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Implementation and Computation of Fuzzy Geographic Objects in Agriculture

1.1. Fuzzy geographic objects

A spatial object is an object with a geographical extension. Like any object, it has properties and connections with other objects, whether they are spatial or not. Furthermore, some more or less complex processes can occur on this object. Often, the processes also have a spatial dimension and occur on the geographical extension. This exact vision of the objects requires that the extension area (the polygon in 2D) of the element is known in order to be positioned on a reference ellipsoid (mathematical model of the Earth) and geolocated. However, there are many cases where the limits of this extension are imprecise (poorly known limits) or even unclear when it is a question, for example, of the probable membership of a point or a geolocated surface to an area. Some imperfections can then be modeled in a fuzzy spatial object. This type of representation makes it possible to identify and manipulate objects with extension zones affected by imperfections. The model proposed in Figure 1.1 shows a fuzzy object composed of a core (the darkest central part) and uncertain parts that appear as concentric zones. The core is the certain part of the object, therefore having a membership degree to the observed process equal to 1. Each concentric zone is associated with a membership degree <1, thus modeling the uncertainty of the phenomenon. These are a-cuts.

Figure 1.1. A representation of a fuzzy geographic object.

Let us look at an example from agriculture. Crops are subject to the pressure of bio-aggressors (weeds, diseases, pests, etc.). To limit production losses, farmers use treatment products such as herbicides, insecticides or fungicides with a sprayer. Often, pests are not distributed homogeneously in the field. They are spatially concentrated in "patches" that grow in size as the infestation progresses. Faced with this situation, the farmer has two strategies: either he applies a product in total coverage on the crop or he treats only the patches. This second strategy, which is part of precision agriculture, requires greater technical skills, but above all it requires a prior map of the patches, a sprayer equipped with a geo-localization system and a control of the application to its position. To simplify the presentation of our examples, we will hereafter concentrate only on the first strategy, that is, applications in total coverage, and in cases of arable crops (wheat, rape, corn, etc.). In this context, the operational goal is to spread the treatment products as homogeneously as possible over the plot in order to avoid concentration peaks in certain areas.

To carry out the application, the product is fragmented into droplets (spraying) by nozzles in order to treat the entire plot. To do this, the agricultural machine must make several swaths spaced by the width of the spray boom. As the nozzles have an angular sector (110° for most sprayers), the swaths at the periphery of the agricultural plot often have an uncertain extension (a-cuts), as shown in Figure 1.1. The same is true at the ends of the swaths because the opening or closing of the spray is not instantaneous. If they are parallel and accurately spaced, the swath interfaces receive similar amounts of active ingredient as the rest of the swath. If this is not the case, if the geometric shape of the plot deviates too much from that of a rectangle, if there are obstacles inside the plot, or if the booms swing, this will result in local under- or overdosing. In addition, if applications are made in windy conditions, the droplets may drift significantly, causing a very significant increase in uncertain spread. The problem of local variations of dosage within the spray core are integrated in the quantity of product used approximated by a fuzzy quantity; we focus on modeling the extension around the global treatment area first and then on the associated possible quantity.

In full coverage applications, the nozzle output must be constant to spread the active ingredient evenly. This quantity can be expressed in ng/m². The formula for calculating the volume per acre of spray mixture applied as a function of nozzle output is as follows:

where:

• V is the spray volume per acre to be applied (L/ha);
• n is the number of nozzles on the boom;
• L is the width of the boom (m);
• Vit is the speed of the device (km/h) during processing;
• Qnozzle is the flow rate of a spray nozzle (L/min).

This formula makes it possible to note that, for fixed working conditions (materialized by the coefficient k), the volume per acre spread is directly proportional to the flow rates of the nozzles.

In practice, the entire spray rarely reaches the target. Indeed, at the beginning of the season, the crops only have a few leaves; only a small part of the spraying volume reaches the crop, and the rest goes to the ground. This translates mathematically into the following relationships:

As the growing season progresses, the leaf mass increases and intercepts more and more of the spray. Correspondingly, the soil receives less and less product. The same relationships can be established for the active ingredient since it is, by dilution, directly proportional to the spray volume per hectare.

The volumes per hectare on the crop (Vcrop) or on the soil (Vsoil) and the deposition of active ingredients on the plant (Dcrop) or on the soil (Dsoil) can be represented by fuzzy numbers. As these four numbers have relations of proportionality or complementarity, we will only be interested in the continuation in the deposit of active ingredient on the crop (Dcrop)1. In practice, when the crop has a very dense leaf mass (e.g. maize), this number can take the value of 1, in the dense zone of the crop, that is, all of the active ingredient from the nozzle is intercepted by the plant and does not reach the soil.

The vision presented here is still a simplified description of the problem. Indeed, other factors can be added, such as the fact that, for treatments during hot periods, part of the deposited products can also evaporate or that, for treatments followed by an unexpected rain, the products are washed off and will run off the ground. These influences make it more difficult to determine the fuzzy spatial object.

In summary, the treatment of an agricultural plot can be represented by a fuzzy spatial object composed of a central part, the core, where the applied dose corresponds to the recommendations of use and of an uncertain spatial extension which receives a lower dose. As the dose received by the crop is a fuzzy number, this leads to the study of the combination of a fuzzy spatial object and a fuzzy number.

1.2. Evaluation of the deposit on crops: formalizing fuzzy data

The issue of treatment of an agricultural plot shows that the knowledge on the area boundaries is sometimes partial or difficult to estimate, and that the information on the dose of active ingredient received by the crop may also be subject to imprecision and uncertainty. To deal with these factors, we can go beyond the use of the classical operators of spatial analysis, which are the calculation of area (m²) and mass per unit area (ng/m²), for the calculation of the total dose of active ingredient applied or the dose of active ingredient per unit area (m²). More generally, the scalar magnitudes of the spatial object and its attribute properties (which can be manipulated in vector and raster modes) need to be reconsidered. Moreover, the associated semantic model must evolve.

The aim is then to formalize the uncertainty according to the knowledge of the context: we try to estimate the active ingredient dose at a geo-localized point Xp or a geographically limited area Xs, according to its membership to one or several treatment zones of agricultural plots.

Several problems related to the available information can be postulated and then formalized:

- The constraints related to the mechanical treatment device sometimes make the quantity of active ingredient applied to the crop an imprecise quantity and it could thus be represented by a fuzzy number. It is then necessary to evaluate the reliability of the total quantity associated with the treatment area. The formalization and implementation of the spatial tools required for this question form the framework and content of this chapter.

- Boundary information for the treated area is sometimes unclear. The treated area of the plot may be a fuzzy spatial object. The treatment depends on the presence or absence of product on the plants. These two problems are complementary and require considering the formalism of uncertainty and imprecision handled in the case of these so-called fuzzy spatial objects and associated quantities, according to the models presented in (Batton-Hubert et al. 2019). Furthermore, it is incumbent to distinguish the formalisms of uncertainty associated with the description of the objects and quantities handled that are required for the construction of new fuzzy quantities.

First of all, we will briefly recall the concepts and formalisms necessary for the representation of these magnitudes and fuzzy geographical objects, and then we will provide the elementary operators necessary for the processing of these objects within the framework of agricultural processing of parcels. The second...