Sampling for weed spatial distribution mapping need not be adaptive

Abstract

Weeds are species of interest for ecologists because they are competitors of the crop for resources but they also play an important role in maintaining biodiversity in agroecosystems. To study their spatial distribution at the field scale, only sampled observations are available due to the cost of sampling. Weeds sampling strategies are static. However, in the domain of spatial sampling, adaptive strategies have also been developed with, for some of them, an important on-line or off-line computational cost. In this article we provide answers to the following question: Are the current adaptive sampling methods efficient enough to motivate a wider use in practice when sampling a weed species at a field scale? We provide a comparison of the behaviour of 8 static strategies and 3 adaptive ones on four criteria: density class estimation, map restoration, spatial aggregation estimation, and sampling duration. From two weeds data sets, we estimated six contrasted Markov Random Field (MRF) models of weed density class spatial distribution and a model for sampling duration. The MRF models were then used to compare the strategies on a large set of simulated maps. Our main finding was that there is no clear gain in using adaptive sampling strategies rather than static ones for the three first criteria, and adaptive strategies were associated to longer sampling duration. This conclusion points out that for weed mapping, it is more important to build a good model of spatial distribution, than to propose complex adaptive sampling strategies.

Appendix

Description of the MRF models used to model the weeds maps The probability of a given map of density classes, $\mathbf {x}=(x(1),\dots ,x(n)) \in \Omega ^n$, is expressed as follow:

$$\begin{aligned} \mathbb {P}\big (x(1),\dots ,x(n)\big )\propto \exp \left[ \sum _{q=1}^n\psi _q(x(q))\ +\ \sum _{(q,p)\in E}\psi _{qp}(x(q),x(p))\right] , \end{aligned}$$

where $\psi _q : \Omega \mapsto \mathbb {R}$ and $\psi _{qp} : \Omega ^2\mapsto \mathbb {R}$ are real-valued functions called respectively first and second order potential functions. E is the set of neighbor quadrats, in other words, $(q,p)\in E$ means that quadrats number q and p share an edge. We proposed several forms of first and second order potential function, which correspond to the following map characteristics.

$\bullet$ Uniform/Non-uniform weights on density classes In a weed map, the empirical proportions of density classes can be sometimes far from uniform. This may be explained by the spatial correlation between classes but no only and in this case we will talk about a priori non uniform weights on the classes. The corresponding order one potential function is $\psi _{q}(k) = \alpha _{k} \in \mathbb {R}$ for all $k \in \Omega$. On the contrary if we assume that the proportions of classes are only the consequence of spatial correlation, we set $\psi _{q}(k) = 0$ for all $k \in \Omega$.

$\bullet$ Abrupt/Smooth spatial variation The following order two potential function leads to maps with abrupt spatial variations of density classes

$$\begin{aligned} \psi _{qp}(x(q),x(p))= \beta \mathbbm {1}_{\{x(q)=x(p)\}} \end{aligned}$$

While this one leads to smoother spatial variations

$$\begin{aligned} \psi _{qp}(x(q),x(p))= \beta \big (1-\frac{|x(q)-x(p)|}{K}\big ). \end{aligned}$$

Parameter $\beta$ is positive since we assume a positive correlation between density classes in nearby quadrats due to weeds spatial propagation by seed dispersal. The second order potential function with abrupt variation assigns the same value (zero) to pairs of neighboring quadrats in different classes, these classes being close (e.g. 0 and 1) or not (e.g. 0 and 3). The maximal value is obtained for pairs of neighboring quadrats in the same class. On the contrary, smooth spatial variation is modeled by a second order potential function whose value decreases when absolute difference between x(q) and x(p) increases.

$\bullet$ Anisotropic/Isotropic spatial repartition Due to soil tillage, one may expect a difference of spatial correlation in tillage direction and in the orthogonal orientation. This anisotropy is modeled by associating different $\beta$ parameter values in the order two potential functions depending on the orientation of the pairs of neighbour quadrats p and q.

$$\begin{aligned} \psi _{qp}(x(q),x(p))= \beta _i\mathbbm {1}_{\{(q,p)\in E_i\}}\psi '_{qp}(x(q),x(p)) \end{aligned}$$

where $i\in \{t,o\}$ and $E_t$ (resp. $E_o$) is the set of pairs of neighboring quadrats along tillage direction (resp. orthogonal to the tillage direction). $\psi '_{qp}$ can be either the order two potential function corresponding to abrupt or to smooth spatial variations. A stronger spatial correlation along the tillage direction corresponds to $\beta _t>\beta _o$. The isotropic model corresponds to $\beta _t = \beta _o = \beta$.

Each of the eight models will be named by three letters encoding their characteristics: spatial variation (S for Smooth or A for Abrupt), classes proportions (U for uniform and N for Non-uniform) and isotropy (I for Isotropic and A for Anisotropic). For example, the model SNA is the model with Smooth spatial variation, Non-uniform weights on density classes and which is Anisotropic. Its joint distribution is proportional to:

$$\begin{aligned} \exp \left[ \sum _{q=1}^n\alpha _{x(q)}\ +\ \sum _{(q,p)\in E_t}\beta _t\left( 1-\frac{|x(q)-x(p)|}{K}\right) \ +\ \sum _{(q,p)\in E_0}\beta _o\left( 1-\frac{|x(q)-x(p)|}{K}\right) \right] \end{aligned}$$