dimension and entropy in the soil-covered landscape

EARTH SURFACE PROCESSES AND LANDFORMS, VOL. 1 3 , 6 1 9 4 8 (1988)

DIMENSION AND ENTROPY IN THE SOIL-COVERED LANDSCAPE

W. E. H. CULLING 28. Under Ffrydd Wood, Knighton. Powys, LD? IEF, U.K.

Received 19 February 1987 Revised 3 August 1987

ABSTRACT

Both the Hausdorff dimension and the K-entropy supply a measure of the irregularity of the landspace surface. The relationship between the two measures is investigated over a variety of terrains in Britain and a method of calculating the entropy is checked against an independent estimate of the dimension with reasonable agreement. The calculation of the K-entropy requires that the landscape surface be represented by an homogenous ergodic random field. This condition is satisfied by the tendency of soil-covered terrains to progressively approximate to a form well represented by a Gaussian field.

Gaussian random fields can either be very smooth, possessing derivatives of all orders at every point or they are highly irregular and non-differentiable everywhere. Within the regular conceptualization the Rice-Kac theory is used to predict the numbers of crossing points and the extent of excursion sets. These predictions are tested against an example terrain from the High Weald of East Sussex with very good agreement, apart from predictions of local maxima. A worked example of the calculation of the K-entropy is given as an appendix.

The potential role of information theory in geomorphology extends beyond the use made of entropy in this investigation. In particular ergodic theory has important practical and theoretical implications.

KEY WORDS Soil-covered landscapes Gaussian random fields K-entropy Hausdorff dimension Excursion sets Ergodic theory Davisian denudation theory

DIMENSION AND ENTROPY IN THE LANDSCAPE

The close connection between the Hausdorff dimension and the entropy of a linear traverse across a landscape surface has already been noted (Culling, 1986a; Culling and Datko, 1987). The additive properties of the dimensions of Cartesian products of sets of points allows for the reduction of problems relating to the dimension of the landscape surface to those expressible in terms of the set of points produced by the intersection of a vertical plane and a level (contour) set. The set of points so produced can be used to code the sequence of values tracing the profile of the intersection as a number on the unit interval. The entropy of this number sequence is intimately related to the dimension of the landscape surface.

Theoretical support resides in the intersection of the theories of probability, ergodicity, entropy, and dimension. The application of these ideas to the study of landscape geometry is not straightforward and can lead to contradiction and paradox. A reasonably rigorous infrastructure is now available, whereby, with care, the various contradictions of a naive approach can be turned to advantage and in effect enable the geomorphologist to get the best of both worlds. The worlds in question being, on the one hand, a regular smooth idealization of the landscape surface and on the other a highly erratic (fractal) viewpoint. The term fractal (Mandelbrot, 1975) is used solely in this general sense as the opposite to smooth. It is not well defined (Falconer, 1985) and we prefer to deal directly with the various properties associated with the term,

0197-9337/88/070619-30$15.00 0 1988 by John Wiley & Sons, Ltd.

620 W. E. H. CULLING

particularly the Hausdorff measure and dimension, which are well defined. The mathematical background has been built up over a series of papers (Culling, 1986a, b, 1987a, b; Culling and Datko, 1987) and is outlined in an introductory section.

The main concern of this paper is the empirical test of certain predictions. The central investigation involves the comparison between the values for the Hausdorff dimension derived from a calculation of the entropy and values derived from independent estimates. This has proceeded in three stages. First existing work on 15 sites from Southern England (numbered 1-15 on Figure 3) and reported in Culling and Datko, 1987 has been supplemented by a calculation of the entropy value. Then for this present exercise a further 35 sites were chosen and both the entropy and the Hausdorff dimension calculated separately. Of these the first 25 (1640 on the map of Figure 3) were considered to be well-developed degradation1 terrains such that predictions made in Culling (1986a) were applicable. This was held not to apply, or at least be doubtful, about the extension to the latter 10 sites (41-50 on the map) from Upland Britain.

In the earlier paper (Culling and Datko, 1987), one site, the Burwash-Brightling area of East Sussex, was selected for more detailed treatment. This has been repeated here in two connections. One covers the robustness of estimates derived from O.S. 1 : 25 000 maps and the problems related to the meso-relief. The second is a test of the predictions of the Rice-Kac theory of level crossing points and excursion sets. For this latter an extended traverse of 30 km based upon the Burwash sheet was investigated. As well as providing material for testing the predictions of the theory the improved data base enabled a sharper estimate of the entropy and this has been given as a worked example in an appendix.

A final section reviews the use of information theory in geomorphology, ranging from the application of the methods of this investigation to other aspects of the subject to more fundamental and recondite matters relating to ergodic theory and the geometrization of Davisian denudation theory.

THEORETICAL BACKGROUND

The Hausdorff dimension and the K-entropy share a similar origin in the need for a discriminant between what had hitherto been indistinguishable sets of points. Hausdorff measure theory puts a size to sets of discrete points, in particular Cantor sets which all have a Lebesgue measure zero. The K-entropy dis- tinguishes between Bernouilli automorphisms. These are the most strongly mixing class of dynamical systems, the simplest example of which is the flow of all sets of sequences of coin-tossing (Bernouilli) trials. In both cases distinction is achieved by defining structures that previously slipped through the conceptual net. In this way and in terms of the degree of irregularity of the surface these two properties provide for a finer conception of the landscape than the customary intuitive approach.

The power of this conception is realized upon a return to the situation posed in the second sentence of the first paragraph and illustrated in Figure 1. Under certain conditions it is possible to derive from the set of intersection points the entropy of the profile sequence, then the dimension of the profile, and in turn, of the surface as a whole. It is also possible to predict the mean number of crossing points and the mean length of the excursion set for any given level as well as the probability of occurrence of isolated peaks. In fact, in the planar case, which is not covered by this paper, given a e value of the elevation at a point on the landscape surface and some knowledge about the derivatives at that point, it is possible to calculate the probability of occurrence of an isolated peak occurring within a given distance from that point.

In order to make any progress with the problem of Figure 1 some structure will have to be put into the space between the intersection points. In this there is little choice; it has to be of a Gaussian nature. The given landscape is regarded as the sole realization of the set of all possible landscapes for that area. This set plus an appropriate probability measure will constitute an ensemble and this is assumed to be a Gaussian field. Although there is only one sample function from this ensemble, the actual realized landscape, later in the paper we return to the possibility of isomorphic landscapes presenting more than one sample. The landscape surface itself can be regarded as an ensemble comprising the set of all possible profiles plus the appropriate probability measure. If the surface, assumed homogenous, is represented by a Gaussian random field then a generic profile curve will comprise a sample Gaussian line graph. The construction of the probability apparatus to put such an ensemble on a proper footing is given in Culling (1987a).

DIMENSION AND ENTROPY I N THE SOIL-COVERED LANDSCAPE 62 1

Figure 1. Set of intersection points between a contour and a vertical plane

The restriction to Gaussian random fields turns out not to be so arbitrary as at first sight appears. If the landscape is subject to a diffusion type degradation (or more generally a Davisian downwasting regime), then provided it has reached a mature stage the distribution of gradient (increments) along a representative traverse will satisfy the conditions of the Central Limit Theorem and will tend, with sample size, to a Gaussian distribution about a zero mean. This implies that the line graph will be Brownian, but because of the dependency in the landscape it will not also be Markovian, in which case the Brownian motion is fractional. The choice of traverse being arbitrary these properties extend to the planar case where the landscape is well represented by a fractional N-parameter Levy Brownian surface.

To summarize, the soil-covered landscape tends to straighten out its curvature (Culling, 1965), progressively smoothing the surface and reducing the value of the Hausdorff dimension. This prediction that the soil-covered landscape tends to a Gaussian form of low dimension is bourne out on investigation (Culling and Datko, 1987) with further reinforcement from the results recorded in this paper. It is fortunate that the geometry of the soil-covered landscape is that of Gaussian random fields for, once outside the Gaussian realm, the results available are inadequate for the construction of a useful model of the landscape. The standard reference for Gaussian random fields is Adler (1981).

The most important property of Gaussian fields for our purposes is that they can be divided into two distinct and separate classes. They are either very smooth or very irregular. This dichotomy, due to Belyaev (1961), is sketched in Culling and Datko (1987) and dealt with more fully in Culling (1987b). A geomorphic version of the criterion for regularity, the sufficient regularity of Adler (1981, p. 40), is that a continuous contour line can be drawn on the surface to represent the level set for any level within the range of the surface. If a Gaussian surface is not regular then it is so irregular that a continuous contour line does not exist anywhere on the surface. In a rough sense such a surface is fractal. More precisely there exists a well-defined class of irregular Gaussian fields, the so-called index+ Gaussian fields. The index /I first conceived by Orey (1970) is of central importance. A definition in terms of Holder conditions is given by Adler (1981, p. 200).


The index /? is related to the Hausdorff dimension via the complement (1 -/?), (OO (Hida, 1980, p. 108). The implications for the application of Brownian surfaces to the landscape are discussed further in Culling (1987a). This selection procedure means that we never deal with the totality of the random variable. Aberrant sample functions are neglected in much the same manner as 'accidental' regularities are omitted from published tables of random numbers.

