analysis of three-component soil structures

Analysis of three-component .a soil structures _I. S. Hewitt and A. R. Dexter

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

Impregnated soil blocks were collected from a tillage trial at Condobolin, New South Wales. The object of the trial was to examine the consequences of performing the first tillage operation of the season on different dates. It was apparent that the soil broke up on tillage into the usual range of aggregate sizes plus a dust which was presumably composed of micro- aggregates. This was apparent on sections through the blocks which showed three components: voids, aggregates and dust.

A statistical method for quantifying soil structure, developed earlier, is extended to include three components. Since this involves a large number of transition probabilities, two approximate simplified methods were tested. One of these, here called the ‘grey-state method’, could be widely applied in structure studies using automatic scanning devices where features brighter or darker than some adjustable threshold level are detected.

The results show that the most recently tilled plot had the coarsest structure, but that the earlier two treatments were not significantly different from each other. The usual trend of decreasing aggregate size and decreasing porosity with increasing depth was not observed. Instead, there appeared to be some inversion of the usual layering of tilled soil which is attributed to the use of a motor-driven rod weeder.

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

The problems of when to till the soil and of how much tillage is necessary have been with us for a long time. In Australia, in particular, the practice of ‘bare fallowing’ - that is, tilling the soil and either leaving it or, more usually, repeatedly tilling it until the sowing of the next crop - has always been a subject of controversy. These fallowing practices have been the subject of a recent extensive review.’ The advantages to be had from bare fallowing are good weed control, increased infiltration and storage of rain water, and increased nitrogen mineralization; disadvantages are the high cost in fuel, machinery and man-hours, and the serious decline in soil aggregate stability and increase in soil erodibility which often occur in the long-term.

The experiment described here had the aim of investi- gating the effects of different dates of fist tillage or, in other words, different lengths of bare fallow period before the sowing of the next crop. The treatment which received its first tillage of the season earliest may be expected to have the finest structure, partly because it receives more tillage operations to break down the larger aggregates,2

0307-904X/84/02089-04/@3.00 o 1984 Butterworth & Co. (Publishers) Ltd.

and partly because it receives more weathering, and hence mellowing, of the tilled soil3 An additional effect of most tillage operations is sorting, which tends to put the smaller aggregates deeper in the tilled layer and the larger aggregates nearer to the surface.4’5

Soils and tillage

The soil samples were collected at the Agricultural Research Station at Condobolin, NSW (33” 04’S, 147” 14’E, altitude 195 m) on 25 May 1979. The soils on the areas sampled varied from gradational (e.g. Gn 2.12) to duplex (e.g. Dr 2.33) on Northcote’s classification.6 The O-200 mm layer is typically composed of 22% clay, 13% silt, 44% fine sand and 19% sand. The organic carbon content is 1.2- 1.4%.

The three treatments sampled had received their first tillage of the season on different dates. Treatment 1 was earliest and treatment 3 was latest. Thus treatment 1 had received the most tillage and the most weathering by rain-

Appl. Math. Modelling, 1984, Vol. 8, April 89

Analysis of three-component soil structures: J. S. Hewitt and A. R. Dexter

fall since its first tillage. Details of the tillage treatments, together with the amounts of rainfall between tillage operations, are given in Table la.

Additionally, paddock E3 was sampled. This had a long history of intensive annual tillage for wheat breeding trials and was said to be in ‘poor condition’. Details of the tillage and rainfall experienced by this paddock since its first tillage of the season are given in Table 1 b.

AU the soils sampled had carried a barrel medic/lucerne/ barley grass pasture for the previous four years.

The tillage implements used were as follows:

(1) Disc plough: Shearer 18 disc plough, working approximately 75 mm deep. (2) Scarifier: Horwood Bagshaw 19 tyne scarifier, with 200 mm ‘duckfoot’ points 70-90 mm deep. (3) Harrows: a heavy ‘stump-jump’ design. (4) Rod weeder: Morris driven rod of 25 mm diameter, approximately 50 mm deep. (5) Combine: in Australia, the word ‘combine’ is used to describe a combined scarifier and seed drill.

