Analysis and Mapping of the Spatial Spread of African Cassava Mosaic Virus Using Geostatistics and the Kriging Technique

Lecoustre, R., Fargette, D., Fauquet, C., and de Reffye, P. 1989. Analysis and mapping of the spatial spread of African cassava mosaic virus using geostatistics and the kriging technique. Phytopathology 79: 913-920. Theories of regionalized variables and kriging were used to assess the incidence along the wind-exposed southwest field borders, disease spatial pattern of African cassava mosaic virus (ACMV). A linearlike gradients, and other less obvious features. Up to 60% of the total variance semivariogram without a range characterizes the ACMV distribution and was reconstructed from a 7% sample. Kriging was successfully applied indicates a strongly spatially dependent structure with limited random to characterize the spatial pattern of spread in cassava fields differing variation. Oriented semivariograms reveal a strong anisotropy in relation in planting date, size, arrangement, orientation, and method of sampling. to the prevailing wind direction. Further features of the semivariogram This technique was also efficient when the pattern of spread was and comparisons of semivariograms between fields and between surveys heterogenous, although more intensive surveys were then required. provide additional information and support various hypotheses on the Practical applications of geostatistics and kriging in epidemiology are pattern of spread. From a sample of limited size, kriging reproduced discussed. the main characteristics of the spatial pattern of spread, including higher Spatial patterns of disease level can provide important clues and pedology (21). to the ecology of disease (e.g., direction and distance of spread, African cassava mosaic disease is caused by a whitefly-borne importance and proximity of the sources of virus and vectors, geminivirus (2). The spatial patterns of spread of this disease vector mobility) (20). Patterns must be considered in designing have been studied intensively in the Ivory Coast (5,6) and are sampling methods and sound control measures (8,20). Hence, mainly characterized by gradients oriented in the direction of efficient methods of analysis and interpretation of spatial pattern the prevailing southwest wind. In this article, the theory of are needed to provide the greatest possible information in relation regionalized variables is used to assess the spatial patterns of to the time and effort involved. Past studies of spatial patterns the spread of African cassava mosaic virus (ACMV) in various have relied mostly on methods based on the examination of the cassava fields that differ in total area, subplot size, planting dates, mean and variance or on the frequency distribution of observed and orientation. We also describe the application of kriging to disease incidence (17). However, these methods do not incorporate reconstruct the spatial patterns of spread of ACMV within information on the location of the samples and in particular they plantings, using data from a limited number of sample points. fail to consider the degree of dependency between neighboring observations (i.e., spatial dependence). Recently, methods that MATERIALS AND METHODS recognize such dependency have been proposed (4,18,19). Geostatistics, as introduced by geologists, quantifies the spatial The theory of regionalized variables. A variable is "regionalized" dependence and has been applied successfully in agroforestry, when its values depend on its spatial position (14). A simple agronomy, and entomology (3,12,13,16). It has been proposed example (15) illustrates this concept. Two series of measurements recently to analyze the spatial spread of plant pathogens (4,12). made of a hypothetical variable at regular intervals along a row Geostatistics uses the theory of regionalized variables and only in a field gave the following numerical sequences: requires an assumption that the variance between samples is a function of the distance of separation. ("Semivariance," as defined A: 1-2-3-4-5-6-5-4-3-2-1, subsequently, is a measure of the expected squared difference B: 1-4-3-6-1-5-3-4-2-5-2. between all values separated by the same distance.) The semivariogram plots the semivariances versus distance and Sequence A has a clearly defined symmetry, whereas any illustrates the spatial variation. structure for sequence B is irregular and difficult to define. Monitoring the incidence and spread of plant virus diseases Nevertheless, the two series of 11 measurements have the same requires extensive and repeated surveys, which are timemean and variance; thus it is impossible to adequately describe consuming, expensive, and inconvenient. Furthermore, very large the detailed spatial distribution of the variable by using only these plantings may be impracticable to survey. An efficient way of two parameters. A regionalized variable arises from the mapping the spatial pattern of spread, based on a sample of limited combination of two contrasting aspects. The first is a random size but able to reconstruct the main characteristics of disease effect, as the studied variable presents irregularities in space that distribution, would be a useful tool in phytopathology. Kriging are not predictable from point to point. The second is a structural is such a technique; it makes optimal, unbiased estimates of aspect, characteristic of a regionalized phenomenon, where the regionalized variables at unsampled locations using the structural data are organized in space. Mineral content, water resource, properties of the semivariogram and the initial set of data values insect numbers, and disease incidence may be considered (9). The technique has been widely used for mapping in geology regionalized variables. Semivariograms. Geostatistics detects spatial dependence by measuring the variation of regionalized variables among samples © 1989 The American Phytopathological Society separated by the same distance. The semivariance is the average Vol. 79, No. 9, 1989 913 of the squared differences in values between pairs of samples elsewhere and is typical of ACMV spread in large cassava plantings separated by a given distance h. The analytical tool is a subject to edge effects (6). Field 2 was square, of 0.49 ha, planted semivariogram G(h), which plots the semivariance versus distance. in July 1983, and oriented with the upwind margin across the It is defined for any distance h: direction of the prevailing southwest wind. Disease incidence was recorded weekly in each plot, and diseased cassava plants were G(h) = [1/( 2 Nh)] >[F(xi + h) F(xi)] , removed after they had been recorded. Disease incidence was assessed initially in the 196 subplots of 25 plants, then recalculated where xi is the position of one sample of the pair, xi + h is in the 49 plots of 100 plants by combining four adjacent subplots. the position of another sample h units away, F(x) is the measure Field 3, of 4.0 ha, was planted in October 1984 as four blocks of a value at location xi, and Nh is the number of pairs (xi, xi of 1.0 ha, each separated by a path 3 m wide. Disease