This irreconcilable divorce between the mathematical and physical viewpoint need cause the geomorphologist little dismay. The discrepancy ultimately defeats any rigorous justification, but it does not preclude a mathematical explanation of physical phenomena. The original explanation of Brownian motion by Einstein and Smoluchowski ran into the same difficulty, basing a second order diffusion equation on a function possessing no finite derivative. The scale is vastly different but the principle remains the same, a physical application of local mathematical properties cannot be taken to the limit. A halt has to be made somewhere, whether at lo2' collisions a second or 50 m.

To return to the dimensions of index-fl Gaussian fields; if /?#+ then the Brownian motion is fractional. For the soil-covered landscape /? takes values > 0.65 and so the dimension of the 2-parameter Brownian surface representing the landscape has a value between 2.0 and 2.35, whereas the microrelief of the landscape surface or certain soil properties are so irregular that /? takes a value - 0 2 and therefore the dimension rises to - 2.8 (Culling, 1986b).

The integer relationship between the dimension values for different values of the parameter N holds for all Gaussian fields whether Brownian or not. In more general terms it can be expressed in terms of the additivity of the component dimensions of Cartesian productsrof sets of points

Dim(A x B)=Dim(A)+Dim(B) (4)

This is familiar in the integral case, the dimension of a plane set is 1 + 1 , but it also applies to sets of fractional dimension. Inversely, a Brownian surface of dimension 2.5 can be divided into a planar set of dimension 2 and a fractal set of points of dimension 0-5. The division by a line set can be made horizontally as well as vertically to give for the profile and the contour set a dimension of 1.5.

Armed with this knowledge half the problem posed by Figure 1 is solved. If the dimension of the set of intersection points can be calculated then the dimensions of the contour, the profile, and the surface are immediately available. Furthermore an estimation of the dimension using entropy methods can be checked against values based independently on the contour, the profile, or the surface. This presupposes a Gaussian surface but there is considerable robustness; Equation 4 applies to a wider class than Gaussian processes.

DIMENSION AND ENTROPY IN THE SOIL-COVERED LANDSCAPE 623

To summarize this section, the theory of Gaussian random fields is employed to impart precision to the idealization that is put between the observer and the landscape to make the latter conceivable. This applies equally to the conventional regular conception as to the less familiar, irregular, fractal model. The distinction between the regular and irregular modes of the Belyaev dichotomy rests upon local properties of Gaussian fields but as in all physical applications the mathematical model stops far short of the limit. The physical surface of the landscape is not as smooth as a contour map implies nor is it so irregular in detail as an index-P Gaussian field.

DIMENSION AND ENTROPY

If M represents the set of points of intersection of a function f, a profile sequence of discrete values, and h(t) is the K-entropy of M

where r is the partition number; the number of subdivisions of the range of the function. The K or Kolmogorov entropy (Kolmogorov, 1958, 1959), apart from one aspect, is identical with the

Shannon (1948) entropy of information theory. Knowing the value of the Hausdorff dimension and providing certain conditions are met, the K-entropy for any finite partition is readily calculable. This is of value in the study of dynamical systems; of more immediate and practical importance to geomorphology is the reverse procedure whereby an independent estimate of the entropy is converted into a value for the dimension. This is possible using a procedure termed one-dimensional Markovisation (Aizawa, 1984; Culling, 1987a), based upon the work of Billingsley (1960, 1965).

The original Shannon scheme dealt with discrete Markov sources and Billingsley was able to show rigorously that Equation 5 applies to Markov automorphisms where

dim ( M ) = l/ln rh(t) ( 5 )

h(t) = -CPiCPijIn Pij (6 1 where the pi are the stationary probabilities and the pij the elements of the transition probability matrix; the sums being taken over the partition number r (Cornfeld et al., 1982, p. 246). A less than rigorous but nevertheless usable theoretical justification of the extension of such a formulation to stationary ergodic sources and fractional Brownian motions and therefore suitable for application to the soil-covered landscape is given in Culling (1987a).

The calculation of the entropy is based upon the coding of the profile as a source. In the terminology of information theory, the partition number is then the number of letters in the alphabet. For the most part we use the simplest binary case but for the Burwash traverse and the worked example the partition number is 11, this being dictated by the range and the contour interval. In Figure 2 the profile of the line of intersection of Figure 1 is taken as an example of binary coding, the partition in this case being either above or below the mean value. The coding transforms the profile sequence to a single number on the unit interval to the base r.

The binary case of Equation 6 is

h(t) = - { Po[ Po0 In Po0 + Po 1 In Po 11 + PI CPlO In PlO + P* 1 In P I 11 1 (7) The stationary probabilities, po and p1 shodd be equal and as C p , = 1, of value t , but this is rarely the case

0.11 001 110 o..... Figure 2. Binary coding of a profile sequence


from an empirical count. The transition probabilities are also estimated from a count of like successions, pii or unlike successions, p i j .

The relationship between the Hausdorff dimension and the K-entropy for the binary case is illustrated in Figure 3. Apart from the dimension scale this is a replica of the graph given originally by Shannon (1948, p. 394, Figure 7). With increase of the transition probability from zero to 05, the value of the entropy rises from zero, rapidly at first and then more gradually as the maximum is reached at p i = * . This maximum value is in all cases l / lnr, when the value for the Hausdorff dimension reaches a maximum of unity. Thus the greater the number of transitions the greater the value of the entropy. On the other hand the longer the strings of similar letters in the sequence the smoother the profile and the less the value of the entropy.

There is a problem involved In the coding of a profile sequence. The coding procedure outlined above assumes that the sequence is discrete. Although in practice any measured set of values will certainly be so, the essential continuity of the landscape profile renders the calculated K-entropy dependent upon the choice of coordinate system.

In relation to the h dimension, the value of the K-entropy is a function of the partition number and with increased resolution the value of the K-entropy increases. With respect to the time dimension the K-entropy, like the Reynold's number, is indeterminate to a length. Consequently, given a continuous source. by manipulating the magnitude of the time interval it is possible to arrange for any entropy value from zero to the maximum given by In r .

843

"9 8 45

848

.47

.815 14 8l

822 .d,

Figure 3. Location of sites investigated


In calculating the Hausdorff dimension, Equation 5, the entropy value is divided by the maximum, In r, effectively scaling the measure over the unit interval and thereby resolving the difficulty in the h dimension. In the language of statistical mechanics, entropy is an extensive property and varies, in general, with the size of the system; whereas dimension is intensive and scale invariant.

In the time dimension a convention about the unit length has to be made before comparisons are valid, as is the case with Reynolds numbers. The choice has settled on 50 m. This was dictated, in the first place, by the available data sets, (O.S. 1 : 25 000 maps), where a greater precision is difficult to apply or justify. More importantly 50 m was held to mark the upper limit to the effect of the meso-relief. This includes rills and minor gullies, scars and other bare rock outcrops, hollows and minor eminences, and all other minor disturbances to the surface both natural and artificial. The meso-relief will tend to increase the surface roughness and therefore also the entropy. The onset of meso-relief marks the transition between two fractal bands. In taking 50 m as the unit interval we calculate the practical minimum entropy value. There is also a possible upper limit to the fractal band marking the onset of the effect of the drainage net upon the landscape.

The choice of 50 m was made before entropy was considered as an irregularity parameter of the landscape. However, from results recorded later it appears to work quite well and this success needs explaining. Why should an ostensibly conventional choice of a unit interval tie in so well with results derived from Richardson plots of the dimension where no such conventions are required? A rigorous justification will require a full scale measure theoretic approach which cannot be entered into here, but later, in the appendix, an heueristic explanation is given as to why the choice of 50 m is appropriate in the present circumstances. It is clear, however, that the extensive character of entropy implies care in its use and the stating of the basis of comparison (i.e. the coordinate system) on all occasions.

THEORY OF LEVEL CROSSING POINTS AND EXCURSION SETS

In Figure 1 the set of intersection points form the set of crossing points for the level of the contour. The set of points at or above the level of the contour form the excursion set. In general and for the linear case, the excursion set comprises a number of closed intervals on the real line, the closure being the set of crossing points.

Since the pioneering work of Rice (1944) and the allied work of Kac (1943a, b), the theory of level crossing points has attracted a great deal of attention. The latest paper appears to be Farahmand (1986). The standard reference for one-dimensional processes is still Cramer and Leadbetter (1967, Chs. 10, 13, and 14), and although we will not follow here, the difficult extension to N dimensions is discussed in Adler (1981).

A straightforward use of crossing point theory presupposes the existence of suitably regular (Gaussian) functions and therefore any geomorphic application works within the idealization of a regular model of the landscape. If this restriction is not followed then the number of crossing points becomes infinite at any level within the range. If Figure 1 represented an index-/? Gaussian surface, then upon resolution each intersection point would reveal a series of intersection sets, each one self-similar and resembling Figure 1 . The contour line itself will disintegrate into a set of discrete points and the very idea of an intersection point is lost.

What we aim to do is to work within the regular model and use the Rice-Kac theory to calculate the mean number of crossing points and then to use this information to calculate the K-entropy and the Hausdorff dimension, which is within the domain of the irregular model. We are able to do so because the apparent contradiction occurs at the local level and will take place beyond the orbit of the landscape models. In practice this limit is set by such as the accuracy of the data set, the choice of unit interval, and so on, and not by theoretical considerations, for the mathematical models can be pushed much further towards the limiting condition than will be necessary for any physical application. This justification for an apparent contradiction applies not only to crossing point theory and entropy calculations but also to any use of contour maps and fractal concepts by geomorphologists.

