A statistical measure of the structure of tilled soil

.I.agric.Engng Res. (1977) 22, 101-104

A Statistical

Measure

of the Structure

of Tilled

Soil

A. R. DEXTER*

The measurement of the structure of tilled soil has always been a problem. A method is presented in which sections through impregnated blocks of tilled soil are analysed. Lines on the sections are represented by strings of O’s and l’s depending on whether they pass through void spaces or soil aggregates respectively. The entropy of these strings is calculated as a statistic with the same mathematical form as a Markov information source and is found to be a sensitive indicator of soil structure. The method is illustrated with comparisons of the structures produced by 3 different implements and with a study of the effect of time on the structure of tilled soil.

One of the problems confronting agricultural and environmental scientists is the enormous spatial variability of most physical properties of tilled soil. The physical properties of soil are but adequate measures of structure are generally lacking. all governed by the “structure”, Bulk density or porosity measurements are useful, but they give no indication of the relative arrangements or sizes of the aggregates or pore spaces within the soil. Sieving can give an idea of aggregate size distribution, but some aggregate break-up occurs, and the results give no information about the relative arrangements of the aggregates in the undisturbed tilled soil. Since variability is clearly one of the principal physical features of tilled soil, it seemed that some measure of structural variability could be a useful soil physical parameter. In physics and chemistry, entropy is used as a measure of the degree of disorganization of a system, and it was decided to investigate the structural entropy of tilled soil. The soil used was a red-brown earth’ and was tilled at a water content just drier than the plastic limit. Tillage was performed with a mouldboard plough to a depth of 100 mm. with a set of spring tines with narrow points to a depth of 70 mm, and with a rotary cultivator to a depth of 80 mm. Steel moulds 300 mm square and 180 mm deep were pressed into the tilled soil. Samples of tilled soil were impregnated by pouring molten paraffin wax (melting point 60°C) into the moulds. When the wax had set, the moulds containing the samples were taken to the laboratory. The sides of the moulds were heated to melt the adjacent wax, and the impregnated samples were removed. Paraffin wax is found to fill all pores down to about 1 mm dia, irrespective of the initial water content of the soil. Sections were cut in the impregnated samples at right-angles to the direction of tillage using machine hack-saw blades. The sections were analysed along horizontal lines at five different levels. Level 0 was 25 mm above the depth of tillage, and each successive level was 10 mm higher. The structure of the tilled soil was analysed from measurements of the distributions of soil aggregates and pores (wax-filled) at the 5 levels. Every millimeter along each line the section was observed. If there was a pore at that point, a 0 was written, and if there was an aggregate, a 1 was written. These values were called elements. Thus a string of O’s and l’s was written which represented the internal macro-structure of the tilled soil at that level. The length of the strings was 230 elements. The 35 mm at each end of each section was not analysed because of possible disturbance when the steel moulds were pressed into the soil. The procedures outlined abov havee been described in greater detail elsewhere.* *Department Recewed

of Agricultural 1 June

1976;

accepted

Biochemistry in revised

and form

Soil

Science,

7 October

Waite

Agricultural

1976

101

Research

Institute.

Glen

Osmond,

South

Aurtralaa

iOM

102

STRUCTURE

The mean entropy per element, H, of the soil macro-structure developed in communications theory.3 H = -

was calculated

OF

TILLED

SOIL

using an equation

;

C IV, C p(x, i) log, p(x, i). . ..(I) I x where m is the number of elements considered. The p(x,i) are the probabilities of the symbol i following the precursor x. Here the symbol i can only take the value 0 or 1, andp(x,O) = I -p(x, 1). The precursor x can be either the single element immediately preceding the element being considered, or can be a string of the n preceding elements. With only two possible element values, the maximum number of precursor strings is 2”. The number of occurrences of a precursor string x is N,. It will be seen that if the soil has no macro-pores, the entire string of measured element values will be composed of 1’s. The entropy will then be zero irrespective of the length of the precursor string. Similarly, just above the tilled layer the line passes entirely through air and all the elements are 0’s. In between these two extremes, in the tilled layer of the soil, the entropy takes values intermediate between 0 and 1 depending on the structural state.

o’5-

0.1

I 0123456

I

I

I

Precursor

Fig. 1. Effect of length of precursor

I string

I

0 7 length,

I 6

I 9

, 10

n

string on the calculated value of the erltropy

The value of the entropy obtained depends on the length n of the precursor string. then each element is considered to be independent of the preceding elements, and