Sample collection and preparation

Samples of the tilled layer of the surface soil were impregnated as follows. Rectangular steel moulds of 270 x 140 x 100 mm internal dimensions were pressed into the

Tab/e 7 Tillage treatments applied at Condobolin

(a) Date of tillage trial

Date

Treatment number Rainfall

between 1 2 3 dates (mm)

17 Aug 78

3 Ott 78

16 Nov 78

22 Nov 78

9 Feb 79

3 Apr 79

1 May 79

18 May 79

25 May 79

Disc plough -

Scarifier -

- Disc plough

Scarifier -

Rod weeder Rod weeder

Scarifier Scarifier

Scarifier Scarifier

- Rod weeder

Soil sampled

53.6

93.8 -

16.4

64.6

31.2 Disc plough

37.2 Scarifier

15.6 Rod weeder

1 .o

(b) Paddock E3

Date Treatment

Rainfall between dates (mm)

23 Aug 78 Disc ploughed 50.2

2 Ott 76 Scarifier 121.8

7 Dee 78 Scarifier 53.8

7 Feb 79 Rod weeder 29.4

2 Apr 79 Scarifier and harrows attached 39.0

30 Apr 79 Combine and harrows attached 16.6

25 May 79 Soil sampled

soil down to the depth of tillage. The longest dimension of the moulds was perpendicular to the direction of tillage. Epoxy resin (Ciba: 10 parts LC191, 1 part HY95 1, and 1 part white pigment DWO 111) was poured into the

moulds to impregnate the enclosed tilled soil. The white pigment was added to improve the contrast between soil aggregates and voids on the sawn sections. Four blocks were collected from each tillage treatment.

The blocks were later sectioned lengthways in the vertical plane by using a diamond.saw. Two sections, 50 mm apart, were cut through each impregnated block for analysis of the structure at different depths.

Structure analysis

The macrostructure of the soil was measured by the method due to Dexter.’ Thin lines, horizontal in the original soil, were drawn across the sections at depth of 20,40 and 60 mm fortreatments I,2 and 3, and at 10 mm depth for paddock E3. The 25 mm at each end of each line was not analysed because of the possibility of disturbance when the moulds were inserted. These lines were analysed at points separated by 0.5 mm intervals. The points were called elements and were given the value 0 if there was a void (indicated by epoxy resin) at the point, the value 1 if an aggregate was present at the point, and a value 2 if the fine soil matrix, henceforth called ‘mix’ occurred. The mix was composed of features less than 0.5 mm across. In this way, data strings of length 440 elements were obtained which represented the structure of the soil at that depth. Since four samples were taken from each treatment, and two sections taken in a sample, the effective data string length for each depth of each treatment was 3520 elements.

The value of each element in turn is considered to be influenced by the preceding four elements, which are called the precursor of the element in question. There are 34 = 81 precursor states, labelled 0000 = 1,0001 = 2, 0002 = 3, etc. The total string is considered to be the end-product of a Markov process, in which each element is generated by the preceding four elements. Each of the 8 1 precursors will have two transition probabilities associated with it; the probability of a 0 occurring, po,i, and the probability of a 1 occurring, pi,i, for the ith precursor. The probability of a 2 occurring will then be pa,i = l- pe,i - pi,i. The theory of this type of process has been considered in detail elsewhere’ for a process with two states, 0 or 1, representing void or aggregate. Extensions to this theory to accommodate a third component, 2, are as follows.

Transitions from state i can occur to the three states (3(i - 1) + 1,2,3)1 (modulo 81). Some of these transitions or the states themselves may not occur in practice, but this does not detract from the general theory. Only three states can be self-transitioning: namely, state 1,OOOO; state 41, 1111; and state 81,2222. This self-transition is never in practice mandatory, so that the Markov chain is regular, not absorbing. The transition matrix P = {pii} will be an 8 1 x 8 1 matrix with 243 non-zero elements, namely :

=PO,i j = 3i - 2 hod.81

Pij =Pl,i i = 3i - 1 Imod.

= 1-PO,i-Pl,i i = 3ihnod.81

\=o otherwise

(1)

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


The probability ui of the occurrence of state i is determined by:9

uP=u

21 = (Ul, 4, u3, *. . , k31) (2)

Solution of equations (2) is generally by standard numer- cal methods, given P.