If {X(t), t >O} is a stationary, zero-mean Gaussian process, with covariance function, r ( t ) = cov(X(O), X(t)} and spectral density function f, where

(8 )


then the mean value for the number of up-crossings, C,, of the level u, of a regular Gaussian process in the interval [0, T I is

This formula first given by Rice (1944) under a number of unnecessary assumptions has since been improved upon by many workers and Ito (1964) has supplied minimal conditions. It is fundamental in the practical application of Gaussian processes. The implications with respect to excursion sets bulk large in reliability theory where it is of great importance to be able to state the probability that a variable will not exceed a certain value over a given interval. There are obvious applications to the whole question of thresholds in geomorpholog y.

A further important application concerns the estimation of the parameter 1,. This is the only quantity in Equation 9 that relates to the sample function and given the variance

@(u) = 2710 exp(u2/20Z) C, (10) provides an unbiased statistic of 1, from a realization of the process. It is to be expected that the collation of estimates of A2 from a variety of values of u will increase the amount of information employed and therefore of the precision of the estimator. This aspect has been taken up by, among others, Hashofer and Sharpe (1969) and Lindgren (1974). Furthermore Sharpe (1978) has suggested that the number of crossing points can be used as a diagnostic for a Gaussian structure.

In practice interest attaches to the estimation of the mean frequency y, given by the bracketed term of Eauation 9

where 1, is the second spectral moment. Now (Lindgren, 1974, p. 401),

so that

and

Normalizing to ,lo = 1, then

and

1 Z k - - (-)kr(2k)(0)=var{dkX(t)/dtk)

y = ( - r(O)/r(0))1/2

A, = E { IX(t)(2} =cZ =r(O)

1, = y 2 = E [X(t)Z]

T {x(t)2}ddt

T o

provides an unbiased (integral) estimator of the mean square frequency. Conditions for the interpretation of X ( t ) as a sample path derivative are given by Cramer and Leadbetter (1967, p. 188).

( 1 In the discrete case, for interval length h

where N h = T. The estimator is biassed

with variance E{y^,Z} = l/h2E{(Xl -X,))


where u k is defined v k = I/h cov { (xk+ 1 -x,), ( X I -XO))

= 2/h2 [ r { ( k - 1)h) - 2r(kh)+ Y { ( k + 1)h}] (20) (Lindgren, 1974, p. 417).

of the mean frequency

The sampling properties of f, as an estimator of y were given in a closed and complicated form by Rice (1944). The first manageable expression for the variance was given by Steinberg et al. (1955)

In many cases it is found much simpler to use a count of crossings to give an unbiased (crossing) estimator

f c = 271/ TC (0, T ) (21)

var { fc} = (271/T var { co(o, T )} = (471/r 1 { { Cd T ) } {co( T ) - I} ] - E { C , ( T ) ) {E[Co( T)I- 1)) (22)

The three types of estimator are discussed and compared by Lindgren (1974) who shows that the zero- crossing estimator can often compete with the integral estimator in efficiency and can be improved significantly by including information from non-zero crossings.

Turning to excursion sets, provided the Gaussian process is ergodic (which in practice means de-trending if necessary and that points sufficiently separated are independent), then the mean length of the excursion sets above a level u is given quite simply by

p { X ( 0 ) > u}p- (23) where p is the mean number of up-crossings of the level u per unit time. This appears intuitively right; the division by the number of crossings is qualified by the probability of the function equalling or exceeding the value u. Details of this and related quantities are found in Cramer and Leadbetter (1967) and a review in Culling and Datko (1987).

Finally we instance the question of local maxima. Considerable results are available on maximal and extremal values of Gaussian processes. These will be of much greater value when we come to deal with the planar case. In the meantime tests in the one-dimensional case from the Burwash traverse have been found disappointing. In studying sample functions of a Gaussian process it is not entirely clear what near a local maximum of level u implies. Lindgren (1970) discusses the problem and gives an ergodic definition that has the important practical outcome that interpretations can be made from a single realization.

A local maximum will occur on the trace of a function f(t) whenever the derivative f(t) has a down-crossing of the zero level. The derivative of a Gaussian process is also Gaussian and the results of crossing theory apply equally to f(t). However, there is an increase in complexity in relating these results back to f(t). In raising the order of the derivative by one, the order of the parameters is doubled. Thus f(t) has variance I , and a second spectral moment A4; the expectation of the down-crossings of zero by the derivative being

It can be shown that for a stationary stochastic process a distribution function does exist for the conditional probability that the value of f(t) does not exceed a value u, at a point in [ t o - h, t o ] , given that a local maximum occurs within that interval for sufficiently small h. If, furthermore, the process is ergodic then this distribution function tends to a limit as T-co with probability one. Only in the case of a Gaussian process can the distribution function be expressed explicitly and then in much more complicated form than Equation 9 (Rice, 1944, pp. 77-81; Cramer and Leadbetter, 1967, pp. 242-248; Lindgren, 1970).

RESULTS AND DISCUSSION

Observations are entirely subservient to the theory. Sites have been selected so as to give an adequate test of the predictions. Provided this requirement is met the precise location is of less consequence. Nevertheless the


ENTROPY & HAUSOORFF DIMENSION

Figure 4. Relationship between K-entropy and Hausdorff dimension for a binary partition

sites are marked on Figure 3 which is an extension of the map and the numbering of Culling and Datko, 1987, Figure 1.

The sites are based upon O.S. 1 : 25 000 sheets and in most cases take advantage of the entire sheet area. It is desirable that topography be as homogenous and isotropic as is practicable. This is mainly a matter of geology and as in the earlier paper this is listed against each site in Table I . The relationship with the main geological outcrops can be followed on the G.S. Ten Mile (1 : 625 000) sheets on which the 1 : 25000 sheet boundaries are marked. The selected terrain needs to satisfy certain conditions. It must supply adequate relief with a sufficient contour density to capture the landscape form. Large rivers, alluvial tracts, and areas of low relief are to be avoided. The check on the entropy calculations is based upon Richardson plots. As this method records the dependence of the measured length of a contour line upon the divider distance it is essential that a sufficient length of contour near to the mean elevation is available. This rules out areas of isolated features and built up areas where the contour line may be obscured. The ideal site is a remote, mature, soil-covered landscape of adequate relief and usually to be found in the upper reaches of a drainage bash ;I not straddLng the watershed. Perhaps it should come as no surprise that sheets satisfying all the criteria are not plentiful.

Reliance upon map data is error prone. However, the entropy calculations are based upon intersections and the choice of contour is robust to reasonable deviation from a symmetrical partitioning so many of the sources of error are eliminated.

The investigation falls under two main headings. One based upon 50 sites deals with the assessment of the entropy method of calculating the dimension. The other based upon the single Burwash sheet and the associated traverse is concerned mainly with the application of crossing point theory but also with ancilliary problems arising out of the first study; matters of robustness, homogeneity, and isotropy and with meso- relief.

ASSESSMENT OF ENTROPY MEASURES There are two questions to be answered:

1. Does a binary coding of the profile sequence provide sufficient information for the determination of the K- entropy? In other words, is it possible to calculate the K-entropy from the situation depicted in Figure l? This will depend upon the boundedness of the landscape profile and in this particular case, on its ergodic Gaussian nature.

2. Is the estimate of the Hausdorff dimension derived from the calculated K-entropy consistent with values derived by other methods? For the most part these are based on Richardson similarity methods but for the first 15 sites also on variogram methods.

In addition certain information is gained relating to the dimensional aspects of the evolution of soil covered terrains.


Method The history of the investigation of the 50 sites falls into three stages. The first 15 had been part of an earliei

study into the Hausdorff dimension (Culling and Datko, 1987, Table 1) and it needed only to estimate the entropy.

A further 25 sites (16-40 in Figure 3) were selected solely to test the entropy estimation procedure. The coverage of Southern Britain was considerably increased and a preponderance of Chalk terrains reduced. In each case a binary coding of a transect profile was used together with a Richardson plot of an appropriate contour. Where it was not possible to employ a diagonal transect one running north-south or east-west was substituted.

In three cases the sheet area has been split. In SK 24, Brailsford, only the southern Triassic country is examined. In SN 42, Carmarthen, the coastal plateaux region has been excluded, while in SN 73 the sheet area has been divided into two and both areas recorded. To the northeast the Burcombe Park ridge of Exmoor was treated separately from the dissected area of the headwaters of the Mole.

Finally the survey was extended to a further lO(41-50) sites from Upland Britain. These were regarded as just falling outside the category of terrains dominated by a diffusion type degradation. To a variable extent the control of landscape form by processes operative through the soil cover fails; soils are thin and bed rock outcrops. Procedure was as for the preceeding 25 sites. The Sedbergh site (SD 69) covers only the northeast section of the sheet; part of the Howgill Fells.

In addition, advantage was taken of an earlier ancilliary study designed to check the use of map data with that supplied by O.S. digital tape data of the northeast quadrant of SO 39, Shelve, sheet. Transects were north-south and taken at 500m intervals. The independent estimate of the dimension was based on variogram methods, detailed in Culling (1986b).