incidence + h). When the distance becomes great, the sample values may was recorded in January 1985 in plots of 100 plants, and diseased become independent of one another and then G(h) tends towards plants were left in place. a maximum value. The value a of h, corresponding to this Methodology. The first step was to analyze the experimental maximum, is called the semivariogram range and corresponds semivariogram and to fit a model. The validity of the fit was to the distance at which correlation between values taken at the evaluated by calculating the correlation coefficient between sample points is negligible. observed semivariogram values and the model predictions. The The shapes of the experimental semivariograms may be highly nonoriented semivariogram was studied first. To analyze the variable. The semivariogram immediately takes its maximum anisotropy of the variable, we also studied the semivariograms value if there is no correlation and signifies that the phenomenon oriented in four principal directions. The precision of the estimates is completely random. It is represented by a flat semivariogram: depends not only on the quality of the adjustment between the the "pure nugget" effect. This depends on microstructure and observed semivariogram and the modeled semivariogram, but also is usually superimposed on other structures. The observed on the density and distribution of the samples. Then, the second semivariogram can be adjusted to several theoretical models, step was to determine the sampling characteristics: density and including spherical, exponential, Gaussian, and linear (14). The distribution of the samples and size of the window. The third linear model does not have a plateau and may be considered step was to investigate whether the kriging technique used with to be the beginning of the spherical or exponential model (14). the established sampling procedures could reproduce the observed Its equation is G(h) = Go bh. pattern of spread within the different cassava fields. The calculated Anisotropy characterizes a regionalized variable that does not patterns of spread were then compared with those observed in have the same properties in all directions. Semivariograms can fields 1, 2, and 3 by comparing the maps of spread and by be calculated for all directions combined or for specific directions calculating the correlation coefficient between calculated and to test for anisotropy. If the structure cannot be demonstrated observed values. in one particular direction, it suggests that the structure is oriented along an axis perpendicular to that direction (14). RESULTS Kriging. If the adjusted semivariogram describing a given variable for a selected model is known, a local estimate can be Experimental and adjusted semivariograms. Figure 1 A presents made of the regionalized variable from a sample collected the experimental semivariogram for field 1, 7 mo after planting. experimentally. Kriging is the estimation method. This method The experimental semivariogram could be fitted closely to a linear is termed unbiased as, unlike other more simple methods, it plots model (r = 0.97, df 11). Such a linear relationship appears the mean and variance of the phenomenon, restores the values to be typical of ACMV spread in our experiments, as it was measured at sample points, and ensures that the estimation also observed in field 3 (r = 0.93, df25, Fig. 1 B). Semivariograms variance is minimized. The size of the "window" defines the square for field 2 were calculated 6, 7, and 8 mo after planting, area centered on the point to be estimated, the width of which corresponding to increasing levels of infection. Disease incidence maximally equals V72 times the practical range of the was calculated in plots of 100 plants. Each semivariogram could semivariogram. An estimate of the value F(xo) at any point x0 be described adequately by a linear model (Fig. IC); correlation surrounded by n points sampled, is a linear combination of coefficients were 0.98, 0.99, and 0.98 (df = 6) at 6, 7, and 8 experimental values. mo after planting, respectively. The semivariograms exhibited several characteristics. All had F(xo) = ZLiF(xi), nonzero semivariances as h tended towards zero. This is the "nugget variance" and represents unexplained or "random" where F(xi) designates, as before, the value of the variable at variance. In the fields surveyed, nugget variance was limited, which point xi, and Li is the weighted coefficient of the sample xi. The indicates that the spatial pattern of ACMV spread had a strongly Li values are calculated with the modeled semivariogram (3) so spatially dependent structure with limited random variation. that the expected variance value at point Xo is minimum and Actually, with the linear model, it is the high ratio--slope of with •L 1. It is inappropriate to use sample points from the regression line divided by nugget variance--that quantifies distances greater than the semivariogram range to estimate the precisely the spatial component of the structure of the spread value F(x0 ) at any point. In the case of the linear semivariogram, (R. Lecoustre, unpublished). In all fields, the semivariance there is no practical range. Then, the size of the window is limited increased continuously without showing a definite range. This only by the size of the smallest dimension of the field. At least indicates that the greater the separation of the samples, the greater two points are required within a window, as a single point leads the difference in disease incidence. However, the systematic to a linear estimation. deviations of the experimental points from the regression line Field surveys. Analyses were made of data from three field indicated that the semivariance was not strictly proportional to trials at the Adiopodoume Experimental Station of ORSTOM, the distance between points. These deviations further 20 km west of Abidj an, Ivory Coast. The plantings were of healthy characterized the ACMV pattern of spread. For example, cassava cuttings (cultivar CB) obtained from the Toumodi concavities observed for distances between points 40-60 m apart Experimental Station in the savannah region, 200 km north of for field 1 (Fig. IA) and 20-30 m for field 2 (Fig. 1C) were Abidjan. Disease incidence was assessed in plots of 100 plants likely to be related to border effects, which were very pronounced (arranged 10 X 10 at 1 X 1 m spacing) in fields 1 and 3 and at these distances in their respective fields. In field 3, a change in subplots of 25 (5 X 5 at I X I m spacing) in field 2. In each of slope was observed for distances between points around 100 trial, disease incidence was assessed visually. Field 1 of 1.0 ha m (Fig. 1B), which may reflect the fact that this field consisted was planted in October 1982. Disease incidence was recorded of four distinct blocks of 100 X 100 m each separated by a 3every 2 wk for 8 moo. Diseased plants were labeled and left in m wide path with high incidence on each side of this path. place. The pattern of spread in field I is described in detail Oriented semivariograms. With oriented variables, the