In this way classical conservative dynamics can give rise to motion as irregular as that of chaotic dissipative systems or of random fluctuations.

Whether we deal in geomorphology with Hamiltonian or dissipative systems, instability in the sense of exponential divergence, does not necessarily mean the collapse of the system but allows for self- generated stochasticity. Macroscopically, the flow gives the appearance of random disorder, but if it could be observed in detail it would prove to be completely determined. Although some degree of noise is inevitable it plays no necessary part in the production of chaotic irregularity.

Despite the relative unfamiliarity of the idea of pattern without regularity non-linear chaos is not an exceptional event from the museum of mathematical pathology.

Generally speaking chaos happens more frequently than order, just as there are more irrational numbers than rationals (Bai-Lin, 1984, p. 66).

Geomorphology is not excluded from the scope of this statement. Particulate flows, whether of soil particles, fluvial sediment or aeolian dust, can exhibit behaviour usually associated with solids, liquids or gases and, therefore, supply examples of systems with many degrees of freedom for which irregularity, either probabilistic or chaotic, is the rule.

In view of the normality of irregularity in the landscape, whether of process, transport system or landform, there is implicit a programme for geomorphology that is a microcosm of the programme of the physical sciences but allowance must be made for the immaturity of the subject. On the aspects under discussion-flows, ergodicity, homogenous Chaos-




the programme is running at least half a century late. On wider issues, the subject suffers from a lack of scientific infra-structure. Borrowing a phrase from Bentham: geomorphology has witnessed her Bacon and Locke in Gilbert and Davis but her Newton is yet to come. However, a programme that seeks to bridge Kepler, Newton, Birkhoff and Kolmogorov is like a third world country attempting all stages of growth at the same time. There is bound to be confusion, inertia and 'pigeon leagues', as Galileo would say, but the only alternative appears to be stagnation.

We will conclude by summarizing the two examples. In the first, the theory of Gaussian random fields is used to derive properties of an ensemble of landscape profiles. Inevitably the start has to be made with homogenous surfaces, though it is apparent that no natural example will be completely homogenous. How much tolerance can be allowed is not known but from studies of terrains in southern Britain (Culling and Datko, 1987; Culling, 1988), some natural examples approach sufficiently near to justify ergodic expectations.

In the second example, the concern is not so much with the existence of mean values as with their mis- use. The most important practical outcome is the re- iteration of the commonplace that the mean value merely fixes the position of a probability distribution on the continuum of real numbers. Only in exceptional cases can the mean value be taken into mathematical relationships as the single value rep- resentative of the entire behaviour of the variable. It is now known that geomorphic properties can possess a highly variable nature that is not to be ascribed to a regular signal hidden by noise. Unless it can be shown that a parameter is of a kind with temperature there are obvious difficulties over its role in any mathematical relationship. The proper treatment for highly irregular landscape parameters is to regard them as random variables and so prepare them for a respec- table role in well defined relationships. The general applicability of random theory in the earth sciences is, of course, no new idea. Over a decade ago Barucha-Reid (1975), in an introduction to a sym- posium on 'Random processes in geology', claimed that 'the preliminary groundwork had been laid', and that we could now proceed responsibly with the application. What was then a good idea now seems imperative.

Finally, returning to the initial discussion of the two complementary idealizations of the landscape, we do well to remember that an idealization is a simplification and, as Turing (1952) remarks, is a 'falsi-

fication'. The hope is that in forging something simple and conceptuably manageable nothing of importance has been left out. In applying the theory of Gaussian random fields to the landscape a certain amount of tolerance has been left about the approach to the limit, so enabling the geomorphologist to take advantage of both regular and irregular viewpoints without running into contradictions. But, as well as leaving something out, it is also necessary to put something in. This may be a sophisticated mathematical construct or an everyday Euclidean perspective but, hopefully, it will not introduce an unacceptable amount of distortion. For the geomorphologist, as for everyone else, the way we view the landscape is not absolute or immutable; Amiel's dictum applies

The landscape is a frame of mind

REFERENCES

ABRAHAM, R. and MARSDEN, J. E. (1978) Foundations of mechanics (Benjamin, Reading, Mass.) 2nd edition

ANDREWS, D. J. (1981) 'A stochastic fault model. 2. Time dependent case', J. Geophys. Res. 86: 10821-34

