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Agronomic Modeling

Spatial Analysis of Cranberry Yield at Three Scales

^a P.E. Marucci Center for Blue/Cranberry Res. and Ext., Rutgers Univ., 125A Lake Oswego Rd., Chatsworth, NJ 08019-2006
^b Dep. of Environ. Sciences, Rutgers Univ., New Brunswick, NJ

^* Corresponding author (gimenez{at}envsci.rutgers.edu)

Received for publication December 17, 2003.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Cranberry (Vaccinium macrocarpon Ait.) is an intensively managed perennial crop. Patches of disease, local variation in soil properties, and regional changes in soil type and hydrology cause its yield to vary spatially at several scales. We evaluated the spatial variability of cranberry yield with two support sizes and covering three scales: (i) 500 contiguous 0.09-m² samples covering a 6 by 7.5 m area (small scale, SS), (ii) an average number of 100 variably spaced 0.09-m² samples from each of 21 fields (medium scale, MS), and (iii) 534 fields (16830 m² average area) each characterized with a single value of total yield (large scale, LS). Differences in yield calculated from points separated by incremental distances h were raised to power values q (from 0 to 4 in steps of 0.1). The q = 2 data were fitted to either spherical (SS and LS) or exponential (MS) semivariogram models. The logarithm of average differences plotted vs. log h were characterized by their slope, (q). Structure functions [(q) vs. q] were fitted with the universal multifractal model containing three parameters (C, , and H). Small scale and LS data had nonlinear structure functions typical of multiscale phenomena. Spatial properties of cranberry yield at MS were: (i) better defined in cranberry fields with more than 12 yr in production (small range and nugget variance), and (ii) influenced by multiscale factors (nonlinear structure functions). Younger fields had greater range and nugget variance and a linear structure function. Precision agriculture in perennial crops should consider temporal changes in the spatial structure of crop yield.

Abbreviations: Lng_Rng, long range • PRR, Phytophthora Root Rot • Srt_Rng, short range

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
PERENNIAL CROPS are likely to develop persistent spatial variability in yield in response to biological and edaphic factors by developing genotypic heterogeneity over the life span of a plant (Novy et al., 1994, 1996). However, very limited research was conducted on spatial variability of yield and diseases of perennial crops (Timmer et al., 1989; Horner and Wilcox, 1996; Turechek and Maddan, 1999; Oudemans et al., 2002; Pozdnyakova et al., 2002). Cranberry (Vaccinium macrocarpon Ait.) is a perennial high value horticultural crop that grows on wetlands as a complete coverage of vines. Although commercial cranberry production is intensively managed, a high degree of variability has been identified within cranberry fields (beds), both by remote sensing as well as by ground based measurements (Pozdnyakova et al., 2002). Considering the characteristics of cranberry production, enhancement of profitability should be achieved through precision management rather than hectarage expansion. A necessary first step for developing a precision agriculture system for cranberries is to adequately understand the spatial variation of soil properties and crop characteristics and any possible changes of those spatial patterns over time.