Distributions of void, aggregate and mix lengths are calculated as follows. Voids are considered to be generated by any precursor with the two right-hand-most elements 10 or 20, taking string generation from left to right; similarly aggregates by 01 or 21, and mix lengths by 02 or 12. The 18 void generating precursors are then 14 or 7 + 9i, i = 0, 1 ,..*, 8}, the 18 aggregate precursors are (2 or 8 + 9i, i=O, 1, . . . . 8) and the 18 mix generating precursors are (3or6+9i,i=O,l,..., 8). Let V,,i be the probability of a ‘void’ length i of type j, where j = 0 for a void, 1 for an aggregate and 2 for a mix length. and let Si,i be the set of precursor states which generate voids of type j and length i. For i = 1, the sets Si,i are listed above. For i = 2 or 3:

&,a= (10, 19,37,46,64,73)

S i, a = {5,23,32,50,59,77)

&a = (9, 18,36,45,63, 72)

Se, s = C&55) (3)

S 1, s = (14,681

S a, s = (27,541

and for i > 4, Si, i has one value only, respectively 1,4 1 and 81. The ‘voids’ of length i (i = 1,2,3) are formed by the occurrence of an element not of that type after a state of the ‘void’ length i generating type. Thus:

(4)

From this, the mean void, aggregate and mix lengths and their variances can be calculated; for the mean ‘void’ length of type j, Li, we have (for 1 E Si,,):

Lj = 2 iVj,i i=l

= i ivj,i +P~~V~,4[(l-~~,~)-2-1-22pi,~-33p~I] i=l

(7)

Transition probabilities and their approximation

The transition probabilities for the 81 precursors of length 4 elements are estimated from the raw data strings by counting the 243 5-element strings in these by computer. Then ratios are taken to give the transition probabilities, which affect a complete description of the full Markov process.

It is usual to average the occurrences of mirror reversed element strings, as experience has shown that no natural left- or right-hand bias is evident. Hence, for example, the number of occurrences of 00001 is taken as the average of the number of occurrences of 00001 and 10000. The transition probabilities po, i , p 1, 1 and pa, 1 are calculated in terms of the occurrences of the strings 00000, 00001 and 00002 as:

n(OOOOO)

PoJ = n(OOOOO) + n(OOOO1) + n(00002)

n(OOOO1)

P1, l= n(OOOOO) + n(OOOO1) + n(00002) (8)

n(00002)

P2P l= n(OOOOO) + n(OOOO1) + n(00002)

The higher transition probabilities are calculated similarly. It is evident that 243 5-element string occurrences, or

162 transition probabilities, is an unnecessarily large amount of information to describe a process, particularly since many (frequently more than half) of the states will either not occur or will comprise less than 0.1% of the sample string. This description and listing of transition probabilities will apply not just once, but perhaps 20 times for a variety of soil treatments. It is desirable to have an accurate description of the occurrences or transition

I

where

1ESj.i

Di=cuk k E Sj, I k

Because the generating state is the same for ‘void’ lengths of length 4 or more, then:

vi,4 = uztl -P~,I)*/L+ =$,4

vj,i = vj,,i-l&J i>4 (6)

Tab/e 2 Percentage occurrences of the 27 most important 5-element strings for Condobolin treatments (T)

Tl State no. State type (20 mm) :dO mm)

Tl T2 T2 T2 T3 T3 T3 T3 (20 mm) (20 mm) (40mm) (60 mm) (20mm) (40 mm) (60 mm) (10 mm)

1 00000 15.83 2.24 15.63 13.76 1.95 14.71 32.00 14.97 11.84 5.19 2, 82 00001 0.80 0.16 0.29 1.29 0.30 0.10 1.39 0.69 0.20 0.29 3,163 00002 0.56 0.20 0.19 0.53 0.29 0.30 0.87 1.12 0.29 0.20 5,109 00011 0.92 0.24 0.34 1.63 0.30 0.10 1.55 0.80 0.29 0.39 9,217 00022 0.76 0.36 0.21 0.04 0.39 0.30 0.90 1.16 0.36 0.23 14,118 00111 1.10 0.30 0.34 1.59 0.39 0.23 1.51 0.83 0.36 0.36 27, 235 00222 1.13 0.67 0.31 0.62 0.44 0.34 1.02 1.10 0.56 0.43 41,121 01111 1.10 0.43 0.63 1.61 0.47 0.27 1.51 0.87 0.37 0.27 81,241 02222 1.19 1.09 0.46 0.80 0.52 0.29 0.97 1.25 0.72 0.66 122 11111 19.38 33.26 45.95 33.08 41.97 53.01 23.88 19.90 28.50 19.38 123,203 11112 1.62 1.96 1.24 0.41 2.05 0.52 0.75 1.81 1.85 2.41 126,230 11122 1.85 2.28 1.38 1.32 2.31 1.71 0.76 2.01 1.92 2.85 135,239 11222 2.15 2.61 1.38 1.51 2.81 1.65 0.85 2.11 2.18 3.73 162,242 12222 1.91 2.51 1.22 1.46 2.71 1.45 0.75 2.02 2.05 3.67 243 22222 23.62 30.45 16.97 14.25 20.61 11.96 12.10 25.77 32.31 34.06



probabilities of the more commonly occurring states, which are also those to whose transition probabilities the Markov chain process is most sensitive; and to have a few parameters to describe the occurrences of the remaining states. The 27 most important and commonly occurring Selement strings and their percentage occurrences are listed in Table 2 for the 10 Condobolin samples. These are effec- tively 15, after the elimination of mirror images. It can be seen which strings dominate in the generation of long strings of similar elements.