Results These are listed in Table I. In the first column the numbering extends that of the earlier paper. Then the

National Grid reference, a location name, and the geology (with subsidiary outcrops bracketed) again follows the pattern of Culling and Datko (1987, Table 1). The calculated K-entropy, h(t) is found in column 4 and in the next the derived estimate of the Hausdorff dimension, DHe, a mere matter of multiplying by l/ln 2 = 1.4427 . . . The value from a Richardson plot, DH,, is given in column 6. In many cases a second structure of higher dimension can be detected and this, if present, is indicated in brackets. In the final column the discrepancy between the two estimates of the dimension is given, based upon the value of D,, as a datum. Using the same format the data from the Shelve area are listed in Table 11.

Discussion From the column of discrepancies, 33 are less than 5 per cent and a further six lie between 5-10 per cent.

This leaves 11 where the comparison is poor. A likely explanation of these large discrepancies, which are invariably positive, lies in the strong presence of more than one structure in the landscape. Where two (or possibly more) structures of differing dimension are present the lower value corresponds to the smooth degradational surface, while the upper value (1.35-1.6) represents the influence of the drainage pattern. The drainage net itself usually has a dimension approaching 1.5-1.6. The fact that the second value in column 6 is usually less than this reflects the amelioration due to degradational processes and to the inevitable smoothing in the cartography.

Rejuvenation tends to oppose the smoothing effect of degradation so raising the value of the dimension towards that of the drainage. Given adequate resolution and precision a Richardson plot can distinguish between different dimensioned structures in the landscape. With digital data the precision is remarkable (Culling and Datko, 1987, Figure 4). In contrast the K-entropy based upon a binary coding presents just one number that collates the influence of all structures present. This value is likely to be intermediate and that this is so is instanced in Table I and illustrated in Figure 3 of Culling and Datko (1987). The K-entropy thus gives an overall measure of the balance between rejuvenation and degradation.

A further major source of error arises in areas of low relief particularly if a surveyed contour is used. A noticeable increase in irregularity is due to the picking up of the influence of the meso-relief. Every hollow

Table I. Entropy and dimension of selected sites in Britain

W ) DIfe D H , Discrepancy

1 TQ62 2 TQ02 3 SU81 4 SU89

5 s u 5 4 6 SU37 7 SY 59

8 sx 77 9 s x 4 7

10 SP 82 11 SP 12 12 SK 70 13 SO 77 14 SO 28 15 SO 39 16 SE 26 17 SE95 18 SJ 64 19 SK 24 20 SK 65 21 SN 42

22 SO 14 23 SP 43

24 SP I 1 25 SP 85

26 SS 60 21 ss 73 28 SS 73 29 ST 02 30 St 85

31 ST 99 32 SU 01

33 sw 75 34 SX 17 35 T F 37

36 TF 83 37 TG 12 38 TL43 39 TL 16 40 TM 16 41 NN 90 42 NT 05 43 NT81 44 NX47 45 NY 67 46 NZ 60 47 SD 65 48 SD 69 49 SE 09 50 SH 62

Burwash, East Sussex. Hasings Beds. Billingshurst, West Sussex. Weald Clay Singleton, West Sussex. Chalk. High Wycombe, Bucks. Chalk (C.W.~; Reading

Beds; London Clay). Popham, Hants. Chalk. Lambourn, Berks. Chalk. Egdon Heath, Dorset. Chalk (Gault, Grt. Oolite,

Dartmoor, Devon. Granite, (Culm) Tavistock, Devon. Culm, U. Devonian. (Granite,

Stewkley, Bucks. Upper Jurassic, Gault. Stow, Gloucs. Inf. and Grt. Oolite (Lias) Ribbesdon, Leics. Liassic. Forest of Wyre, Worcs. Coal Measures. Clun Forest, Salop. Ludlow, Downtonian. Shelve, Salop. Pre-Cambrian, Cam., Ordovician. Ripley, N. Yorks. Millstone Grit. (Mag. Limst.) Wetwang, Humbs. Chalk, Quaternary. Audlem, Cheshire. Keuper. (L. Lias.) Brailsford, Derbys. Triassic. Southwell, Notts. Triassic. Carmarthen, Dyfed.

Painscastle, Powys. Ludlow, O.R.S. Hook Norton, Oxon. Inf. and Grt. Oolite, U.

Brixworth, Northants. Inf. Oolite, U. Lias. Yardley Chase, Northants. Grt. Oolite. (Inf.

North Tawton, Devon. Culm. (Permian). Exmoor, Devon. Devonian. (Culm). North Molton, Devon. Culm. Wiveliscombe, Devon. Devonian. (Culm). Trowbridge, Wilts. Oxfordian. (Grt. Oolite,

Rodmarton, Gloucs. Grt. Oolite. Cranbourne Chase, Dorset. Chalk. (Reading

Chacewater, Cornwall. Devonian. Bodmin Moor, Cornwall. Granite. Somersby, Lincs. Kimmeridge. (Lower

Sculthorpe, Norfolk. Chalk, Boulder CJay. Aylsham, Norfolk. Crags. Boulder Clay. Clavering, Suffolk. Chalk, Boulder Clay. Higham, Suffolk. Chalk, Boulder Clay. Debenham, Suffolk. Crags, Boulder Clay. Gleneagles, Tayside. New Granite. Cobbinshaw, Lothian. O.R.S., Calciferous Sandst. Cocklaw, Northumbs. Upper O.R.S. Minnigaff, Galloway. Llandovery. Spadeadam, Cumbs. Carboniferous Limst. Westerdale, N. Yorks. Grt. Oolite, Lias. Bowland, Lancs. Millstone Grit, Carb. Limst. Sedbergh, Cumbria. Ludlow, Wenlock. Redmire, W. Yorks. Carb. Limst. Harlech Dome, Gwynedd. Cambrian.

U. Lias.)

Spilite)

Up. Ordovician, Llandovery.

and M. Lias.

Oolite, U. Lias)

Kimmeridge, Gault)

Beds, London Clay)

Greensand).

0.188 0.163 0.180

0.162 0.121 0.175

0.109 0.080

0.160 0.141 0.125 0.142 0.176 0.177 0.1 19 0.076 0.129 0.086 0.185 0.165

0.176 0.1 16

0.104 0.07 1

0.098 0.238 0.121 0.210 0.165

0.195 0.175

0.164 0.208 0.1 16

0.141 0.142 0.151 0.129 0.104 0.104 0.1 16 0.151 0.103 0.140 0.117 0.098 0.166 0.173 0.104 0.075

1.275 1.114 (1.387) 1.235 1.193 1.260 1.036 (1.239)

1.234 1.099 (1.350) 1.175 1.073 (1.295) 1.225 1.109 (1.287)

1.155 1.186 (1.532) 1.095 1.059 (1.292)

1.235 1.179 (1.369) 1.200 1-148 (1.381) 1.185 1.147 1.205 1.260 1.255 1.092 (1.337) 1.255 1.043 (1.263) 1.170 1.183 (1.297) 1,110 1.114 (1.600) 1.186 1.160 (1.320) 1.124 1.150 1.267 1.290 1.238 1.260 (1.375)

1.254 1.275 (1.420) 1.167 1.189

1.150 1.125 .(1.310) 1.103 1.176 (1.318)

1.141 1.138 (1.480) 1.344 1.314 1.175 1.107 1.303 1.300 1,238 1-242 (1.620)

1.281 1.160 1.253 1.196

1.231 1.190 1.300 1.037 (1.490) 1.167 1.170 (1.380)

1.203 1.182 (1.375) 1.205 1.176 (1.318) 1.217 1.236 (1-425) 1.187 1.217 1.149 1.176 1.149 1.100 (1.550) 1.167 1.250 1.218 1.100 (1.317) 1.149 1.150 (1.600) 1.202 1.062 (1.400) 1.169 1.167 (1.380) 1.141 1.158 (1.350) 1.239 1.062 (1.375) 1.249 1.250 (1.450) 1.149 1.087 1.108 1.133

+0.161 + 0.042 + 0.224 +0.135 +0.102 +0.116

- 0.03 1 + 0.036 + 0.056 + 0.052 + 0.038 -0.055 +0.163 +0.212 -0.013 - 0.004 + 0.026 - 0.026 - 0'023 - 0.022

- 0.02 1 - 0.022

+ 0.025 - 0'073

+ 0.003 + 0.030 + 0.068 + 0.003 - 0.004

+0.121 + 0.057 + 0.053 + 0.263 - 0.003

+ 0.02 1 + 0.073 - 0.046 - 0.030 - 0.027 + 0.049 - 0.083 +0.118 - 0.00 1 +0.140 + 0.002 -0.017 +0.177 -0.001 + 0.062 - 0'025

DIMENSION AND ENTROPY IN THE SOIL-COVERED LANDSCAPE 63 1

Table 11. Shelve SO 39 NE

Transect h( t ) D,, D", Discrepancy

1 2 3 4 5 6 7 8 9 -

0.074 0.125 0072 0.072 0.058 0.071 0.147 0.124 0.124

1.107 1.180 1.104 1.104 1.084 1.102 1.212 1.179 1.179

1.09 (1.29) 1.16 (1.34) 1.08 (1.27) 1.07 (1.29) 1.08 (1.37) 1.10 (1.38) 1.05 (1.34) 1.12 (1.36) 1.05 (1.27)

+0.017 + 0.020 + 0.024 + 0.025 + 0.004 + 0.002 +0.162 + 0.059 +0.129

and rise in the ground, whether natural or artificial, will register. Being of greater irregularity the effect of the meso-relief is to raise the measured value of the entropy. To eliminate this effect is one reason for choosing 50 m as the unit interval, so introducing a high frequency filter. Even so some influence is likely to remain and affect the course of the contour line. If a transecihappens to run along the general direction of such a contour the number of intersections is increased. This topic is investigated further in a detailed study of the Burwash sheet.