Preliminary studies of the spatial variability in cranberry yield indicate that factors influencing variability differ with scale. At the small scale, those factors are likely to be soil-born fungal diseases, such as Fairy Ring and Phytophthora Root Rot (PRR). Several causal pathogens have been described in the literature as related to Fairy Ring, including basidiomycetous fungus Psilocybe agrariella as well as Phialophora and Rhizoctonia spp. (Caruso and Ramsdell, 1995; Zuckerman et al., 1968; Chang, 1989). Phytophthora cinnamomi Rands is a primarily causal agent of PRR in New Jersey (Caruso and Ramsdell, 1995). Fairy Ring expands outward from the point of infection in all directions at a rate of 0.3 to 0.4 m per growing season, whereas PRR is found as discrete patches ranging in size from <1 m² to about 100 m² (Caruso and Ramsdell, 1995). The diseases reduce root biomass and vine density leading to lower yields (Oudemans et al., 2000). Vine density can also be influenced by the genetic diversity found within a cultivar. For example, following planting, the vines will "fill in" the planted area and as the field matures (beds may remain in production for as much as 100 yr) specific spatial patterns of genetic variation in cultivar can develop. Development of rogue genotypes with low yield potential can spread into die-back patches during the life span of the field. At the field (medium) scale, cranberry yield is correlated with vine density and multiple soil properties, such as water content, infiltration rate, and temperature, as well as with elevation (Pozdnyakova et al., 2002). At the regional (large) scale yields from entire fields are correlated (R² = 0.27–0.39) with field geometry, namely with the ratio of bed perimeter to bed area (Pozdnyakova, unpublished data, 2001). The correlation between field geometry and cranberry yield might be caused by better drainage in beds with large perimeter/area ratios since all the beds typically have drainage ditches at the edges. Beds of similar geometry also tend to be clustered (Fig. 1), resulting in a spatial dependence in yield. Other factors that might contribute to yield autocorrelation at the large scale are soil type, microclimate, and regional hydrology.

View larger version (38K): [in this window] [in a new window]   Fig. 1. Major part of the study area near Chatsworth, NJ. Total yield for entire fields was used to evaluate large scale variability. Shaded are 21 fields with dense sampling to account for within field variability (medium scale).

The objective of the present study was to evaluate spatial variability of cranberry yield at three scales with semivariogram models and with a generalized semivariogram (structure function) approach. The results can be applied to develop sampling and spatial analysis strategies for the precision agriculture practices on cranberries, as a model system for perennial crops.

	Theory

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Semivariogram
A semivariogram represents the scaling of the second moment (variance) of the fluctuations (or increments) of measured values, z_i, taken at increasingly longer distances, h:

[1]

where (h) is the semivariance, whose values typically increase with an increase in h reaching a maximum (sill) at a value of h defined as the range (Isaaks and Srivastava, 1989). The intercept of the semivariance at h = 0 is defined as the nugget. Positive values of the nugget represent a combination of experimental error and of unresolved spatial variability occurring at scales smaller than inter-sampling lag distance (Burrough, 1993). The fraction of the sill that is not nugget variance constitutes the structural variance with values approaching unity in a strongly spatially structured system (Robertson and Freckman, 1995), or zero in a system that has little spatial structure at the scale of observation (Morris, 1999). In other cases, (h) values can increase without evidence of reaching a sill. Unbounded semivariograms are typically the result of spatial trends, and in some cases result in a power-law semivariogram of the form (Burrough, 1983; Neuman, 1994):

[2]

where H is the Hurst exponent, a real number with values between 0 < H < 1. The value of H = 0.5 indicates noncorrelated increments (white noise), whereas values of H < 0.5 indicates negative correlation (phenomena dominated by short-range variability), and H > 0.5 represent positive correlation (phenomena dominated by long-range variability).

Semivariograms spanning wide range of scales may exhibit different domains of variation each dominated by a local condition. This situation is referred as to a multiscaled pattern of variation (Burrough, 1983; Neuman, 1994).

Structure Function
A generalization of Eq. [2] consists of extending the analysis of the spatial dependence of the increments to higher and lower moments of order q (Davies et al., 1994):

[3]

where (q) is an exponent, and . and |.| indicate statistical average and absolute value, respectively. The various moments of order q weight the distribution of increments differently and provide information on the spatial dependence of increments with small and large values. For instance, high order moments emphasize the largest values in a distribution of increments, whereas q = 1 reproduces their original distribution. The shape of the structure function (q) vs. q defines the complexity of a distribution. If (q) is a linear function of q, its value reduces to qH (single scaling), and the Hurst exponent H (see Eq. [2]) is sufficient to characterize it. On the other hand, a nonlinear structure function (multiscaling) requires more than one parameter for its characterization (Davies et al., 1994).