transition probabilities. Then, given the mean ranking order, the transition probabilities of the process for each sample can be determined from 17 parameters.

The mean ranking order for treatments 1,2 and 3, obtained from the total of 101 occurring 5-element strings, and the ranking order for paddock E3 with 52 occurring 5-element strings, are given in Table 3. It was found that the mean deviation in ranking order for the 5-element strings of the individual samples from the mean ranking order was 8.3.

The remaining S-element strings occur, for the Condo- bolin samples, in about 8% of the sample strings. These number 2 16, which is reduced to 120 after the removal of mirror images. The occurrence of these can be approximated as follows.

Equation (9) was fitted to the occurrences of 5-element strings for all of the plots and depths. A Newton’s method approach to the determination of I/ and D so as to mini- mize:

Zipf postulated’* that word occurrence in a language can be described by:

Pj = Po(i + V) -m (9)

where Pi is the probability of occurrence of a word which is rankedith(i= 1, . . . , N) in frequency of occurrence, and I/ and D are parameters determined by the word sample taken. PO is a normalizing factor such that:

C (Pi - Po(i + V)-yD)2 i

was used. This summation was taken over distinct (non- mirror image) strings, whereas PO is determined by:

PO = Pall states (i + TV -’ (12) A fit coefficient of determination R2 can be defined as:

R2= l-C(Pi-Po(i + V)-1’D)2/C(Pi-F)2 (13) i i

It was found that the goodness of fit was relatively in- sensitive to the parameter V, which also varied in the range (0,600) for the best fit to the different samples. V was eventually taken as 15.0 throughout for treatments 1,2 and 3, which resulted in an increase of up to 5% of the fit parameter (lo), and a decrease in R2 of not more than 1%. The values of D, PO and R2 are given in Table 4 for V = 15.0. For paddock E3, optimum fit parameters V, D, PO and R2 are also given in Table 4.

$Pi=l (10) i=l

and, although a function of V and D, is best given additionally rather than recalculated. Equation (9) was applied to describe the occurrence of the 120 least common 5- element strings in this Markov process.

The ranking order must also be determined; however, for the purpose of describing the 5-element strings from a number of similar soil samples or treatments, a mean ranking order can be taken. It is assumed that the ranking order of the least commonly occurring strings is such that devia-

tion from the mean is insufficient to affect the parameters V and D, and hence the approximate occurrences and

(11)

A second method can be used to reduce the quantity of data needed to describe a three-component type process. This method is denoted the ‘grey state method’ and is as follows.

Tabled Ranking order of occurring 5-element strings fortheaverage of treatments 1.2 and 3 (T123) and paddock E3

String no. String no. String no. String no. ~-

Rank T123 E3 Rank T123 E3 Rank T123 Rank T123

1 207 207 27 80 68 53 144 79 24 2 204 204 28 117 108 54 150 80 43 3 171 216 29 4 120 55 104 81 52 4 161 234 30 45 132 56 116 82 61 5 225 161 31 120 153 57 18 83 78

6 189 189 32 10 8 58 31 84 89 7 134 225 33 7 19 59 129 85 92 8 113 134 34 108 57 60 177 86 132 9 216 131 35 15 63 61 198 87 186 10 234 171 36 68 75 62 6 88 16

11 86 125 37 23 84 63 21 89 33

12 95 13 38 42 96 64 56 90 44

13 83 77 39 8 107 65 69 91 50

14 77 80 40 11 113 66 96 92 53

15 13 40 41 75 144 67 153 93 66 16 25 45 42 156 159 68 159 94 71 17 165 90 43 57 165 69 180 95 87 18 111 111 44 32 183 70 59 96 93

19 26 26 45 54 39 71 183 97 98 20 90 117 46 74 69 72 213 98 99 21 131 150 47 29 72 73 20 99 102

