Variabilities of measured and simulated soil structures A. R. Dexter and J. S. Hewitt

Waite Agricultural Research Institute, Glen Osmond, South Australia 5064 (Received January 1983)

The macro-structure of tilled soil varies significantly between replicate samples collected from the field. This is illustrated with data from a grey swelling clay from Victoria, Australia. Soil structure was quantified statistically from studies of linear transects on sections cut through impregnated blocks of undisturbed soil as described previously.

Simulated soil structures were generated using the mean parameters measured from actual soil structures. The simulated and actual soil struc- tures exhibited similar variabilities. Not all of the statistical parameters are independent, and some covariances of these are examined.

The variabilities of some derived structural quantities and dissimilarity coefficients between replicate soil structures are also examined as func- tions of sample size. Minimum sample sizes are set which are necessary to distinguish between soil structure of various degrees of similarity.

Key words: mathematical modelling, soil science, soil structure, agricultural engineering

A statistical method for quantifying soil macro-structure along linear transects through soil has been developed and has been described in detail elsewhere.rA3 This method has been applied in a number of field studies of practical im- portance.4-6

In this method, blocks of soil are impregnated with resin in the field and are then sectioned in the laboratory. Thin lines, horizontal in the original soil, are drawn or scratched on the sections at the depths of interest. These lines are then analysed at equally spaced points which are spaced at intervals of about one-fifth of the size of the voids or aggregates along the line. This gives spacings of about 0.5 mm for seed-beds or 1 .O mm for coarser structures. For each sampling point, a 0 is written if there is a void at the point and a 1 is written if there is a soil aggregate at the point. The zeros and ones are called elements. In this way, the soil structure along the line is represented by a string of zeros and ones. The statistical distribution of voids and aggregates along the line can then be described in terms of the statistical distribution of zeros and ones in the string.

Whereas the soil macro-porosity is simply the propor- tion of zeros in the string, a true measure of soil structure must also take account of the relative positions of zeros and ones in the string. The method adopted involves calculation of the probabilities of zeros appearing in the string depend- ing on the values (0 or 1) of the several elements immedia- tely to the left of the point being considered. In practice

it has been found that it is sufficient to consider only the preceding four elements. Since each element can take only two values (0 or l), this gives rise to 24 = 16 different combinations of elements or precursors of a point. These 16 precursors give rise to 16 transition probabilities, pi, of zeros following them. These 16 transition probabilities provide a good representation of the statistical structure of the string, and therefore also of the soil structure as observed on the line on the soil section.

Examples of three sets of transition probabilities are given in Table 1. These are for three depths in a self- mulching soil which, at the time of sampling, had received 34 years of a fallow-wheat rotation with conventional tillage with straw stubble incorporation. Details of this trial, and structural comparisons between this treatment and other treatments with various forms of stubble manage- ment and weed control, have been given elsewhere.’

Simulation

The above method of describing soil structure has the advantage that the sets of transition probabilities, such as those given in Table I, can be used to simulate soil macro- structures in a computer at any later date for further research purposes. An example of this is a model for root growth in structured soil where such ‘simulated soil’ is used.4 The process of simulation has been described pre- viously,’ but is outlined again here for completeness.

0307-904X/84/02093-08/$03.00 0 1984 Butterworth & Co. (Publishers) Ltd. Appl. Math. Modelling, 1984, Vol. 8, April 97

Variables of soil structures: A. R. Dexter and J. S. Hewitt

Table 1 Soil structural parameters for depths of 10, 20 and 40 mm. The pi are probabilities of a 0 following the precursor i

p; for depth

i Precursor IOmm 20 mm 40 mm

1 0000 0.682 0.470 0.444 2 0001 0.333 0.383 0.365

3 0010 0.600 0.469 0.585 4 0011 0.310 0.365 0.266 5 0100 0.540 0.512 0.500 6 0101 0.349 0.398 0.414 7 0110 0.484 0.546 0.455 8 0111 0.289 0.269 0.259 9 1000 0.510 0.526 0.569

10 1001 0.358 0.380 0.333 11 1010 0.476 0.426 0.352 12 1011 0.313 0.238 0.223 13 1100 0.566 0.510 0.465 14 1101 0.273 0.361 0.258 15 1110 0.488 0.395 0.397 16 1111 0.196 0.217 0.165

A structure is generated using a set of 16 transition probabilities, pi, and a source of random numbers in the range O-l. The process is conveniently carried out with a computer. It consists of comparing a random number R with the value of pi appropriate to the precursor. The string is then modified as follows:

if R <pi, a 0 is added to the string, or

if R > pi, a 1 is added to the string

Adding, for example, a 0 to8 precursor (say) 1100, generates a new precursor, 1000, which can be used in the next step. This procedure can clearly be continued in- definitely to generate strings of zeros and ones of any desired length.