The structure function can be characterized with three parameters using the universal multifractal model (Schmitt et al., 1995):

[4]

where H, defined as (1) = H, characterizes the spatial correlation of the average absolute increment, C is the codimension that characterizes the inhomogeneity of the mean of the process, and describes the degree of multifractality; = 0 is single scaling and = 2 is the lognormal multifractal case. In the case of single scaling either C or is equal to zero and the structure function is a linear function of q, and only defined by H. Larger H values result in better defined spatial correlation (Tennekoon and Boufadel, 2003). Nonlinear structure functions typical of multifractal processes are characterized by values of C and larger than zero. The larger the value of C the sparser the occurrence of any given increment, whereas larger values of imply than few high increment values dominate a distribution (Seuront et al., 1999).

To date, the semivariogram remains as a standard method to quantify spatial structure of soil/crop properties and a large body of knowledge on its application exists, making this technique also appropriate as a standard for comparison of new approaches. However, using the same raw data, Eq. [3] provides a better insight on the complexity of a field than either the semivariogram or the power law semivariogram (note that Eq. [2] cannot discriminate between single and multiscaling spatial patterns because it only quantifies the second order moment). The disadvantage of Eq. [3] resides in the potentially large number of parameters needed to quantify the heterogeneity, which can be partially overcome by using the universal multifractal model (Eq. [4]), which represents the structure function with three parameters (Tennekoon and Boufadel, 2003).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Data Collection
We analyzed the spatial variation of cranberry yield data obtained with two sample support sizes at three different scales. For the first two sampling scales (small and medium) yield was collected using a square support of 0.3 m side (0.09 m²). Sampling squares were placed on the vines, and berries within a square were counted. Yield data included number of harvestable fruit, number of rotted fruit, as well as nonharvestable fruit. In addition, average berry weights were determined so that yields could be estimated and expressed in kg/m².

Small scale sampling comprised 500 contiguous (0.09 m²) squares arranged in a 6 by 7.5 m area, where yield was recorded in 2001 from a location with Fairy Ring disease in a field planted with Ben Lear cultivar. The medium scale sampling was designed to characterize intrafield variability in 21 fields (Fig. 1) planted with three cultivars (Ben Lear, Early Black, and Stevens), which were sampled in the fall of 2002. In each field, an average number of 100 locations, separated by distances ranging from 3 to 200 m, were set before harvest using a stratified random sampling method (Sample 3.03 extension, ArcView 3.2, ESRI, Redlands, CA), and later sampled on a support of 0.09 m². Sampling density varied from a minimum of 21 to a maximum of 91 sampling areas per hectare, with an average of 57. Field 31 (Fig. 1) planted with Stevens cultivar and having a large area (1118 m²) of PRR disease was sampled in 2000 by combining sampling strategies for the small and medium scales. A total of 148 randomly spaced points were set up within the field and sampled on a 0.09-m² support (Table 2, F31-1). In addition, 23 locations were sampled with 25 contiguous 0.09-m² samples arranged in a 1.5 by 1.5 m area. The small and medium scale data were pooled and analyzed together (Table 2, F31-2).

Data Analysis
Descriptive statistics, which included mean, median, standard deviation, minimum and maximum values, were calculated with STATISTICA (StatSoft, Tulsa, OK). Skewness, kurtosis, and Shapiro-Wilk's index were used to test for normality. Shapiro-Wilk's parameter was estimated with the algorithm proposed by Royston (1982), as implemented in STATISTICA (StatSoft, Tulsa, OK).