The findings from the 50 sites were supplemented by the study of nine transects from the Shelve (SO 39) sheet. These are based upon digital data and the sheet area was originally chosen solely to provide a check on the use of map contour data. According to the criteria listed above, the Shelve area does not qualify and this increases the stringency of the test. That the results give reasonable confirmation is reassuring and this confirmation is found to extend to entropy calculations.

The northeast quadrant of the Shelve area (SO 39) was chosen from the small area of the Powys-Salop border covered by O.S. digital contour tapes. It is an elevated tract, mostly above 350m and covers the watershed between the two arms of the Upper Onny draining south and the Habberley and Minsterley Brooks draining northwards. Roughly parallel outcrops align NNE-SSW following the Pontesford-Linley fault which controlled deposition in the area during the Ordovician. Nearly half the area is underlain by Longmyndian, forming the western margin of the Longmynd itself (c. 380 m), the valley of the Upper East Onny at Bridges (c. 260 m), and the slopes rising westwards towards the Stiperstones ridge (536 m). Below the ridge the slopes are of poorly exposed, heather and bilberry clad, Habberley (Tremadoc) shales. The Stiperstones Quartzite (Arenig) is much faulted giving rise to the celebrated tors and clitter slopes. The ridge top is uneven and cut by a col at 425 m. To the northwest the lower (320-350 m), featureless area of Pennerley and The Bog is underlain by grits and flags of the Mytton (Arenig) member and the shales of the Hope (Llanvirn) member. In the extreme northwest the shales are intruded by the dolerite of Roundhill. To the north, Tankerville Hollow is one of the 'batches'; steep, unwooded valeys, the slope at the head often reaching 35" and possibly overdeepened by glacial action (Whittard, 1979; Earp and Hains, 1971). The Mytton member contains mineral veins and the country northwest of the Stiperstones is dotted with old mine workings (Dines, 1958).

In Table 11, apart from two cases, the discrepancies are less than 5 per cent. Some contamination of the two aberrant values is a possibility. The nature of the area raises the questions of homogeneity and isotropy and the level of consistency between the transects is more than could reasonably be expected. This topic is taken up again in the section on the Burwash sheet.

Apart from the exceptions noted, the agreement between the two sets of values for the dimension, from the 50 sites and the Shelve transects, is reasonable. This not only justifies the use of entropy methods to estimate the Hausdorff dimension it also supports the use of a binary coding.

Within a wider conception the results in Tables I and I1 reinforce and supplement the findings of the preceding papers (Culling, 1986a; Culling and Datko, 1987). The contention that a well-developed (mature) soil-covered landscape will present a surface of low dimension number is borne out. For the 50 sites the range


of D,, is 1.095-1.344. The mean value is 1.202 (a =0.0568) and in only three cases is a value of 2 1-3 recorded. These all stem from the Southwest Peninsula; SS 60, North Tawton (1.344); SS 73, North Molton (1.303) and SS 74, Chacewater (1.300). In each case strong rejuvenation by a dense drainage net is present in an area of adequate relief. These values can be contrasted with SS 73, Exmoor (1.175), SX 17, Bodmin Moor (1.167), and SX 77, Dartmoor (1.095).

Provided they can supply sufficient relief the clay vales, with high drainage density, show high values for the dimension: TQ 02, Billingshurst (1.235), SU 70, Ribbesdon (1.205), SK 65, Southwell (1.238), and ST 85, Trowbridge (1.281). High values also occur in Chalk country where the dry valley system is well incised; SU 81, Singleton (1.260), SU 89, High Wycombe (1.234), SU 37, Lambourn (1.225), and SU 01, Cranbourne Chase (1-237); contrasting with those of SU 54, Popham (1.175), where they are not. High values also occur where they would be expected, in those areas near to or affected by the rejuvenation of a large river; SO 77, Forest of Wyre (1.255), SO 28, Clun Forest (1.255) SK 24, Brailsford (1.267), and ST 99, Rodmarton (1.253). The high values recorded for T F 83, Sculthorpe (1.205) and TG 12, Aylsham (1.217) appear to be due to contamination by meso-relief. In the former the mean 200 contour can be seen to pursue an erratic course across the airfield.

Upland Britain The investigation was extended to a further 10 sites (41-50), in order to sample terrains that were

considered to lie beyond but not too far outside the orbit of the theory of soil-covered landscapes. Regularity and irregularity, entropy and dimension are not restricted to well-behaved landscapes, nor indeed to the existence of a theory of degradation. But problems are to be expected over scaling and the employment of a binary coding, as sample sites diverge from the soil-covered category and in particular can no longer be regarded as tending to approach a Gaussian form.

In the event discrepancies between the two estimates of dimension are low; lower in fact than for the mean of Table 1. The values themselves are also akin to those of the first 40 sites. This is not what would be expected. As the surface form ceases to be solely due to processes within the soil cover, as the relief increases, and as the drainage pattern coarsens the surface can no longer be regarded as Gaussian. The estimated dimensions would be expected to be significantly low with respect to the entropy measures and higher when on the basis of contour line calculations. In such a terrain a binary coding sacrifices too much information. The scaling is such that the sample number of crossing points is inadequate with the result that the calculated entropy is spuriously low. On the other hand the increased relief and an entrenched drainage will tend to raise the value of a dimension calculated from a Richardson plot.

Apart from the Harlech (SH 62) sheet, the remaining nine sites are just at or beyond the boundary of the category of soil-covered landscapes. They mark the commencement of a transition zone that ends in the bare rock slopes of mountainous terrains. For certain of these sites the low values may reflect the rounded or flat topped nature of much of the higher ground of Northern England and Southern Scotland. This suggests that, so far as dimension and entropy measurements can detect in these selected terrains, surface evolution is still predominantly controlled by the protective and transport properties of the soil cover, despite its being thin and intermittent. However, it must be remembered. that the theory of landscape development is based upon the presence of an exponential convergency factor izthe solution of a diffusion equation. It therefore includes all the linear variants that can be reduced to the standard diffusion equation, which cover a mass transport term (Hirano, 1968) and distance dependent functions giving Bessel or Legendre type solutions (Trofimov and Moskovkin, 1976), Culling (1986a, p. 233). It is by no means limited to soil creep but will certainly include terrains fashioned by slope wash or indeed any other process that can be approximated by a linear (first order) function of the slope gradient (Kirkby, 1971).

The low values recorded for the Harlech sheet are surprising. The region is to be classed as well outside the category of soil-covered terrains. However, the ruggedness of this area is most likely subjective and influenced by the prospect of an irregular skyline, the total relief, and the uneveness underfoot. Once the micro and meso-relief is filtered out, the uplands of Southern Britain show little unexpectedness; certainly less than parts of Devon and Cornwall and even of areas in the clay vales. The greater relief and coarser terrain mean that once having started to climb a slope it is likely to continue to rise for some distance. All this leads

.


to a lower value of the entropy but the value for the Harlech sheet is still too low and for which there are other reasons. The profile curve between crossing points will be far from Gaussian and the loss of information in using a binary coding will be prohibitive, while the coarser terrain reduces the number of crossing points to below an adequate sample. These questions are returned to later.

Apart from the Harlech sheet, the prime facie evidence from this foray outside the class of soil-covered terrains is to reinforce the consistency between the two methods of dimension estimation and to support the binary approximations of the previous 40 sample sites. The empirical results as a whole verify the theoretical justification for the extension of the methods of one-dimensional Markovisation to the study of soil-covered landscapes.

HOMOGENEITY AND ISOTROPY

In order to make any progress theoretically attention has been restricted to homogenous and isotropic Gaussian random fields. This is a gross simplification. Measurements taken are samples from the ensemble of all possible transects of the given area. Some of these will be non-generic, e.g. those following river valleys, and are to be rejected. Others can be contaminated by the meso-relief. There are also the effects of inhomogeneity and anisotropy.

To investigate these matters further the Burwash (TQ 62) sheet was re-examined. The sheet area lies immediately southeast of Tunbridge Wells in the High Weald just north of the axis taken through the Purbeck inliers. Elevations reach 625' (190 m) and the topography is finely dissected and maturely degraded. The grain of the region is roughly north of west to south of east, reflecting the faulted outcrops of Wadhurst Clay and Ashdown Sand, occupying most of the southern half, while the northern half is underlain by Tunbridge Wells Sand; all from the Wealden Series (Neocomian), (Gallois, 1965). The Rother provides an alluvial tract east-west across the northern third of the sheet area. To the south Dallington Forest forms the watershed with the drainage south to Pevensey Bay. The region falls into the area of the Wealden Island of Wooldridge and Linton (1939), (Jones, 1981, p. 106), and the drainage is well adjusted and of trellis form. Erosion surfaces can be traced upon the interfluve crests and on valley side spurs (Bird, 1958).