If n = 0,

. ..(2) H = -(VL log, IIL+(~ -VL) log, (1 -vL)). Here, qL is the linear porosity of the soil which is defined as the proportion of the elements in a line at a given level which have the value 0. The maximum possible value of the entropy, H == 1.O. p(x,i) in Eqn (1) depart further is obtained when rr. = O-5. When n = 1, the probabilities from the value 0.5 and the entropy value drops. It can be shown that H is a monotonically decreasing function of n, with the most accurate estimate of the entropy being obtained as n approaches infinity.3 In practice, H decreases somewhat erratically with increasing n because of the finite length of the initial measured data string. This effect is illustrated in Fig. I, which shows the decrease in H with increasing n for one of the replicates for level 2 of the tilth produced by the rotary cultivator. With data strings of finite length, underestimates of H are obtained as n becomes large. This is because many of the possible precursor strings will occur only once and give rise to probabilities of 0 or 1 when perhaps intermediate probabilities would be obtained with longer data strings. In other words, N, should only rarely be allowed to drop below 4. For this work, the precursor string length was set at n = 5. Thus, the calculated entropies include structural features extending over the range of sizes l-6 mm. Results for different implements are shown in Fig. 2 (top). Differences in the structures produced by the implements are clearly evident, as are the variations in the structures with depth. The error bars give an indication of the limits of variation of the values of H calculated from different replicates. The change in entropy with time after tillage is shown in Fig. 2 (bottom).

A.

R.

103

DEYTER

For this graph, the values for all three implements were averaged, although similar trends occurred In the first 51 days after tillage, 69 mm of rain were recorded, with the individual implements. and the structure was affected only on levels 2, 3 and 4. In the following 54 days, 139 mm of rain fell, and the structure at all 5 levels appears to have been affected equally. Consideration of the changes in the individual probabilities p(x,O) have enabled these changes to be described in terms of changes in the distributions of aggregates and pore spaces.2 After a considerable time. the value of H will decrease to an equilibrium value, depending on the amount of shrinkage For the examples given ‘in cracking of the soil and on the level of macro-biological activity.

4-

(b)

‘. t ‘.

6 Entropy,

H

Fig. 2. Entropy produced at different levels by a mouldboardplough (---), spring tines (- - -) and a rotary cultivator (. .) (top), and the entropy at the time qf tillage (----), and at 51 days (- - - -) and 105 days (. .) afrw tillage (bottom)

Fig. 2, there were no significant trends or differences in the values of the linear porosity, q,<, and it is reasonable to assume that there would likewise have been no significant changes in the bulk density or the volumetric porosity of the tilled layers. The absolute magnitude of H depends not only on the soil structure, but also on the spacing between elements. If the spacing is much smaller than the size of the structural features, H will tend toward zero because successive elements will occur within long strings of O’s and I’s If the spacing is much larger than the size of the structural features, H will tend toward the value given by Eqn (2) because the values of successive elements will be independent. In order to test the sensitivity of entropy to element spacing, measurements were taken on a section of a bed of aggregates sieved in the range 5.1-9.5 mm dia. Entropies were calculated with n =:: 4, with element spacings ranging from 0.5 mm to 5 mm in steps of 0.5 mm. The results are shown in Fig. 3 where it can be seen that there was only a small dependence of entropy on element spacing in the range 1 mm to 4.5 mm. The spacing of 1 mm was chosen for this work because of its

104

STRUCTURE

OF

TILLED

SOIL

relevance to plant growth and its suitability for tillage studies. Russell” indicates that the most suitable seedbed for the initial growth of plants is composed of aggregates having diameters in the range l-5 mm. This was another reason why precursor string lengths greater than 5 elements were not considered. By changing the element spacing, soil structural features on any size-scale could be investigated.

. ..I_. Element

spacing (mm)

Fig. 3. The effect of element spacing on entropy for a bed of sieved aggregates

of 7.3 mm mean diameter

The entropy of tilled soil, measured as described above, has proved to be a sensitive indicator of soil structure. Entropy is able to distinguish between soils when measurements of bulk density or porosity would show no significant difference. Entropy values provide information on the structure of soil in situ and give equal weighting to the spatial arrangements of the pores and the aggregates. The entropy is a measure of the variability of structure, and hence of the range of different micro-environments experienced by, for example, germinating seeds in a seed-bed. soil parameter which could be a Entropy, therefore, seems to be a sensitive “single-value” useful tool in tillage implement development and in studies of plant response to soil structure. REFERENCES ’ State, H. C. T. et al. A Handbook of Australian Soils. Rellim Tech. Publs., Glenside, S. Australia, 1968 2 Dexter, A, R. Internal structure of tilled soil. J. Soil Sci., 1976 27 (3) 267 3 Shannon, C. E.; Weaver, W. The Mathematical Theory of Communication. Urbana: Illinois University

Press, 1949 4 Russell, E. W. Soil Conditions and Plant Growth. London:

Longman, Green & Co., 1961