Semivariogram
Semivariograms (Eq. [1]) were estimated using GS+ (Gamma Design Software, Plainwell, MI) (Robertson, 2000). Sample semivariograms were standardized by the sample variance for comparison across data sets. Omnidirectional semivariograms were calculated for each of the data sets because of the relatively limited number (about 100) of sampling locations (Isaaks and Srivastava, 1989). The maximum lag distance varied from 5 m with 0.5-m increments (small scale) to 100 m with 10-m increments (medium scale), resulting in 10 points to which to fit a semivariogram model for these two scales. The large-scale semivariogram was calculated for a maximum lag distance of 3000 m with 150-m increments, resulting in 20 points to which to fit a semivariogram model.

Spherical and exponential models were fitted to the experimental semivariograms (Isaaks and Srivastava, 1989). The spherical semivariogram model is defined by:

[5]

where C₀ is the nugget, C₀ + C₁ is the sill, and a is the range. The exponential semivariogram is similar to the spherical in that it approaches the sill, but it never reaches it (the range is usually assumed to be the point at which the model reaches 95% of the sill). The equation for an exponential model is:

[6]

The best model for a data set was chosen based on the maximum coefficient of determination and minimum residual sum of squares for the fit as well as through cross-validation, in which every known data point is estimated using only neighboring values (Isaaks and Srivastava, 1989). The regression coefficient between actual and estimated values and the proportion of the variation explained by the model should be as close to unity as possible.

Structure Function
The program GAMV provided in GSLIB (Deutsch and Journel, 1998) was modified to calculate multifractal spectra using Eq. [3], by raising the absolute value of the fluctuations (or increments) to q values between 0 and 4 at increments of 0.1. The output of the program included pairs of log |Z_x – Z_x+h|^q and log(h) values over the q values considered. Values of (q) were estimated from the slope of the linear regression of the output pairs over the linear region of the function (Fig. 2) using the same maximal and incremental lag distances than for the corresponding semivariogram models. The functions (q) vs. q were fitted with Eq. [4] using nonlinear regression to obtain values of parameters C and , while values of H were obtained as (q = 1) = H (Liu and Molz, 1997).

View larger version (17K): [in this window] [in a new window]   Fig. 2. Example of the fit of the log |Zx – Zx+h|q vs. log (h) for different q, data for small scale yield variability.

View larger version (43K): [in this window] [in a new window]   Fig. 3. Histogram of cranberry yield data for the small scale variation.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

Sampling support size affected the statistics of yield distribution more than the spatial scale at which measurements were made (Table 1). The mean and the standard deviation of yield decreased markedly with the increase in support size. The mean of the observations taken with a 0.09-m² sampling support was similar at small and medium scales, but data were more variable at the former than at the latter scale, probably because measurements at the small scale included approximately equal areas with high and low yield.

Semivariogram
Spatial variability of cranberry yield showed differences at the three scales of the study (Tables 1 and 2). At the small and large scales, semivariances reached a well-defined sill and were best characterized by the spherical semivariogram model (Table 1). The difference in the range at the small and large scales suggests that different factors operate at each of these scales (Gajem et al., 1981; Burrough, 1993; Gelhar, 1993; Neuman, 1994). At the small scale, yield variability was largest but its spatial structure was completely resolved by the semiovariogram function, which is to be expected given the high sampling density (continuous sampling) used at this scale (Fig. 4, Table 1). On the other hand, fitted values of the nugget, sill, and structural variance at the large scale were intermediate between the ones obtained at the small and medium scales (Fig. 5, Table 1). At the large scale, the nugget effect was probably caused by a significant intrafield variability that could not be resolved by the large sampling support size. At the medium scale, the semivariogram of all 21 fields combined had the smallest structural variance, largest nugget, and smallest sill values among the scales. The best semivariogram model at the medium scale was the exponential one, which is typical of properties showing overlapping scales of variation and when the zone of transition between scales is not defined (Burrough, 1983, 1993).

View larger version (10K): [in this window] [in a new window]   Fig. 4. Semivariogram of cranberry yield at the small scale.

View larger version (13K): [in this window] [in a new window]   Fig. 5. Semivariogram of cranberry yield at the large scale.