At each 1 km National Grid line, both north-south and east-west, the calculation of the entropy has been repeated. The results are listed in Table 111 together with a count of mean level up-crossings. The surface is not homogenous and certain values are abnormally high. Furthermore, the surface is anisotropic, the values from the east-west transects being significantly higher than those from the north-south examples. Each of the major east-west adjusted tributaries of the Rother has been rejuvenated and plays host to a relatively dense set of north-south minor tributary valleys. Where the mean contour is such as to pick out these valleys the number of intersections is enhanced and the value of the entropy thereby raised. The east-west transects at references 20,25,27 and 30 are cases in point with values well in excess of the mean. They mark the regions of rejuvenation but no transect escapes and contrary to expectation the east-west graining of the terrain results in higher values parallel to rather than across the grain.

The entropy value for one of the north-south transects, (66), is abnormally high. This is an example of contamination by the meso-relief. The mean contour (225') roughly follows the transect which thus picks up the increased number of intersections.

From this analysis of the Burwash sheet it appears that if inhomogeneity or anisotropy is present to any great extent it should be recognizable by its effect on the entropy value and in practice on the number of mean crossing points. Contamination by the meso-relief can give just as large an effect but equally should be recognizable during the analysis. In this way, with care, entropy calculations can obviate the difficulty of meso-relief contamination which is liable to beset and to remain unrecognized during the contour based Richardson procedures.

There exists the possibility of turning the difficulties associated with inhomogeneity to advantage. Variation in the measured values can put a measure on the degree of homogeneity. More importantly it may be possible to use the variation in entropy value to monitor the evolution of the landscape. Ideally the fingering of rejuvenation through a drainage basin is to be delineated, but this is not yet possible except in exceptional circumstances.

'%I .


Table 111. K-entropy h(t), Hausdorff Dimension D,, and number of up-crossings of zero mean level Co for 10 km transects at 1 km National Grid intervals: Burwash Sheet

TQ 62

East-west 20 21 22 23 24 25 26 27 28 29 30

North-south 60 61 62 63 64 65 66 67 68 69 70

0.25 1 1,363 0.168 1.243 0.2 12 1.306 0.167 1.242 0.198 1.286 0.325 1.469 0.198 1.286 0.346 1.499 0.227 1.327 0.183 1.265 0.302 1.436

0135 1.195 0.168 1.242 0.168 1.242 0151 1.219 0.135 1.195 0.198 1.286 0.279 1.402 0.129 1.186 0.135 1.195 0.198 1.286 0.168 1.242

7 4 5.5 4 5

10 5

11 6 4.5 9

3 4 4 3.5 3 5 8 3 3 5 4

CROSSING POINTS AND EXCURSION SETS

In the previous report on the Burwash sheet (Culling and Datko, 1987) illustrations were given of the Gaussian nature of the distribution of increments for a diagonal traverse. With respect to the theory of crossing points and excursion sets it was found that the sample size was inadequate. Otherwise agreement is good and supports the prediction that a well-developed soil covered terrain should present a Gaussian surface.

To remedy this defect and to provide a satisfactory test of the application of the Rice-Kac theory to the landscape, a longer traverse has been taken within the same terrain, the High Weald of the Kent-Sussex border. A diagonal line has been drawn from the nwthwest corner of sheet T Q 42, at Danehill, south of East Grinstead, across sheet T Q 52 (Heathfield), towards the southeast corner of sheet TQ 62 (Burwash) at Brightling. Of this the first 30 kms was taken giving a sample of 600 values at 50 m intervals and to the nearest 5 feet (figure 5). Within the Burwash sheet the traverse is almost entirely on the Wadhurst Clay; in the two sheets to the west it is almost entirely on the Ashdown Sand, returning to the Wadhurst Clay and the Tunbridge Wells Sand in the far northwest corner. Once again it must be stressed the values are taken from map data and are subject to all the usual sources of error.

The sample has been used to check:

1. The Gaussian nature of the distributions of elevation and of increments. 2. The predictions of the Rice-Kac theory with respect to the mean number of crossing points and mean

length of excursion sets. 3. To examine the distribution of local maxima.

DIMENSION A N D ENTROPY IN THE SOIL-COVERED LANDSCAPE 635

800

700

600

500

I 4 0 0

300

200

- L

1 I 1 I 5 10 15 20 25 30

kilometres

Figure 5. Profile of Danehill-Heathfield-Brightling Traverse

4. As a separate exercise, to calculate the entropy using all the contour data, the details of the calculation being given in the appendix.

Gaussian landscape The distributions of elevation and of increments both follow roughly a Gaussian form but with

disturbance. The former (Figure 6) is affected by the lack of available relief resulting in the lower limb being bulked out and truncated. As for planar excursion sets (Culling and Datko, 1987, Figure lo), where a similar situation occurs, the upper limb shows reasonable agreement. The distribution of increments (Figure 7) reveals an unexpected asymmetry. In the forward (southeast) direction there is a shortage of lower values when compared to a Gaussian distribution, compensated by an over-endowment of higher values, implying that in general the northwest facing slopes are steeper. Otherwise the agreement is good.

Level crossing points

process, in the interval [O, TI, is given by the Rice-Kac formula Equation 9, The mean number of up-crossing points C,(O, T), of a given level u, by a stationary regular Gaussian

120 1

Figure 6. Distribution of elevations D-H-B Traverse

636

1201

W. E. H. CULLING

Figure 7. Distribution of increments D-H-B Traverse

As a preliminary trial the second order differences of the covariance function were taken and the plotted values were extrapolated (BFS) to give an approximate value of 200 for -r"(O). Upon substitution and employing the empirical value of o2 = 11245.067, the value for the mean number of crossings of the zero mean level, Co(O, T ) is calculated as 12-735.

From the distribution of increments and using the integral estimator Equation 17:

?;= 178.625 p,, = 13.365 The standard deviation of the mean square frequency from Equation 19 is 00116.

The observed value is 14. Taking this value for qC gives -r"(O)=241.699 and this is the value taken as the basis for the calculations of the mean number of crossing points. The difference between the two estimated values of y amounts to a factor of 1.0475 in the calculations.

The mean value of the sample of 353.508 is sufficiently near 350 for this latter value to be taken as the mean value contour thus simplifying things considerably. Predicted and observed values are compared in Figure 8 and in the circumstances the agreement is good. As well as providing support for the application of the Rice-Kac theory this result adds further confirmation that the landscape is well represented by a Gaussian random field.

In order to test the formula for the variance of yc, Equation 22, we return to the data from the Burwash Sheet. From the numbers of up-crossings listed in Table I11 for the north-south and east-west transects of the

Figure 8. Distribution of up-crossings D-H-B Traverse


Table IV. Variance of the mean frequency of zero level upcrossings $,: Burwash Sheet TQ 62

N-S (60-70) E-W (20-30)

n 11 11 E { C , ( T I } 4.1 364 6.4545 uc; 0.001 978 0.005494 G 0.04447 0.074 12

sheet at 1 km intervals we arrive at the results of Table IV. In assessing these resL..s it s..ould be remembered that the abnormally high values discussed previously have been included in the calculations. The anisotropy is even more apparent.

Excursion sets Given that a process is stationary, regular, and Gaussian, a knowledge of the variance enables a

calculation of the probability that the value at any point is at or above a given value u, i.e. P ( X ( t ) a u } or alternatively an empirical estimate can be made of the time spent by a sample function at or above a given level. In Table V the predicted and observed values are compaed. Once again agreement is good until the lower values are reached whereupon the two sets of values diverge. This is likely to be due to the incidence of alluvium.

The mean length of excursion sets above a level u is given by P { X ( 0 ) 2 u } p- ', where p is the mean number of up-crossings of the level u per unit time. Calculations have been compared with observed values in Table VI. The agreement is not as good as we have come to expect. Outside the range 300450, with low and probably inadequate sample numbers, it is poor.

Local maxima From Figure 9 where the local maxima are plotted it can be seen that they do not follow a Gaussian

distribution. As before this may be due to inadequate sample size. On the other hand it could be the result of inadequacies in method or theory. The interpolation of local maxima from map data is more than usually prone to error. If this is the case resolution of the problem will require the greater precision of digital data or, if necessary, a survey on the ground. Of course the results may be due to the erosion surfaces known to be present. If it is a case of inadequate sample size then the discrepancies of this and the previous section may disappear when we come to deal with the planar case.

Table V. Time spent above a level u, P ( X ( O ) > u } . Danehill-Reathfield-Brightling traverse

U Predicted Observed

600 550 500 450 400 350 300 250 200 150

0.0 10 0.032 0.084 0.182 0.330 0.512 0.709 0.836 0.926 0-973

0.015 0.055 0.092 0.182 0.340 0.528 0.685 0.8 17 0.937 0.997

638

6-

4-

W. E. H. CULLING

Table VI. Mean excursion set lengths. Danehill- Heathfield-Brightling traverse

U Predicted 0 bserved n

600 550 500 450 400 350 300 250 200

6.90 8.14 9.79

12.17 16.28 22.64 32.80 54.63

108.55

9 33 18.33 10.90 17.00 24.38 41.10 70.00

111.80

1 1 3

10 12 13 10 I 5

Figure 9. Distribution of local maxima D-H-B Traverse

Entropy Basing the calculation of the K-entropy of a profile sequence on a binary coding sacrifices a great deal of

information. Where the profile is well approximated by a Gaussian function of low dimension, such as a profile across a soil-covered landscape, there does not appear to be a serious loss. With a large range, a non- Gaussian profile and where the terrain is coarsely subdivided by a low density drainage net, (as exampled in the Harlech sheet), then the number of mean level intersections will be low. A binary coding sacrifices too much information. This gives a meaningless value for the K-entropy; the scaling is wrong, the process is not ergodic, and the method is misapplied. There were indications that this was beginning to be apparent in the final 10 sites from Upland Britain and almost certainly in the Harlech sheet.