22 40 83 48 38 129 74 30 100 114

23 125 86 49 39 138 75 36 101 168 24 84 15 50 63 156 76 72 25 19 23 51 107 177 77 12 26 79 42 52 138 186 78 22

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


Tab/e 4 Zipf’s law parameters for different treatments, T, and sampling depths

Sample V D p0 R’

Tl,20mm 15.0 0.589 0.405 0.80 Tl.40mm 15.0 0.377 8.193 0.89 T1,60mm 15.0 0.516 0.473 0.65

T2,20 mm 15.0 0.566 0.501 0.79 T2,40 mm 15.0 0.363 13.00 0.75 T2,60 mm 15.0 0.355 6.698 0.63

T3,20 mm 15.0 0.616 0.184 0.64 T3.40 mm 15.0 0.498 0.866 0.74 T3.60 mm 15.0 0.395 3.283 0.81

E3,lO mm 1.325 0.798 0.0323 0.99

Consider that the third state, the 2 state, is an intermediate (grey) state which can be seen as either a void (white) or aggregate (black). This process can be compared to viewing the structure on a screen where the contrast can be altered so as to make the grey either white or black. Firstly, if this is taken as the void state, i.e. 2s are seen as OS, then the transition probabilities to zeros are calculated for the two-state process from the original data strings. Call these probabilities pi (i = 1, . . . , 16). Secondly, consider that the 2s are seen as 1 s, and a second set of transition probabilities to zeros, qi (i = 1, . . . , 16), is found. The two sets of 81 transition probabilities for the three-component process can then be approximated from the two sets of 16 transition probabilities.

Suppose that a 4-precursor state k in the threecom- ponent type process is seen as state i in the-process where 2s are seen as OS, and as state j when they are seen as 1s (e.g. a partition of the 81 states into this second state process is given by equation (14)). Then the probability of a 0 after state k will be approximated by qi, a 1 by 1 -pi, and a 2 by pi - qi. The probability pi - qi should not be negative. Loss of information in less commonly occurring states very occasionally leads to pi - qi, being slightly negative. The criterion adopted to correct this is that the transition probability considered accurate is that for the same element as the right-hand most element of the precursor, the most likely transition, e.g. the transition probability for a 1 to occur after 2011 is taken as correct The remaining pi or qi is adjusted accordingly.

Suppose that it is not necessary to adjust any probabilities in obtaining the grey-state Markov chain. Then the 81 precursor states can be partitioned into 16 sets of states in two useful ways. Call these Ai (i = 1, . . . , 16) and Bi (i = 1,. . . , 16). Then:

A I= (0000)

A2 = {0001,0002}

As = {0010,0020}

Aq={OOll, 0012,0021,0022}

As = {0100,0200}

A6 = {0101,0102,0201,0202}

AT= {0110,0120,0210,0220)

A~={0111,0112,0121,0122,0211,0212,0221,0222}

Ag = {1000,2000}

Are = {1001,1002,2001,2002}

All = ~1010,1020,2010,2020)

Ara={lOll, 1012,1021,1022,2011,2012,2021,2022}

A 13 = ~1100,1200,2100,2200}

Ar~={1101,1102,1201,1202,2101,2102,2201,2202}

A~~={1110,1120,1210,1220,2110,2120,2210,2220}

Ar~={1111,1112,1121,1122,1211,1212,1221,1222, 2111,2112,2121,2122,2211,2212,2221,2222}

(14)

i.e. the sets Ai consist of states with equivalent OS. Simi- larly the sets Bi are a partition of the 18 states into states with equivalent IS. All states in set Ai (Bi) will form states of the same set Aj (Bi) under the same transitions. Each state in set Ai has a transition probability of a zero occurring of qi, and seach state in the set Bi has a transition probability of a 1 occurring of 1 -pi. This is a necessary and sufficient condition for the grey-state Markov chain to be lumpable into either a Markov chain (the Ai) consisting of OS and elements not OS, or a Markov chain (the Bi) consisting of 1s and elements not 1s (see Chapter 6 in reference 9).

The Markov chain with states Ai will then give the exact linear porosity and distribution of void lengths, and the Markov chain with states Bi will give the exact linear proportion of aggregate occurrence and distribution of aggregate lengths. So, hence, will the grey-state Markov chain, and obviously the linear proportion of mix will be exact. Occurrences of the 81 states and the distribution of mix lengths will not necessarily be equivalent to those of the full process.