ARMSTRONG, A. C. (1986) 'On the fractal dimension of some transient soil properties', J. Soil Sci 37: 641-52

ARNOLD, V. I. (1963) 'Proof of a theorem of A. N. Kolmogorov on the invariants of quasi-periodic motions under small perturbations of the Hamiltonian', Russian Mathematical Surveys 18: 9-36

ARNOLD, V. I. and AVEZ, A. (1968) Ergodic problems of classical mechanics (Benjamin, New York)

BAI-LIN, H. (1984) Chaos. (World Scientific, Singapore) BARUCHA-REID, A. T. (1975) 'Introduction', in

MERRIAM, D. F. (ed.) Random processes in geology (Springer, New York)

BATCHELOR, G. K. (1953) The theory of homogenous turbulence (Cambridge University Press, Cambridge)

BECKETT, P. M. T. and WEBSTER, R. (1971) 'Soil variability-a review', Soils and Fertilisers 34: 1-15

BERRY, M. V. (1978) 'Regular and irregular motion', in JORNA, S. (ed.) Topics in nonlinear dynamics: a tribute to Sir Edward Bullard (Am. Inst. Physics, New York) pp. 16-120

BILLINGSLEY, P. (1965) Ergodic theory and information (Wiley, New York)

BRINKMAN, R. (1987) 'Angles on the tip of a needle and pH measurements', Catena 14: 145

BURROUGH, P. A. (1981) 'The fractal dimensions of landscape and other data', Nature 294: 240-2

CARTWRIGHT, D. E. and LONGUET-HIGGINS, M. S. (1956) 'The statistical distribution of the maxima of a random function', Proc. Roy. Soc. A237: 212-32




CRAMER, H. (1962) Random variables and probability distributions (Cambridge University Press, Cambridge) 2nd edition

CULLING, W. E. H. (1963) 'Soil creep and the development of hillside slopes', ]. Geol. 71: 127-61

CULLING, W. E. H. (1965) 'Theory of erosion on soil covered slopes', ]. Geol. 73: 230-54

CULLING, W. E. H. (1983a) 'Rate process theory in geomorphic soil creep', Catena Suppl. 4: 191-214

CULLING, W. E. H. (1983b) 'Slow particulate flow in con- densed media as an escape mechanism: mean translation distance', Catena Suppl. 4: 161-90

CULLING, W. E. H. (1985a) 'Equifinality: chaos, dimension and pattern', (Discussion Paper NS. 19, Grad. Sch. Geogr., London School of Economics)

CULLING, W. E. H. (1985b) 'A unified theoretical approach to particulate flows in geomorphic settings' (Papers in geography. 2nd series. 2, Dept. of Geogr., Royal Holloway and New Bedford College, Egham)

CULLING, W. E. H. (1986a) 'Highly erratic spatial variability of soil-pH on Iping Common, West Sussex', Catena 13: 81-98

CULLING, W. E. H. (1986b) 'Hurst phenomena in the landscape', Trans. Jap. Geomorph. Union 7-4: 1-23

CULLING, W. E. H. (1987a) 'Equifinality: modem approaches to dynamical systems and their potential for geographical thought', Trans. Inst. Br. Geogr., N.S. 12: 57-72

CULLING, W. E. H. (1987b) 'Sed Contra', Catena 14: 146-7

CULLING, W. E. H. (1987c) 'Ergodicity, entropy and dimension in the soil covered landscape', Trans.] ap. Geo- morph. Union 8-3: 157-74

CULLING, W. E. H. (1988) 'Dimension and entropy in the soil covered landscape', Earth Surf. Proc. and Landforms 13: (in press)

CULLING, W. E. H. and DATKO, M. (1987) 'The fractal geometry of the soil covered landscape', Earth Surf. Proc. and Landforms 12: 369-85

DEXTER, A. R. (1977) 'Effect of rainfall on the surface micro-relief of tilled soil', J. Terramech. 14: 11-22

DRAKE, S. (1970) Galileo studies (Michigan University Press, Ann Arbor, Michigan)

DRAZIN, P. G. and REID, W. H. (1981) Hydrodynamic stability (Cambridge University Press, Cambridge)