By increasing the partition number, more and more of the information inherent in the profile curve can be encoded and the estimate of the K-entropy correspondingly improved. In practice, using map data, the partitioning is limited by the available contour data and a lower limit to the time interval has been set at 50 m. For the Danehill-Heathfield-Brightling trsyerse the partition interval has been taken at 5 0 giving a partition number of eleven. Details of the calculation are given in the appendix.

It is possible to perform the calculations from a knowledge of the probable occupancy of the partition intervals, Pi , and this can be gained from Table V and of the number of crossing points, p i j , obtainable from the Rice-Kac formula. In turn these imply a knowledge of the variance and the number of mean level crossing points. A calculation on this basis gives a value of 1.297 for the Hausdorff dimension.

Alternatively the calculation can be based entirely on empirical data; the occupancy and the number of transition intervals being taken from the coded sequence. Using this method gives a value of 1.281 for the dimension. These two estimates are reasonably close but are high when compared to alternative estimates. However, as already noted, the K-entropy makes no distinctions if a plurality of structures is present, as IS the case here.

Further aspects of the increase in partition number are taken up in the appendix. Meanwhile we return to binary coding and the sensitivity to the choice of contour.


Table VII. Robustness of estimates of entropy to deviation from the mean level and equipartition:

Danehill-Heathfield-Brightling traverse

~ ~ ~ ~

450 0.182 0.8 18 0.137 0.198 400 0.340 0.660 0.154 0.222 350 0.528 0.472 0.168 0.242 300 0-685 0.3 15 0.154 0-222 250 0.8 17 0.183 0.094 0.136

In the procedure based upon a binary coding the contour that most nearly divides the transect equally between those elevations lying above and those below is chosen in lieu of knowledge of the mean level. This practice is tested in Table VII where are listed the proportions above and below the given contour value together with the calculated values for the entropy and dimension. A Gaussian function should tend with time to a symmetrical division of the transect but in practice deviation is to be expected. The chosen proxy mean value contour of 350 subdivides the transect profile in the proportions 0.528/0.472, which is about the accuracy that can reasonably be expected. A value outside the range 0.6/04 would be rejected. From the calculated values for the f 5 0 contours in Table VII, despite proportions outside this range, the discrepancies are less than 10 per cent. The presence of 25' (or in later editions 5 m) contour intervals means that a contour close enough to one giving the ideal 05/0.5 proportions will almost always be available. This being so the technique used for the latter items of Table I (the first 15 use the mean value) is deemed adequate and within the limits of error set by other sources.

The identity of the mean value with an equal amount of time spent either side depends upon a sample function curve symmetrical about the mean value and this is guaranteed by a Gaussian process. There is no need to make this assumption, however; robustness to deviation from the mean can be given an alternative theoretical justification. This is found in the differing sensitivity between different parts of the curve in Figure 4. The entropy value is insensitive to inequality between the constant transition probabilities p + + and p - - compared to a relative sensitivity to the values of the cross transition probabilities p + - and p - + . In other words the entropy is sensitive to the number of crossing points but insensitive to the subdivision of the transect into positive and negative. This difference in sensitivity depends upon the position on the graph of Figure 3; for small values of P , ~ , i#j, which will be the case for soil-covered landscape profiles, the graph rises steeply, whereas towards the mid-point it is almost flat.

These differing sensitivities imply that a mere count of the number of intersections of a contour near to the mean value and a knowledge of the number of units in the traverse is sufficient for a close approximation to the K-entropy value. It does not matter if the chosen contour is away from the mean or from an equal subdivision of the transect so long as the num6kr of intersections is not altered. Then all that is required is the conversion of the number of up-crossings to a fraction of half the total number of units in the transect and the K-entropy and the Hausdorff dimension can be read off the graph of Figure 4.

The fact that the number of crossing points is an integral value means that the calculated values for the entropy will cluster round certain values and some such clustering is apparent in Table I. Increasing the number of sample transects will remove this slight embarrassment and provide an improved estimate of the entropy. As in the case of the Burwash sheet (Table 111), this can be conveniently done by counting the crossing points at each 1 km National Grid line, neglecting the abnormal values, if this can be independently justified, taking the mean value, and applying the statistical analysis as in Table 111. It is doubtful if a simpler and more statistically sound method of calculating the Hausdorff dimension from empirical map data is possible. In addition all the errors involved in the use of map data, apart from gross cartographic mishap, are eliminated.


INFORMATION THEORY IN GEOMORPHOLOGY

Information theory has its origins in the application of the theory of random processes to mathematical models of, in the first place, communication systems, then dynamical systems in general, or as here, with surfaces (more generally manifolds) with a random geometrical structure. Prior to the advent of entropy theory in dynamical systems, the principal invariant was assumed to be the spectrum and ultimately the greatest impact on geomorphology of entropy notions will be associated with its flow invariant character.

Information can be regarded as defined upon a probability measure space with respect to at most a countable partition. The mean value of this function, the entropy, is therefore a function of the partition. In practice we deal with discrete sequences and it is intuitively obvious that in refining the resolution, the amount of information is increased. At the limit, for a continuous source, the maximum entropy is given by a Gaussian-Markov function. From the calculations in the appendix increasing the partition from two to eleven does significantly increase the entropy but it is still far below the maximum. Dependence upon the partition is eliminated by the division of the entropy by In r, scaling the range to the unit interval, and giving the Hausdorff dimension.

Although it is possible to work with logarithms to any base, it is preferable, as here, to work with natural logarithms. It is also customary to assume stationarity. Without going to extremes, any landscape showing unmistakable signs of non-stationarity in the elevation should be de-trended prior to analysis. Non- stationarity with respect to the variance is another matter. Results are available for non-stationary Gaussian processes but for the moment they lie outside the orbit of practicality.

The fundamentals of information theory can best be gleaned from Shannons (1948) memoir, which for a pioneering work is of exceptional clarity. The reprint (Shannon and Weaver, 1963) includes a non-technical introduction by Weaver. The definition of the K-entropy with respect to dynamical systems is given in Cornfeld et al. (1982, pp. 246-247), a more mathematical treatment is given in Parry (1981, pp. 56-73). The Benchmark volume, Gray and Davisson (1977) contains seminal papers by Birkhoff, Breiman, Kolmogorov, Sinai, and by Ornstein.

First this section deals with the implications of this study, then the use of entropy measures in other branches of geomorphology, followed by the consideration of more fundamental aspects, leading up to some concluding paragraphs on the axiomatization of Davisian theory.

Entropy and dimension The purpose of the investigation is purely scientific. It is designed ostensibly to validate certain estimation

procedures, and in fact does so, but there are more fundamental implications. Within the category of soil- covered terrains, no significant inconsistency arises between estimates of the Hausdorff dimension based upon entropy measurements and those based on Richardson plots or in some cases variogram methods. The calculation of the K-entropy is easy to perform, particularly for a binary coding, while the consequent loss of information does not appear to be significant. Indeed a rough estimate can be made from a count of crossing points on a 1 : 25 000 map.

The inherent ambiguity in the K-entropy with respect to the partitioning can be removed upon conversion to the Hausdorff dimension. Both the K-entropy dfid the Hausdorff dimension provide a measure of the irregularity of a landscape surface. Surfaces of different roughness can be distinguished with a precision dependent only upon the quality and density of the data. The two measures have been used in this series of papers to verify the prediction that well-developed soil-covered landscapes become relatively smooth and of low dimension. As degradation proceeds and the landscape becomes smoother both quantities decrease towards zero (apart from the integer component in the dimension). Thus the value of the K-entropy can be used to monitor the progress of a terrain throughout a cycle. Rejuvenation tends to raise the value towards a maximum set by the drainage pattern; while degradation tends to reduce the value towards zero, the value appropriate to the certainty attaching to a plane surface. In a similar manner changes in the regularity of the landscape surface can be used to recognize and measure the influence of structure and process. Variation in roughness, measured as a dimension, has been used (Eliot, 1987) to determine the relative age of glacial moraines. In the same manner distinction should be possible between older and newer drifts or between glaciated and unglaciated terrains.

DIMENSION AND ENTROPY IN THE SOIL-COVERED LANDSCAPE 64 1

Provided certain ergodic conditions are met the K-entropy measures unexpectedness or randomness in any sequence of values. The landscape profile sequence is a concrete example and others from geomorphology include the soil structure, where particle/void patterns can be coded directly (Dexter and Hewitt, 1984); all roughness measures whether hydraulic, of the microrelief, of the vegetation cover, of weathering surfaces, of rock texture, of chemical weathering fronts, and of the form and dynamics of earthquake and landslide surfaces (Andrews, 1981). In the planar case the K-entropy and the Hausdorff dimcnsion can be applied to patterning, whether of the drainage net, of the vegetation cover or of any erratically distributed property.

Abstract applications can be made to any geomorphic property or process that can be recorded as a sequence of values. Thermodynamic aspects of entropy enter largely into the particle dynamics of geomorphic transport processes (Culling, 1985b, 1987a).