Medium- and small-scale yield variability was studied together by combining the data from both scales gathered in Field 31 (Fig. 1). When only 148 widely spaced data points were analyzed, the semivariogram parameters were intermediate between the Srt_Rng and the Lng_Rng groups (Fig. 6a, Table 2). However, when additional closely spaced points were added to the data, increasing the total number of points to 723, the model that described the data best was the spherical one (Fig. 6b). Incorporation of the closely spaced points improved the fit of the semivariogram model at the short lag distances, but the fit at the longer lags was worse (Fig. 6b). Overall, the goodness-of-fit of the semivariogram model decreased (larger residual sum of squares) when the closely spaced points were added to the data (Table 2). Although the range did not change substantially with the addition of close spaced points, the nugget decreased and the structural variance increased because of improved information on short-range variability. The variation at scales smaller than 10 m accounted for about 36% of the total variability in the data, whereas 50% was attributed to variability at the medium scale, and the remaining 14% was undefined.

View larger version (22K): [in this window] [in a new window]   Fig. 6. Semivariograms of cranberry yield in Field 31 (see Fig. 1) sampled with (a) 148 wide spaced points and (b) for 723 wide and close sampled points combined.

Yield data collected at the small and large scales shared multiscale properties (i.e., nonlinear structure functions), while intrafield variability (medium scale) was at a single scale when data from all fields were analyzed together (Fig. 7, Table 1). The fit of the structure function at the small scale showed the best fit for all the q values considered (Fig. 2, Table 1). At this scale, the values of H and C were twice as large, while was 16% smaller than the respective values at the large scale. Relatively large values of H and C imply a more pronounced spatial correlation of the average absolute value of the increments and more patchiness in their spatial distribution, respectively. Relatively smaller values of indicate that the distribution of yield contain extreme variation that is probably associated with Fairy Ring disease and with the related variability in vine density (short range phenomena) that resulted in samples having zero yield (Table 1). At the large scale, the structure function indicates a more uniform spatial distribution of cranberry yield with fewer but larger deviations from the mean increment value that could be related to differences in management practices.

View larger version (19K): [in this window] [in a new window]   Fig. 7. Structure functions (q) of cranberry yield at small, medium (all fields combined), and large scales.

View larger version (25K): [in this window] [in a new window]   Fig. 8. Structure functions (q) for the medium scale variability: (a) fields with short range (Srt_Rng) semivariograms combined, (b) fields with long range (Lng_Rng) semivariograms combined.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The complexity of the spatial variability of cranberry yield was manifested through the analysis of yield data collected with two support sizes and at three spatial scales. Our results suggest that intersampling distance is more important than support size in determining spatial structure of cranberry yield. Sampling schemes aimed to characterize intrafield variation should include sampling at distances of <5 m to capture short-range variation.

Field age is an important consideration when analyzing intrafield spatial variability of cranberry yield. In older fields, the spatial structure of yield is more homogeneous, containing a few areas or patches with very high yield fluctuations. In that sense, practices leading to precision agriculture are probably easier to implement in older than in younger fields. This research revealed that the most important and variable spatial structure in cranberry yield is at the field scale (defined as medium scale in this paper). Management of this variation either by segmented precision agriculture or by implementing long-term measures to bring the whole field to uniformity should inevitably increase overall yield and farmer's profitability. Thus, future research should focus on understanding relationships among management and yield spatial structure in perennial crops and the changes that occur over time.

The structure function is a promising approach to characterize spatial heterogeneity. Further applications of this technique in the context of precision agriculture are the analysis of remote sensing images and the simulation of field heterogeneity.

	ACKNOWLEDGMENTS

 
This paper presents a research supported by Initiative for Future Agriculture and Food Systems financed by USDA and NASA, grant no. 2001-04782 (Enhanced management of agricultural perennial systems using GIS and remote sensing).

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TOP ABSTRACT INTRODUCTION Theory MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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