ECKMANN, J. P. (1981) 'Roads to turbulence in dissipative dynamical systems', Rev. Mod. Phys. 53: 655-71

FLAVEL, W. S. (1986a) 'Field investigation of soil movement: a different approach', Z. Geomorph. Suppl.-Bd. 60: 83-91

FLAVEL, W. S. (1986b) 'Field investigation of a stochastic theory of soil creep', in GARDINER, V. (ed.) International Geomorphology I (Wiley, Chichester) pp. 513-26

GREENWOOD, J. A. and WILLIAMSON, J. B. P. (1966) 'Contact of nominally flat surfaces', Proc. Roy. Soc. A295: 300-19

GUMBEL, E. J. (1958) Statistics of extremes (Columbia University Press, New York)

HANKS, T. C., BUCKNAM, R. C., LAJOIE, K. R. and WALLACE, R. E. (1984) 'Modifications of wave-cut and fault controlled landforms', ]. Geophys. Res. 89: 5771-90

HAUSDORFF, F. (1919) 'Dimension und 5iusseres mass', Math. Annln 79: 15 7-79

HENTSCHEL, H. G. E. and PROCACCIA, I. (1984) 'Rela- tive diffusion in turbulent media: the fractal dimension of clouds', Phys. Rev. A29: 1461-70

HOLLAND, C. H. (1986) 'Does the golden spike still glitter?', J. Geol. Soc. 143: 3-21

HOPF, E. (1948) 'A mathematical example displaying fea- tures of turbulence', Comm. Appl. Math. 1: 303-22. Excerpt (pp. 303-309) reprinted in Bai-Lin, H. Chaos (1984) pp. 112-18

HUTCHINSON, J. E. (1981) 'Fractals and self-similarity', Indiana Univ. Math. Journ. 30: 713-47

JAEGER, R. M. and SCHURING, D. J. (1966) 'Spectrum analysis of terrain of Mare Cognitum', J. Geophys. Res. 71: 2033-43

KOLMOGOROV, A. N. (1933) Foundations of probability. Trans. N. Morrison, Chelsea reprint, 1956 (Chelsea, New York)

KOLMOGOROV, A. N. (1954) 'The general theory of dynamical systems and classical mechanics', in ABRAHAM, R. and MARSDEN, J. E. Foundations of mechanics (Benjamin, Reading, Mass) pp. 741-5 7

KULLBACK, S. (1934) 'An application of characteristic functions to the distribution problem of statistics', Am.]. Math. Statist. 5: 263-307

LANDAU, L. D. (1944) On the problem of turbulence. Trans. Collected Papers (1965) pp. 387-91 Ed. D. ter Haar, reprinted Bai-Lin, H. Chaos (1984) pp. 107-111, reproduced in Landau, L. D. & Lifshitz, E. M. (1959) pp. 103-7

LANDAU, L. D. (1965) Collected Papers (Pergamon, Oxford)

LANDAU, L. D. and LIFSHITZ, E. M. (1959) Fluid mechanics. VI: Course of theoretical physics. Trans. Sykes, J. B. & Reid, W. H. (Pergamon, Oxford)

LICHTENBERG, A. J. and LIEBERMANN, M. A. (1983) Regular and stochastic motion (Springer, New York)

LONGUET-HIGGINS, M. S. (1952) 'On the statistical distribution of the heights of sea waves', J. Marine Res. 11: 245-66

LONGUET-HIGGINS, M. S. (195 7a) 'The statistical analysis of a random moving surface', Phil. Trans. Roy. Soc. A249: 321-87

LONGUET-HIGGINS, M. S. (195 7b) 'Statistical properties of an isotropic random surface', Phil. Trans. Roy. Soc. A250: 157-74

LORENZ, E. N. (1963) 'Deterministic non-periodic flows', J. Atmos. Sci. 20: 130-41

LOVEJOY, S. (1982) 'Area-perimeter relation for rain and cloud areas', Science 216: 185-7




LUMLEY, J. L. (1987) Review of turbulence and random processes in fluid mechanics by M. T. Landahl & E. Mollo- Christensen. J. Fluid Mech. 183: 566-7

McMILLAN, B. and DEEM, G. S. (1977) 'The Wiener programme in statistical physics', in MASANI, P. (ed.) Norbert Wiener: collected works I (M.I.T. Press, Cambridge, Mass.) pp. 654-71