For example, if the pi for the 10 mm depth given in Table 1 are used, and if it is assumed that the initial pre- cursor is 1111, then it can be seen that a sequence of random numbers 0.498,0.232,0.991,0.046,0.382,0.627, 0.357, . . . would add elements 1, 1, 1, 0, 0, 1, 0, . . . to the string, respectively.

Data strings generated in this way will only have stat- istical properties identical to those of the original data strings measured from the soil sections if they are in& nitely long. Owing to the stochastic nature of the sirnula- tion procedure, finite strings exhibit variability about the mean structure. In this paper the structural variabilities of finite samples of ‘simulated soil’ are compared with the structural variabilities observed between finite samples of soil collected from the field.

Variability of transition probabilities

In order to investigate the variability of the transition probabilities between replicate strings, 10 simulated strings were generated using the data for 20 mm depth in Table I for each of the following string lengths: L = 100,250, 500, 1000,2000,2640,4000 and 8000 elements. The length 2640 was included because it was the actual effective measured string length used to determine the soil struc- tures in Table 1 and in reference 7. For each generated string, the transition probabilities pi were calculated, and from these sets of 10 pi estimates of the standard devia- tions CQ of the pi were determined.

98 Appl. Math. Modelling, 1984, Vol. 8, April

In addition, the standard deviations Ui of the pi were calculated a priori from the pi in Table 1, using a theory of limiting variances applicable to Markov chains.j Values of ui for three of the most important pi at the 20 mm depth are given in Table 2, where it can be seen that u decreases as the chain length L increases. The small differences between the values of ui obtained by the two methods are attributable to the scatter resulting from the small number of samples (10 strings) used in the simulation method.

Plots of the logarithm of uj against the logarithm of L are linear, as shown in Figure 1. The slopes of these lines are all very close to -0.5, which indicates that the ui decreases inversely with the square root of the chain length L. This arises because the number of occurrences of particu- lar structural features, such as a particular precursor, is expected to be proportional to L. The standard deviations estimated from the six replicate measured data strings of length 440 elements from the 20 mm depth are shown as dots in Figure 1. The closeness of the dots to the lines indicates that the variabilities of the pi from the measured and simulated data strings were indistinguishable.

Tab/e 2 Variation of standard deviations oi, obtained theoretically and by string simulation of transition probabilities pi, with string length L, using data from 20 mm depth

String length L

100 0.262 0.242 0.160 0.1 JO 250 0.181 0.190 0.100 0.061 500 0.121 0.125 0.070 0.054

1000 0.081 0.056 0.049 0.042 2000 0.056 0.057 0.035 0.029 2640 0.049 0.050 0.030 0.024 4000 0.039 0.037 0.025 0.015 8000 0.028 0.017 0.017 0.015

01 of P, 08 of P* 016 of PI6

theor. sim. theor. sim. theor. sim.

0.087 0.112 0.051 0.051 0.036 0.033 0.025 0.019 0.018 0.017 0.015 0.020 0.013 0.006 0.009 0.008

Figure 1 Standard deviation oi of transition probabilities&. for

i = 1, 8 and 16, as determined from sets of ten simulated strings of length L. Estimates of the vi obtained from sets of six measured strings of length L = 440 elements are shown as dots. Data are for 20 mm depth

Correlations between transition probabilities

It has been shown above that variability of transition probabilities arises as a result of the finite length of the data strings used in their derivation. However, it is also possible that the deviations of the pi from their mean values are correlated. For example, it is not very likely that a soil sample would exist with all of the pi being larger than the mean values of the pi for the soil as a whole (as would be obtained from an infinitely long data string). It is more likely that some of the pi would be larger than the corresponding means, and others would be smaller than the corresponding means.

Also, in our analysis of the data strings, we have routinely made the structures symmetrical. In effect, this has been equivalent to examining each string from left to right and from right to left and averaging the results. The symmetry of soil structure has been discussed pre- viously.8 This symmetrization imposes constraints on some of the pi as followsz2

Any correlations with the sub-sets of four pi in equations (1) and (2) are of little significance in practice.