Adjustments to remove negative probabilities will cause variations from exact representations of linear porosities and void and aggregate distributions, but these variations can be expected to be slight in practice, as negative probabilities tend to be derived for states with least occurrences.

Method comparisons

Comparison of the effectiveness of the Zipf and grey-state approximations of the Markov process with those of the full Markov process for three-component systems is made by both direct comparison of the transition probabilities and state occurrences, and by comparison of the derived quantities - linear proportions of void, aggregate and mix, and distributions and means of void, aggregate and mix lengths. Direct comparison of methods can be made for the linear proportions and mean lengths, but the large number of quantities to be compared for the transition probabilities and length distributions enforce the use of a general goodness of comparison parameter. A similarity coefficient Sr has been defined as:”

sl=C”i~ilPi-P:l/~Ui~i i i

(15)

for comparison of two structures with state occurrences Ui and Vi and transition probabilities pi and pj respectively. This is extended for the three-component structures as:

81 = F UiViI L- t min(pi,i-p(,i)l,/f UiVi (16) i=l j=l i=l

and is interpreted as the probability that the same transition occurs for the two Markov chains, given that they are



in the same state. Sr has been found to take values of O- 0.015 for very similar structures.

Comparison of the void, aggregate and mix length distributions is made using the parameter:

ISq= 5 (Vj,i- Wj,i)”

i=l /

2 (Vti+ Wii) (17) i=l

where Wj,i is the length distribution using the approximation method. The suffix 4 is used in equation (17) to avoid confusion with other measures of soil structural similarity defined in reference 11. S4 will be 0 for identical distributions, and will have a maximum value of 1 for total disjoint distributions. Since the void, aggregate and mix lengths are in geometrical progression for i 2 4, then:

+ vjf4/(1 -Pj, I)’ + wi4/(1 -PjZ)'

-2~j,4wj,4/(1-Pj,Z(lj,Z) (18) where Pj,l and qj,l are the appropriate transition probabilities for ‘void’ type i, and I E Sj, 4.

Results and discussion

The data strings for depths 20,40 and 60 mm for treatments 1,2 and 3, and paddock E3,lO mm depth, were analysed by the three methods. The linear proportions of void, aggregate, and mix, and mean void, aggregate, and mix lengths as determined by the full 162 transition probabilities are given in Table 5.

There are obvious similarities in treatments 1 and 2. Porosity is about 25% at 20 mm depth, decreases to about 7% at 40 mm and increases to 18% at 60 mm. The proportion of aggregates increases with depth, and mix proportions take a maximum at 40 mm and a minimum at 60 mm. With treatment 3, porosity decreases with depth, from 44% at 20 mm to 16% at 60 mm; aggregate proport- tions vary little, and mix proportions increase with depth. Mean void, aggregate and mix lengths are similar for treatments 1 and 2, with the exception that treatment 2 has a much greater mean void length at depth 60 mm, although a lesser porosity. The mean aggregate lengths increase with

Table 5 Void, aggregate and mix proportions and mean lengths for Condobolin treatments 1,2,3 and paddock E3

Sample

Proportions of: Mean lengths (mm) of:

Void Aggregate Mix Void Aggregate Mix

T1.20mm T1.4Omm T1,60 mm

T2,20 mm T2.40 mm T2,60 mm

T3,20 mm T3.40 mm T3,60 mm

E3.10 mm

0.26 0.34 0.40 3.3 3.7 4.0 0.07 0.46 0.47 1.5 5.7 5.1 0.20 0.55 0.25 5.7 10.6 4.9

0.25 0.4% 0.27 3.2 6.0 3.4 0.07 0.56 0.37 1.7 6.0 3.9 0.17 0.62 0.21 10.5 11.9 3.9

0.44 0.35 0.21 6.0 5.3 3.6 0.25 0.33 0.42 3.7 4.3 4.4 0.15 0.40 0.45 4.9 6.4 6.4

0.09 0.36 0.55 2.7 3.2 4.6

0.8

0.6

I 4 Aggregate length (mm)

0.8

0.6

0 0.5 1 2 4 8 16 3;

Void length (mm)

Figure 7 Proportions P of aggregates and voids having intercepted lengths greater than x (mm) in treatment 2 at depths of 20 mm (0),40 mm (0) and 60 mm (A)

depth in treatments 1 and 2, with 100% increases from 40 to 60 mm. Aggregate sizes have much less variation with depth for treatment 3. Paddock E3 is characterized by a low porosity (9%), the largest proportion of mix (55%), and generally small mean void, aggregate and mix lengths which are reasonably comparable with those of treatments 1,2 and 3 at 20 mm depth.

