Cupitt and Whelan (2001) use local block kriging with local variograms within VESPER (Whelan et al., 2001) to predict five attributes namely; yield data for 1996, 1997 and 1998, soil EC and elevation data onto a single 5m grid. From the five interpolated attributes multivariate k-means clustering was then used to delineate three potential management zones. The kriging process provides an estimate of the mean prediction variance (σ2krig) and Cupitt and Whelan (2001) show that the
confidence interval (95% C.I.) surrounding the mean yield estimate within a field (μ) can be calculated according to Equation (2.10).
(
1.96)
. . % 95 = ± 2 × krig I C μ σ (2.10)Cupitt and Whelan (2001) suggest that the absolute difference between the mean zone yields (Yzone1−Yzone2 ) should then follow Equation (2.11) (Moore and
McCabe, 2003) for the potential management zones to be considered representative regions of significantly different yield (p<0.05).
96 . 1 2 2 2 1−Y ≥ × krig× Yzone zone σ (2.11) Whelan and McBratney (2003) also utilise Equations (2.10) and (2.11) to examine differences in yield between potential management zones. This suggests that the
LMU classification zones could be compared in terms of significant difference in yields using the above mentioned methods.
Whelan and McBratney (2000) present a methodology for assessing the temporal variability of a paddock. The variability across time in crop yield at within-field scales can be estimated based on the yield data by Equation (2.12).
1 ) ( 1 2 , 2 , − − =
∑
= n Y Y n j i j i i T σ (2.12) where; 2 ,i Tσ = temporal variance at point i, Yi,j= yield value at point i in each year j and Yi= mean yield value at point i for all years. Fixed points within the field must be used for comparison, whereby the data are obtained from spatial prediction onto a single grid. The estimate should provide an indicator of seasonal influences on crop yield (Whelan and McBratney, 2000). In relation to the LMU classification, a mean yield value could be calculated for each LMU that could be considered,Yi,j. In this way the LMUs could be assessed in terms of their temporal stability. Unfortunately, problems would arise when using different crop types.
Yield map standardisation is essential when combining multiple year and crop type yield data. Stafford et al. (1998) and Basnet et al.(2003) have assessed this issue in different ways. Basnet et al. (2003) delineate management zones using multiple crop yield data. They use four years of consecutive yield that had been interpolated on a 10m grid with VESPER (Minasny et al., 2002). The output yield data were converted to deciles (i.e. divide each data set into ten equal parts) and then rescaled (using linear interpolation between adjacent points) to values between 0 and 1. The four scaled yield layers were then combined spatially using arithmetic operators, such as addition and division. Average yield values were calculated on a cell by cell basis, and thus a map of average cell values created (0-100 percent yield). This was then reclassed into high, medium and low yield zones based on even divisions (i.e. 0 to 0.33 = low). Unfortunately, Basnet’s et al. (2003) output map of three management zones does not produce contiguous zones. Stafford et al. (1998) used a fuzzy classification on the yield data after first standardising the yield for each season to zero mean and unit variance (i.e. each value is given its z-score value).
Cluster analysis was carried out using the FCM algorithm of Bezdeck et al. (1984)(cited in Lark and Stafford, 1997). Basnet et al. (2003) used deciles to get the control points and make some form of standardisation prior to fuzzy input; while, Stafford et al.(1998) use zero mean and unit variance (i.e. z-score) for standardisation prior input to the fuzzy function.
Several researchers (Boydell and McBratney, 1999; Aspinal, 2000; Whelan and McBratney, 2000; Kelly et al., 2002; Basnet et al., 2003) have investigated the use of yield data to examine the spatial variability of yield over time.
The development of stable yield zone estimates was the aim of Boydell and McBratney’s (1999) research. Derived from multi-year yield estimates from mid- season Landsat TM imagery over 11 consecutive years, they wanted to discover the number of consecutive years of yield estimates required to give similar ‘stable’ estimates of yield zones. Boydell and McBratney (1999) used modified fuzzy k- means for predictive classification (FuzME (Minasny and McBratney, 2002)) to cluster the data and determine the most suitable number of groups. Their fields showed a strong degree of temporal stability and the general conclusions drawn were that stable yield zone patterns may emerge from multi-year yield estimates. More specifically, they state that five years of data (+/- 2 years) seem to give reasonable stable estimates of yield zones.
Most previous works have hypothesised that spatial trends in yield data would become more stable over time. However, Blackmore et al. (2003) utilising six years of yield data from four paddocks found that historical yield map trends cannot be used to extrapolate yield patterns in the future. Nonetheless, he suggests that spatial and temporal trend maps can help create homogenous management zones.
Detailing Blackmore’s et al. (2003) approach, the yield data collected in 1995-2000 inclusive was standardised on a 20m grid using a 20m search radius (they found that larger grids sizes tended to smooth the data too much, while smaller grids became too reliant on low number of data points). Blackmore et al. (2003) firstly produced two maps (spatial trend and temporal stability), which were later combined to form a spatial and temporal trend map.
a) Spatial Trend Map: The spatial trend map is designed to show this trend by calculating the arithmetic temporal yield at the same grid point over a number of years, which can subsequently be divided into tonne/ha classes. During this process Blackmore et al.(2003) found a large difference between yields from year to year. They documented this as the Temporal stability: inter-year offset; which was calculated as the difference between the arithmetic mean yield values between two years in the same field. Histograms of yield were computed and an offset judged as the difference between the yearly curves.
b) Temporal stability - temporal variance map: The temporal variance at a point has been identified by Blackmore et al. (2003) when one part of the field yields relatively high in one year and relatively low in another when compared to the mean. The variance from the mean (point yield minus the field mean) over time (1995-2000); is calculated for each year and then divided by the number of years (6 in this case) as follows;
(
)
2 00 95 , 2 6 ∑ − = tt== ti t i Y Y σ (2.13) where: 2 iσ is the temporal variance at grid point i, t is the time in years between 1995 and 2000, Y is the yield in years t at point i, and Yt is the mean of the yield for the whole fields in years t. This variance can be converted to standard deviation and maybe more useful as it is in tonnes/ha. The temporal variance will be small if an area of the field were to always yield close to the mean. This could be considered Stable in Time (SIT), as it would have a small temporal variance. Another area could sometimes yield large and sometimes small relative to the mean, this would be temporally unstable and it would give a large value of temporal variance (Blackmore
et al., 2003). The temporal variance map can be classified into areas that are SIT and
areas considered UNSIT (unstable in time) by setting a particular threshold in the temporal standard deviation data. However as yet, Blackmore et al.(2003) have no conclusive method defined to set this level. Therefore, they suggest looking at the sensitivity of the areas within the map that are deemed UNSIT at several levels. c) Spatial and temporal trend map: The spatial trend map and temporal variance map are brought together to form a single overview of the field and classified into four homogenous classes:
1. high yielding area: above the grand mean (for all years) for the field: 2. low yielding area: below the grand mean (for all years) for the field: 3. stable area – low inter-year spatial variance (arbitrary threshold) 4. unstable area – high inter-year spatial variance (arbitrary threshold)
As such, four possible combinations are possible: i) High and stable (HS); ii) High and unstable (HU); iii) Low and stable (LS) and iv) Low and unstable (LU) (Blackmore et al., 2003).
Blackmore’s et al.(2003) work offers an appealing approach as it provides a spatial layer based on yield data for comparison purposes that takes in consideration the fact that yield patterns do not necessarily become more stable over time. It is envisaged that a spatial and temporal trend map of this kind could be analysed against the map of LMUs using map comparison techniques to determine the appropriateness of LMUs.