MALKUS, W. V. R. (1972) 'Non-periodic convection at high and low Prandtl number', Mem. Soc. Roy. Sci. Liege Series 6, 4: 125-81

MANDELBROT, B. B. (1977) Fractals: form, chance and dimension (Freeman, San Francisco)

MANDELBROT, B. B. (1982) The fractal geometry of nature (Freeman, San Francisco)

MARTIN, P. C. (1979) 'Schwinger and statistical physics: a spin-off success story and some challenging sequels', Physica A 96: 70-88

MORI, H. (1980) 'The fractal dimensions of chaotic flows of autonomous dissipative systems', Prog. Theo. Phys. 63: 1044-7

MOSER, J. (1966) 'On the theory of quasi periodic notion', Siam Rev. 8: 145-72

MUCKENHEIM, W. (1986) 'A review of extended proba- bilities', Physics Reports 133: 337-401

NASH, D. B. (1980) 'Morphological dating of degraded fault scarps', J. Geol. 88: 353-60

NAYAK, P. R. (1971) 'Random process model of rough surfaces', J. Lubric. Technol. 93: 398-407

PEVZNER, Y. M. and TIKHONOV, A. A. (1964) 'Statisti- cal description of the microprofile of automotive roads', i. Terramechanics 1: 5-16

PIERSON, W. J. and MARKS, W. (1952) 'The power spectrum analysis of ocean wave records', J. Geophys. Union 33: 834-44

RICHARDSON, L. F. (1926) 'Atmospheric diffusion shown on a distance-neighbour graph', Proc. Roy. Soc. All0: 709-37

RUELLE, D. (1980) 'Strange attractors', Math. Intelligencer 2: 126-37

RUELLE, D. and TAKENS, F. (1971) 'On the nature of turbulence', Comm. Math. Phys. 20: 167-92; 23: 343-4

SCHUMM, S. A. (1973) 'Geomorphic thresholds and complex response of drainage systems', in MORISAWA, M. (ed.) Proc. 4th Geomorph. Symp. fluvial geomorphology (Binghampton, New York) pp. 299-310

SHAW, R. (1981) 'Strange attractors, chaotic behaviour and information flow', Z. Naturforsch. 36a: 80-112

SHIMADA, I. and NAGASHIMA, T. (1979) 'A numerical approach to ergodic problem of dissipative dynamical systems', Prog. Theo. Phys. 61: 1605-15

SIBBESEN, E. (1986) 'Soil-movement in long term field experiments, Plant & Soil 91: 73-85

STONE, C. J. (1977) 'Consistent nonparametric regression', Ann. Statist. 5: 595-645

STUART, J. T. (1977) 'Bifurcation theory in non-linear hydrodynamic stability', in RABINOWITZ, P. H. (ed.) Applications of bifurcation theory (Academic, New York) pp. 127-47

SWERLING, P. (1962) 'Statistical properties of the contours of random surfaces', I.R.E. Trans. Information Theory IT-8: 315-21

TAYLOR, G. I. (1921) 'Diffusion by continuous movements', Proc. London Math. Soc. 20: 196-212

TAYLOR, G. I. (1935) 'Statistical theory of turbulence', Proc. Roy. Soc. A151: 421-78

TURING, A. M. (1952) 'The chemical basis of morpho- genesis', Proc. Roy. Soc. B237: 37-72

WHITEHOUSE, D. J. and ACHARD, J. F. (1970) 'The properties of random surfaces of significance in their contact', Proc. Roy. Soc. A316: 97-121

WIENER, N. (1923) 'Differential space', J. Math. & Phys. 2: 131-74 Collected Works (1977) I. 23d, 455-98

WIENER, N. (1938) 'The homogenous chaos', Am. J. Math. 60: 897-936 Collected Works (1977) I. 38a, 572-611

WIENER, N. (1939) 'The use of statistical theory in the study of turbulence', Proc. 5th Int. Congr. App. Math. 356-8 Collected Works (1977) I. 39b, 641-51

WIENER, N. (1977) Collected Works I. Ed. P. Masani (M.I.T. Press, Cambridge, Mass)

YALIN, M. S. (1977) Mechanics of sediment transport (Pergamon, Oxford)



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