To test for any correlations between the pi of replicate data strings,,six measured and 10 simulated data strings of length L = 440 elements were analysed. Some of the resulting correlation coefficients are given in TabZe 3. It can be seen that very few correlations were found, even at the 10% level of significance. Where a significant correla- tion appears to exist in the measured strings, it is often found that it is not significant or is even of the opposite

Table 3 Correlation coefficients between the values of pi obtained from replicate data strings of length 440 elements. Correlation coefficients are presented for the pi of six replicate measured data strings and for the pi of 10 replicate simulated data strings. The whole correlation matrices are not shown

Correlation coefficients of pi from:

Measured strings Simulated strings

i 1 8 16 1 8 16

1 1 .oo -0.18 -0.26 1 .oo 0.14 0.22

2 -0.63 -0.63 0.26 -0.34 -0.26 0.43

3 -0.76 -0.02 -0.06 -0.14 -0.19 -0.35 4 0.63 -0.35 -0.81 0.74 0.05 0.12

5 0.42 -0.70 -0.35 -0.14 -0.60 -0.11

6 -0.61 0.18 0.30 -0.44 0.17 0.32

7 0.71 0.02 -0.44 -0.45 0.19 -0.18

8 -0.18 1 .oo -0.04 0.14 1 .oo 0.41

9 0.39 0.43 -0.34 0.24 0.40 0.21 10 -0.00 0.71 0.21 0.28 0.41 0.71

11 0.08 0.20 -0.03 0.01 -0.15 0.10

12 -0.11 -0.07 -0.77 0.55 -0.16 0.09 13 0.92 0.16 -0.41 0.36 -0.32 -0.04

14 -0.14 0.12 0.00 0.16 -0.08 0.47 15 -0.65 0.63 0.01 -0.36 -0.33 -0.16 16 -0.26 -0.04 1 .oo 0.22 0.41 1 .oo

Significance 10% 0.73 10% 0.55 levels 5% 0.81 5% 0.63

1% 0.92 1% 0.77

Variables of soil structures: A. R. Dexter and J. S. Hewitt

Table 4 Some derived quantities: porosity I)L; mean aggregate size 6; and mean pore size 6, and their variabilities as determined from six measured strings and 10 simulated strings. String length is 440 elements and data are from 20 mm depth

Measured Simulated

r)L 6 6 VL 6 6 (mm) (mm) (mm) (mm)

Mean 0.348 1.78 0.95 0.348 1.88 0.99 S.E. (mean) 0.008 0.06 0.03 0.014 0.10 0.03 S.D. 0.020 0.16 0.08 0.045 0.32 0.10 S.E. (S.D.1 0.006 0.05 0.02 0.010 0.07 0.02

sign in the corresponding simulated strings. Possible exceptions may be the significant correlations between p1 and p4, p5 and ps, etc. A similar small proportion of significant correlations exist in the columns for i = 2-7 and i = 9-l 5 which are not shown in Table 3.

In conclusion, there do not appear to be many signifi- cant correlations between the pi in replicate measured or simulated data strings, and for practical purposes the vari- abilities of the pi can probably be considered to be inde- pendent.

Variability of derived quantities

Sets of pi provide a convenient way of storing information about the structure of soils. The pi contain information about the size distributions and relative positions of voids and aggregates along the transects in the original soil. The quantities can be calculated readily from the pi (see reference 2). The most useful of these are probably the porosity, and the mean pore and aggregate sizes. These mean sizes are, of course, mean intercepted lengths and are not equivalent to the more usual sizes as determined by, for example, particle sieving.

The mean values of porosity and void and aggregate sizes for the 20 mm depth are given in Table 4. These were determined from sets of six measured and 10 simulated strings of length 440 elements. Also given are estimates of the standard deviations (S.D.) of these values and esti- mates of the standard errors (S.E.) of the means and of the standard deviations. The estimates of the S.E.s of the S.D.s were obtained from:

SE. (S.D.) = S.D./&2n) (3)

where n is the number of replicates.’ A more rigorous approach would be to assign confidence limits to the stan- dard deviations using the x2 distribution.”

It can be seen from Table 4 that there are no significant differences between the mean values or the standard devia- tions as determined by measurement and by simulation. Only the standard deviations of the porosity and the mean aggregate size come close to being significantly different. Better replication is needed for an accurate assessment of differences between the standard deviations.

