Six 1m^{2} plots were located between 5 and 7 m apart in an east–west direction across an arable field near Bristol, UK (51°25′N, 2°40′W; UK National Grid Reference ST 555703). The locations of these plots, which were used as the source of introduced beads and seeds, were surveyed so that they could be reestablished precisely after each subsequent soil cultivation. Initially, the six square 1m^{2} source areas were excavated to 20 cm and 20 000 plastic yellow beads spread at that depth in each. The soil was replaced, compacted to field level and 20 000 navy blue beads spread on the surface of each plot. The beads, which were the raw material for plastic injection mouldings, were cylindrical with dimensions of c. 1 mm diameter and 3 mm long. Data given by Fogelfors (1985) for a cultivation with a harrow indicated that about 1% of seed might be expected to reach 4·5 m from its origin, giving a bead density of 200 m^{−2} from the source density used in this experiment. The area was mouldboard ploughed on 11 September 1986 (Table 1). The plots were resurveyed and green plastic beads placed on the soil surface of the source plots prior to spring tine cultivation in a northerly direction. Subsequently, the plots were tine cultivated in the southerly direction with purple beads and seeds of barley (cv. Igri) on the soil surface. Purple beads and seeds of triticale (cv. Lasko) were then placed on the soil, before blench harrowing to the north. The final operation was seed drilling winter wheat to the south on 19 September 1986, prior to which skyblue beads were placed on the source squares. The five cultivation operations imposed on the area on a north–south axis, comprising ploughing with a reversible mouldboard plough, two spring tine cultivations in opposite directions, one pass with a blench harrow and seed drilling of wheat, were typical of those used to establish winter cereals on loam soils in this area of the UK. Bead and seed numbers were estimated from mean weights (Table 1). Each plot received every cultivation in the sequence, with different coloured beads placed on the soil prior to each. Thus the skyblue beads were only subject to drilling, but navy beads were subject to all cultivations.
Different coloured beads on the soil surface were initially counted in contiguous 10 × 10cm squares from the centre of the source area in the directions of cultivation (north and south). The counts of beads on the surface were made on 19 September 1986, after the completion of cultivations, to distances of 5 m north and 3 m south of the source area. Beads on the surface were also counted in 0·01m^{2} squares in six other directions (W, NW, NE, E, SE and SW) from the edges of the source areas, to assess movement away from the north–south axis (Fig. 1). The numbers of germinated barley and triticale plants were counted in similar quadrats on 24 October 1986, 5 weeks after sowing. At this time, bead and plant densities were low at distances greater than 3 m south and 5 m north from the source. Therefore, they were counted in larger 0·25 × 1·0m quadrats placed at 0·5m intervals up to 20 m north and 5 m south of the source area.
During the early summer of 1987, buried beads were retrieved from soil samples taken at 1m intervals to distances 15 m north and 5 m south from the first four plots only, involving the transport of in excess of 1·7 t of soil. Three samples were removed from each 1m^{2} source area, one from the centre and two at 0·5 m from the centre, straddling the north and south edges of the source area (Fig. 1). Samples were taken by hammering a 30 × 30cm open steel box into the ground. One side of the box was removed and four soil samples excavated in 5cm deep sections to a depth of 20 cm. Samples were washed out over sieves and the beads recovered and counted. In July 1987, the numbers of barley and triticale inflorescences were counted in 0·25m^{2} quadrats, at 1m intervals away from four of the plots. As severe lodging had occurred, counts were made from 2 m either side of the source areas.
Statistical analysis
As the bead densities recorded at different distances from the source are the result of a series of superimposed separate operations, a novel method of quantifying the effects of individual cultivations was required. This was possible for overall lateral movement but not for vertical movements, where sequential operations might move seeds up or down.
The effect of a cultivation can be quantified in terms of the proportion of seeds reaching any chosen distance from the source. This defines the probability distribution function (p.d.f.) of the distance a seed is moved by a cultivation. Various statistics can be derived from this, such as the mean distance moved and the distance from the source that will contain 50% of the beads.
Two separate approaches were required: first, for the bead profiles resulting from the last cultivation (the skyblue beads), and secondly, for the other coloured beads that had been subject to more than one cultivation. The aim in both cases was to estimate the p.d.f. of the distance a seed is moved by a cultivation.
Estimating the p.d.f. for the final cultivation
The effect of the last cultivation, seed drilling, could be investigated directly using the final positions of the skyblue beads. The observed distribution of the skyblue beads at distance d, g(d), is the convolution of the initial distribution of beads (evenly spread within a 1m square) at distance d, h(d), and the p.d.f. of the distance, x, moved by drilling [denoted by f (x)]. Thus if h(d) is the initial density of beads at distance d from the centre of the 1 m square, then:
 (eqn 1)
where A is the bead density per unit length of a transect of a chosen width. The convolution of f(x) and h(d) predicts g(d) generally as:
 g(x) = ∫h(d) f (dv)dv.(eqn 2)
For the specific h(d) above, g(x) is given by:
 (eqn 3)
A reasonable assumption is that a proportion, p, of the beads is moved, and the distance they are moved has an exponential distribution (i.e. the probability of a bead moving a given distance decreases with the distance moved). This was specifically chosen as a description of the dispersion process as it assumes that after a bead has been moved any chosen distance, the p.d.f. of the distance left to move is still exponential with the same mean, thus incorporating a ‘lack of memory’ property which we think is likely to hold in practice. Then the p.d.f. of the distance moved, x, denoted by f(x), is:
 f (x) = (1p) × δ(x) + p × λe^{λx}, x ≥ 0(eqn 4)
where δ(x) is the Kronecker delta function, having a value of 1 at x = 0 and 0 otherwise, p is the proportion of beads moved by the cultivation, and λ is the reciprocal of the mean distance moved by the ‘mobile’ beads. This is in contrast to Rew & Cussans (1997), who assumed that all seeds in their experiment moved but that the distance moved had the lognormal distribution. When the distribution in equation 4 is substituted in the convolution equation 2 above, the final observed distribution of beads, g(d), is as below:
 (eqn 5)
where d is the distance moved by the cultivation.
The parameters λ, p and A (the number of ‘sown’ beads in a strip of width 2Δ[Δ = 15 cm] in the original 1m^{2} area) were estimated by fitting equation 5 to the observed profile where the fitted counts in the soil core at distance d (with a width of 2Δ) are given in Table 2. The observed counts were assumed to have a Poisson distribution and fitted using maximum likelihood, using Genstat (Genstat Committee 1993).
Table 2. Equations used to predict counts of skyblue beads at different distances for soil cores of width 2Δ Distance (x)  Predicted bead count/A 