The aggregates and mix length distributions are similar in type, the accumulative distributions being approximately linear on a log (size) scale, and displaced according to particle size, e.g. aggregate length distributions for treatment 2 are shown in Figure 1. The cumulative void length distributions have a decaying rate of decrease behaviour on the same scale. Usually more than half of the void lengths are in the 0.5-l mm size range (0.5-2 mm for treatment 3), whereas this would be 0.5-4 mm (sometimes 0.5-8 mm) range for aggregate and mix lengths. The void length distribution for depth 60 mm shows a relative displacement towards larger void sizes for all of treatments 1, 2 and 3. This is also shown in Figure 1 for treatment 2.

The mean percentage changes (about equally positive or negative), and standard deviations, in the linear proportions and mean lengths of void, aggregate and mix, through using the Zipf and grey-state approximations, are given in Table 6. These are obtained by averaging for all depths of treatments 1,2,3 and paddock E3. Two exceptionally large variations do occur for plot 2, at 60 mm depth, and were not included in the averages. There was a Zipf method approximation of 6.4 mm mean void length compared with 10.5 mm, and a 6.3 mm mean mix length approximation by the grey state method as compared with



3.9mm. This happens because a large number of states do not occur in the sample, whereas they are allotted occurrence values/transition probabilities by the Zipf/grey-state method. This causes states which would normally not occur to have a significant part in void or mix length generation. From Table 6 it is seen that the grey-state linear proportions, expected to be exact, vary by up to 3% for the voids, 7% for aggregates, and 16% for the mix, while the grey-state mean void and aggregate lengths, also expected to be exact, are within 1%. The grey-state method nevertheless gives a better approximation than the Zipf method, with the exception of the mix lengths.

The similarity coefficient Sr (comparison values in Table 7) takes a mean value of 0.0008 and a standard deviation of 0.0004 for the Zipf method approximation of the transition probabilities and state occurrences of

Tab/e 6 Percentage changes and standard deviations of percentage in approximation of the Condobolin three-component process by the Zipf and grey-state processes

Zipf SD. Grey-state SD.

Parameter (%I (Zipf) (%) (grey-state)

Linear prop. voids 10.7 7.6 1 .o 1.2

Linear prop. aggregates 2.1 2.1 1.6 2.9 Linear prop. mix 5.0 4.5 4.4 8.2

Total linear prop. 5.9 6.2 2.4 5.1

Mean void length 6.9 5.3* 0.3 0.2 Mean aggregate length 4.5 3.6 0.04 9.76 Mean mix length 3.6 4.0 6.8 9.0” Total mean lengths 4.6 4.3 2.2 5.7

‘One anomalous result not included

Tab/e 7 Values of S, for comparisons between the full-Markov method and the Zipf and grey-state approximations

Sample Zipf Grey-state

Tl,2Omm 0.0013 0.0086

Tl, 40 mm 0.0006 0.0197

T1,6Omm 0.0002 0.0088

T2.20 mm 0.0011 0.0028

T2.40 mm 0.0010 0.0059

T2,60 mm 0.0003 0.0106

T3,20 mm 0.0007 0.0069

T3.40 mm 0.0013 0.0126

T3,60 mm 0.0005 0.0105

E3,lO mm 0.0006 0.0064

treatments 1, 2 and 3; and the grey-state method has a mean value of 0.0086 and a standard deviation of 0.0030. In comparison, Sr has values 0.03-0.035 in comparing similar structures for different samples of treatments 1, 2 and 3, and values of 0.03.5-0.07 for dissimilar structures. The values of Sr for paddock E3 approximations are also low.

Values of the length distribution comparison parameter S4 are listed in Table 8. The grey-state method gives almost exact distributions of void and aggregate lengths, as expected. The values for the Zipf approximation are about 0.027, and the grey-state approximation to the mix length distributions about 0.032. These values indicate good representation, as comparison lengths of different samples gave values of S4 of the order of 0.14, while void-aggregate or void-mix distribution comparisons generally gave values greater than 0.4. Values of S4 for paddock E3 are very low (of the order of 0.001) for the approximation methods.

Conclusions

Examination of the soil structure sampled at Condobolin has shown that aspects of soil structure other than just void or aggregate occurrences sometimes need to be incorporated into a stochastic model. In this case a fine porous soil matrix, presumably composed of micro-aggregates, occurs as a third major component; this third component could equally well be stones or discrete pieces of organic matter.