Variability and structure comparisons

It is often of interest to be able to compare the structures of different soils or of soils which have been subjected to different management treatments. For this purpose, Hewitt and Dexter proposed three different methods.3 Of these, the measure of structural dissimilarity, Sr , has been

Appl. Math. Modelling, 1984, Vol. 8, April 99

Variables of soil structures: A. R. Dexter and J. S. Hewitt

used the most. This is given by:

(4)

where Ui and Vi are the relative frequencies of occurrence of the precursors i in the two structures being compared, and pi and qi are the probabilities of zeros following these,

respectively. On the basis of practical experience, it has been possible

to assign structural comparison criteria as follows:

sr=o identical

0 < S1 < 0.005 almost identical

0.005 < Sr < 0.015 very similar

0.015 <Sr<O.O3 similar

0.03 < SIG 0.05 slightly different

0.05 < Sr < 0.08 different

0.08 <Sr very different

However, the variability of structure between different replicate samples has an influence on the degree of struc- tural similarity which can be tested. We have examined this by simulating ten strings of length L = 100,250, 500, 1000,2000,2640,4000 and 8000 elements. We have then calculated the mean value of Sr for structural comparisons between all pairs of strings of the same length. These results are plotted for the 20 mm depth structure in Figure 2. The results can be described well by:

log,S, = 0.45 - 0.535 log, L (5)

which again illustrates that variability decreases inversely with the square root of the string length. The coefficient of variation within the sets of S1 was fairly consistent at around 0.35 and was independent of string length.

For comparison, we compared the values of Sr obtained in comparisons between the six measured strings of length 440 elements. This gave a mean Sr of 0.072 and a coefti- cient of variation of 0.30. The corresponding mean value of

-2.0

-2.5-

-3.0- vy

Y -3.5-

-4.0 -

-4.5- ’ 100 250 500 1000 2000 4000 8000

Figure 2 Decrease in mean value of the dissimilarity coefficients S, with increasing string length L, as determined from comparisons within sets of ten simulated strings. Mean value of S, determined within six replicate measured strings of length L = 440 elements is shown as a dot. Data are for 20 mm depth.

Sr from equation (5) for simulated strings of L = 440 is Sr = 0.060. This close agreement indicates that simulated strings have a variability which is indistinguishable from that of measured strings.

An implication of equation (5) is that a certain mini- mum length of data strings is necessary to enable certain structural comparisons to be made. These have been calcu- lated by applying equation (5) to the mid-points of the ranges of S1 given above. To distinguish between structures which are:

very different, we require L > 170

different, we require L > 380

slightly different, we require L > 940

similar, we require L > 2 700

very similar, we require L > 12 500

almost identical, we require L > 166 000

identical, we require L = *

Of course, the above figures are only a rough guide and will differ somewhat for different structures.

Conclusions

Soil macro-structure is highly variable in the field. The statistical method described previously for quantification of soil structure enables simulated soil structures to be generated by computer which have a similar variability to the variability in the field. Such simulated structures can be used in studies of the effects of soil structural variability on other properties of soil, or on the behaviour of plants. Large soil samples or a high level of replication is required for accurate determination of soil structure.

Acknowledgements

This work was supported by the Australian Wheat Industry Research Council.

References

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9

10

Dexter, A. R. ‘Internal structure of tilled soil’, J. Soil Sci. 1916,21, 261 Dexter, A. R. and Hewitt, .I. S. ‘The structure of beds of spherical particles’, J. Soil Sci. 1978, 29, 146 Hewitt, J. S. and Dexter, A. R. ‘Measurement and comparison of soil structures’, Appl. Math. ModeZZing 1981,5, 2 Hewitt, J. S. and Dexter, A. R. ‘An improved model of root growth in structured soil’, Plant and Soil 1979, 52, 325. Dexter, A. R. ‘Prediction of soil structures produced by tillage’,J. Terrumech. 1979, 16, 117 Hewitt, J. S. and Dexter, A. R. ‘Effects of tillage and stubble management on the structure of a swelling soil’, J. Soil Sci. 1980,31, 203 Dexter, A. R., Hein, D. and Hewitt, J. S. ‘Macro-structure of the surface layer of a self-mulching clay in relation to cereal stubble management’, Soil TillageRes.m1982, 2, 251 Hewitt, J. S. and Dexter, A. R. ‘Discussion reply to Rimmer, D. L.: “Internal structure of tilled soil”‘, J. Soti Sci. 1978, 29,121 Lambe, C. G. ‘Elements of statistics’, Longman Green, London, 1952 Sokal, R. R. and Rohlf, F. J. ‘Biometry’, W. H. Freeman, San Francisco, 1969

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