 
 
 
 
Estimating the p.d.f. for the other cultivations
The distributions of the beads, other than the skyblue that is produced by drilling alone, are the result of increasing numbers of operations. The distribution of purple beads is that produced by harrowing and drilling, that of orange beads by drilling, harrowing and one tine cultivation, and so on. We denote the p.d.f. of the observed distances away from the centre of the initial square after i cultivations by g_{i}(d), where d is distance from the centre of the square to the north or south. We assume that the i^{th} cultivation will have an associated probability distribution function (p.d.f.) describing the probability of a bead reaching a particular distance, x, denoted by f_{i}(x). Then the p.d.f. of the distribution of beads after i cultivations, g_{i}(d), is the convolution of the distribution after (i – 1) cultivations, and the i^{th} cultivation effect, that is:
 (eqn 6)
Brain & Marshall (1990) developed a novel technique using Fast Fourier Transforms to deconvolute this integral numerically using the pairs of observed profiles, to give point estimates of the underlying cultivation effect p.d.f.s f_{i}(x). This relied on the property that if the Fourier Transform of a function is denoted by f*(s) then:
 g_{i}*(s) = f *(s) × g*_{i1}(s)(eqn 7)
so that the Fourier Transform of the cultivation p.d.f., f*(s), is simply (g*_{i}(s)/g*_{i – 1}(s)). Their technique is presented in more detail in Brain & Marshall 1999). The Fourier Transform of each observed pattern was made, then the Fourier Transform of the cultivation effect could be calculated by dividing the later pattern by that for the earlier one, and the result backtransformed to give an estimate of the underlying cultivation effect. All calculations were made using Mathcad (Anonymous 1993). When these p.d.f.s were plotted against distance moved they often oscillated wildly for distances moved in a negative direction, i.e. in the opposite direction to the cultivation, a counterintuitive result (examples are given in Fig. 2). This is an artefact of the estimation process in the presence of variation, as for negative distances the p.d.f. is effectively zero. The oscillating estimates are thus random variables with an overall mean of zero. However, it is possible to ‘smooth’ these p.d.f.s by multiplying the Fourier Transform of the estimated p.d.f. by a smoothing function before backtransforming. This smoothing effectively takes the average of the p.d.f. over an interval of S either side of the observed point, i.e. by estimating:
 (eqn 8)
rather than f (x). Typical smoothed p.d.f.s are given in Fig. 2, and show that the smoothing has largely removed the oscillations. The exponential distribution, which had worked well for the seed drilling results, generally showed evidence of lack of fit, so the Gamma distribution was used instead, which has the form:
 (eqn 9)
with the mean being (β/α), and standard deviation (Üq1β/α). The estimated means and standard deviations of the distribution are presented in the results for the experiment covered here. The Gamma distribution was chosen as it is a generalization of the exponential distribution, with similar properties to the lognormal distribution used by Rew & Cussans (1997). The lognormal distribution, by contrast, does not include the exponential distribution as a special case. It has been noted previously that the exponential distribution has a ‘lack of memory’ property, so that the probability of a bead being deposited in any time interval is only dependent on the length of the time interval; the Gamma distribution has a similar interpretation, but with several independent exponential stages, corresponding to, for instance, the initial dispersal by the first tine of a harrow, followed by dispersal by a successive tine, etc. The Gamma distribution gave a good description of the harrowing and tining cultivation p.d.f.s for this experiment. In all cases the estimated proportion of the beads not moved by the cultivation was estimated.
To give increased flexibility in estimating the proportion unmoved it was assumed that the ‘unmoved’ beads could be slightly perturbed by a very small random distance in either direction, with the p.d.f. of this random distance being assumed normal with a zero mean and a very small variance. Thus, the p.d.f. of the distance moved by the ‘unmoved’ beads has the p.d.f. f_{U}(x), where:
 (eqn 10)
which gives a zero mean movement with a spread of movement around this mean distance with standard deviation of size σ. In most cases the value of σ was set to be infinitesimally small, when the ‘smoothed’ distribution takes the form of a step of fixed width, i.e. has a value of 1/(2S) for x < S, and 0 otherwise.
Denoting the p.d.f. of the Gamma distribution as f_{E}(x), the final cultivation p.d.f. is given by:
 f (x) = (1p) f_{U}(x)+pf_{E}(x)(eqn 11)
where p is the proportion of beads moved; the smoothed distribution could be readily derived from this. The smoothed p.d.f., f_{S}(x), was fitted directly by least squares, using Genstat (Genstat Committee 1993). In many of the cases there was no evidence that p was less than one, i.e. there was evidence that all beads had moved; where there was evidence of nonmovement the proportion of beads moved was estimated. For the ploughing treatment there was no evidence of movement, so the parameters of the Gamma distribution could not be estimated. Instead the parameters of the normal distribution, corresponding to a slight perturbation of the beads, were estimated where possible.