Most abstract and far reaching, K-entropy and Hausdorff dimension figure in modern developments in the theory of dynamical systems. In certain cases the K-entropy is equal to the sum of the Liapounov exponents. These govern orbit behaviour and where there is divergence, as in strange attractors, the K-entropy rate gives a measure of chaoticity. The characteristic sensitivity to initial conditions found in strange attractors is due to information creation within the flow. This will preclude any prediction or retrodiction beyond the immediate present; truly no vestige of a beginning; no prospect of an end. The immediate relevance for geomorphology lies in the fact that since Lorenz (1963) and Ruelle and Takens (1971), it is fairly certain that a strange attractor lurks within the Navier-Stokes equations and turbulence is governed by information generated by the flow. Customary approaches, and not only in geomorphology, assume a flow controlled by the initial and boundary conditions and of the feasibility of measurements by averages and correlations. This is more suited to laminar conditions but totally inadequate to keep pace with turbulence.

In the spirit of Laplace, it is theoretically possible to examine laminar flow point by point and having realized the total content of information be able to predict the entire future or retrace the whole of the past. This philosophy of nature underpins contemporary thinking and accompanies the geomorphologist into the field but it is an impossible scenario for turbulence. It is not that the information content is vast but that it is continually replenished. Stirring a cup of coffee generates - 1OI2 bits per sec. Turbulence presents the would- be observer with a Tristram Shandy situation. Like Uncle Tobys autobiography which took him two years to cover the first year of life the task can never be completed. If we are ever to gain satisfactory knowledge of the work of rivers then the challenge of turbulence will have to be taken up. It is fairly obvious that this will require new ideas, new methods and new invariants of the flow and it seems highly likely that among these will be entropy or entropy-like measures (Shaw, 1981; Eckmann and Ruelle, 1985) and for a geographical account (Culling, 1985a, 1987d).

It will be apparent that the K-entropy and Hausdorff dimension are not to be regarded as yet two more examples of ad hoc, parochial, geomorphic indicies (Culling, 1986b, p 95). They are, or can serve duty as, flow invariants. Their power is such that they cover the whole range. At the lowest level they can serve merely as a quantitative measure of surface irregularity or roughness, the surface regarded as a static concrete entity. At the other extreme, as suggested above, they will also serve in the understanding of the ultimate complexity of fully mixing turbulence. Somewhere in betweqg is the position adopted here, where the landscape is regarded either as the sole sample realization of an ensemble or as supplying the manifold upon which the set of all generic profiles constitute an ensemble. In either case, and with adequate safeguards, the K-entropy acts as an invariant. We return to the question of ergodicity later, meanwhile we discuss the representation of the soil- covered landscape by Gaussian random fields.

Gaussian landscape The most important implication for geomorphology arising out of the investigation relates to the Gaussian

nature of the soil-covered landscape surface. It provides the means whereby one section of geomorphology can be put on a firm mathematical basis. That the theory of Gaussian random fields is the appropriate mathematics for the study of the soil covered landscape is as fortunate as it is fundamental. Not only does it provide entry into the extensive and well-developed results of Gaussian theory but also through them into other aspects of probability theory, to ergodic theory and to information theory. The Belyaev dichotomy


enables both a regular and an irregular description of the landscape to be encompassed within one theoretical construct. Davisian degradational theory can be given a respectable mathematical footing. Any dimensional property can be deduced from a set of intersection points, for in essence a Gaussian structure fills the space in between the points.

The ability to predict expectations of excursion sets, of isolated maxima, and of extremal values confers considerable knowledge about the landscape. To derive this from a set of intersection points represents considerable scientific leverage.

Ergodic theory The equality of sample and ensemble averages plays a large part in Shannons information theory.

Introducing the idea of an information function of a random process, he proved an ergodic theorem for the sample entropy of Markov sources. Ergodic concepts are prerequisite for the application of entropy theory and the entire analysis of this paper depends upon the soil-covered landscape surface being not only Gaussian but also ergodic. However, the implications of ergodic theory for geomorphology range far beyond the K-entropy and the Hausdorff dimension.

Ergodicity implies the existence of a mean value as T+co. It involves more than stationarity in that for an ergodic system there exists no subset with a probability other than 0 or 1. This implies that an average over the ensemble is equal with probability one to a time average of a particular sample function. Roughly any sample function can be expected to pass arbitrarily close to all available states, with an appropriate frequency, given sufficient time. Thus a sample function can be regarded as fully representative of the whole ensemble and in a certain sense ergodicity confers statistical homogeneity.

For geomorphology, the landscape surface of a particular terrain is the sole sample realization of the ensemble comprising the set of all possible landscape surfaces for that particular terrain plus the apparatus of a probability measure. Obviously it is highly desirable to be able to take a given landscape as representative of the ensemble. In practice the representation process is taken a stage further. A part of the landscape is sampled and this could be a profile, a select area or a set of chosen point samples. If the profile is long enough, the area large enough, or the set of points numerous enough, then they can be regarded as fully representative of the surface, now comprising the ensemble of all possible profiles, areas, and sets of points, if and only if, the ensemble is ergodic. This will not only apply to the elevation but to any other geomorphic measurement made in the landscape. Without ergodic justification measurements in the landscape are acts of faith.

Brownian surfaces, or more precisely N-parameter Levy Brownian fields are ergodic whatever the value of the index fl. It follows that well-developed (mature) soil-covered landscape surfaces are ergodic. Provided the terrain is reasonably homogenous an arbitrarily directed profile will supply an estimate of the entropy that will apply to the whole ensemble of the surface. Any profile, area, or set of points from the surface can be justifiably compared to any other. Inversely if the measured K-entropies are the same the compared profiles can be regarded as from the same ensemble. The question arises, can comparison on such a rigorous basis be made between profiles from different terrains? If the answer is yes then geomorphologists will be presented with the possibility of more than one realization of a landscape ensemble. This will allow the rigorous comparison of terrains on the basis of identical K-entropy. On the other hand for terrains regarded as equivalent upon some other basis, variation in K-entropy can be used as a measure of degradational development.

The comparison aimed at here is not an elementary comparison by juxtaposition and inspection, nor naive measurement and statistical analysis, but an isomorphism. Two dynamical systems, or random processes, or as here, two landscape surfaces considered as random fields, are isomorphic if there exists between them a stationary, invertible mapping. In the case of the landscape the most natural example would be a simple scaling. From the viewpoint of information theory, isomorphism provides for the existence of stationary codes mapping one source into another. The problem is to construct a simple, computable criterion for isomorphism and was partially solved by Kolmogorov (1958, 1959) and Sinai (1959) with the definition of K- entropy and the proof that equivalence was a necessary condition for isomorphism. But is it also sufficient?

It turns out that matters are not as simple as Kolmogorov and Sinai were lead to believe. There are classes of randomness; at the lower end, ergodicity and mixing, where the latter includes the former but not vice


versa and above a series of further hierarchical classes. Ornstein (1970, 1973) solved the problem and showed that for a class of stationary and ergodic processes called B-processes (the B is for Bernouilli), equal entropy was both necessary and sufficient for isomorphism. For the next most general class, the K-processes (the K is for Kolmogorov), equal entropy is necessary but not sufficient for isomorphism.

A B-process is one that can be approximated by a roulette wheel, whereas a K-process is such that no event can be predicted from the distant past. It would be expected that a landscape could possibly be a K-process but not a B-process, which appears far too random. However, Ornstein quotes as examples of B-processes, Brownian motion, presumably the standard Markov case ( f i = +), but also multi-step Markov processes. Now any stationary process can be approximated arbitrarily closely by n-step Markov processes and this would include profiles across landscapes represented by Gaussian random fields. As Ornstein says (1973, p. 50): there is much work to be done in this direction but the possible returns are high (Gray and Davisson, 1977, 229-236; Cornfeld et al., 1982, 258-291; Lichtenberg and Lieberman, 1983, 268-274).

It is desirable to be able to place comparable landscapes in a developmental sequence. A progressive smoothing or Davisian downwasting of the landscape will witness an ultimate decline of the K-entropy towards zero. If terrains can be classed as comparable, and this could be at varying degrees of rigour, then the variation of a variable of interest could be graphed against K-entropy to give the behaviour throughout the Davisian cycle. For example, drainage basins can be classed according to the dimension of the drainage pattern. Given a sample of terrains with equal drainage dimension, the variable of interest, say, total sediment transport out of the terrain (suitably standardized and scaled if necessary) could be plotted against K-entropy to give an indication of how the property behaves throughout the cycle.

Axiomat ics The first duty of a theoretical scientist is to select a viewpoint from which perspective things fall into place.

The second task is to fashion the investigative method, which for almost all cases in physical science is the appropriate mathematical theory. The selected standpoint here is the geometrization of Davisian denu- dational theory and the appropriate mathematical tool, the theory of Gaussian random fields.

Before a subject can take off scientifically it must acquire order, clarity, and simplicity, three qualities embodied in the axiomatic method championed by Hilbert. As he was fond of saying physics is too difficult/important for the physicists and embarked upon his programme to axiomatize physics (Reid, 1970, p. 127). It is now known that this was far too ambitious but the lasting value of the axiomatic method has become part of the repertoire of the theoretical physicist.

In the same vein geomorphology is too difficult/important to be left to the geomorphologists. Not that in an Hilbertian manner we wish to codify existing knowledge but in the spirit of Emmy Noet

dimension and entropy in the soil-covered landscape

Documents