The inclusion of this third component increases the number of parameters necessary to specify the Markov chain model from 16 to 162. Many of these are relatively unnecessary. Some approximation methods are needed to reduce this number, particularly if any two-dimensional analysis is to be made on the structure, when the number of transition probabilities may increase from, say, 26 = 64 to 2 x 36 = 1458.

The two approximation methods examined both have drawbacks. The Zipf method is useful only when a large number of similar structures are being compared, so that a mean ranking order of the least common states can be specified; otherwise no saving in information is made. More analysis is needed to determine the Zipf parameters. The grey-state method approximates any single structure with less analysis by 32 transition probabilities. The Zipf method produces a better approximation than the grey-state method for transition probabilities and state occurrences, but the grey-state method (theoretically) gives exact porosities and void and aggregate length distributions. The main drawback is that the mix length distributions are less accurate. Both methods give a closer approximation to a

Table 8 Values of S, for comparisons between the full-Markov method and the Zipf and grey-state approximations

Sample Zipf void Zipf agg. Zipf mix Grey-state Grey-state Grey-state length length length void length agg. length mix length

Tl, 20 mm 0.018 0.019 0.042 T1,4Omm 0.037 0.002 0.011 T1.6Omm 0.022 0.059 0.006

T2.20 mm 0.042 0.013 0.003 T2.40 mm 0.019 0.030 0.010 T2.60 mm 0.101 0.027 0.053

T3.20 mm 0.068 0.016 0.016 T3.40 mm 0.011 0.014 0.016 T3,60 mm 0.006 0.050 0.014

E3,lO mm 0.0006 0.001 0.003

0.0 0.0 0.0001

0.0 0.0 0.0

0.0 0.0 0.0

0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0

0.011 0.068 0.047

0.013 0.034 0.042

0.016 0.038 0.017

0.0006



structure than would be obtained by using a similar structure; however, it has been, seen that very occasionally a mean component length can vary by up to 50%.

On this basis it is judged that the grey-state approximation is better, both for simplicity and general accuracy. Here, the name was derived by considering the mix state to be an intermediate (grey) state between the void and aggregate states. There is no need to abide by this. Generally, the least commonly occurring state in a three- component structure should be considered to be the grey state. For example, in low porosity structures (e.g. treatments 1 and 2, depth 40 mm) it woild be feasible to take the voids as the grey state, so as to obtain the greatest information about the mix and aggregate states, which (aboul equally) comprise 93% of the structure.

They grey-state method was devised so as to be com- patible with low-cost automatic scanning equipment. Such scanners can give as outputs the coordinates (x, r) of all picture elements which are brighter than some arbitrarily adjustable threshold value. For example, information may be needed about the spatial distribution of a third (prey) type of feature on some scanned sections. Firstly, the threshold can be set so that these features are not detected. The coordinate set of the detected features can then be used to generate a set of transition probabilities pi. Secondly, the threshold can be set so that the grey features are detected. The new coordinate set can then produce a second set of transition probabilities qi. The pi and qi together contain approximate information about the relative sizes and dispositions of all three types of feature.

The Condobolin soil appeared to break up on tillage into large aggregates and a dust which is presumably composed of relatively stable micro-aggregates. The large amount of tillage which most of the plots had received, together with the weathering since the first tillage of the season, resulted in this dust occupying a large proportion (21-55%) of the tilled layer in all of the treatments. The dust appeared as a fine soil matrix or ‘mix’ on the sawn sections of the impregnated soil.

The two treatments (1 and 2) which received their first tillage earliest had very similar structures and had, perhaps, attained an equilibrium structural state. The most recently tilled treatment (3) still had a coarse structure as it had received less tillage and less weathering. Although paddock E3 had received a comparable amount of tillage and

weathering to treatment 1, the structure was much finer and the macro-porosity was very small. This difference illustrates the detrimental effects of a long history of excessive tillage, which even the previous four years of pasture had not been able to rectify.

The profiles of the tilled layers were very interesting and unusual. The expected gradation from the coarsest structure are the surface to the finest structure at the base did not occur. Instead, there was usually some inversion of this expected trend. This might have been because the weathering effect was much nearer to the surface or, more prob- ably, because of the use of a motor-driven rod weeder.

Acknowledgements

The authors would like to thank Neil Fettell for his help and for allowing them to collect samples from his experiment at the Agricultural Research Station, Condobolin, New South Wales. The work was supported by the Australian Wheat Industry Research Council.

References

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