Attribution of River-Sourced Floating Plastic in the South Atlantic Ocean Using Bayesian Inference

15 Most marine plastic pollution originates on land. However, once plastic is at sea, it is 16 difficult to determine its origin. Here we present a Bayesian inference framework to com17 pute the probability that a piece of plastic found at sea came from a particular source. 18 This framework combines information about plastic emitted by rivers with a Lagrangian 19 simulation, and yields maps indicating the probability that a particle sampled somewhere 20 in the ocean originates from a particular source. We applied the framework to the South 21 Atlantic Ocean, focusing on floating river-sourced plastic. We computed the probabil22 ity as a function of the particle age, at three locations, showing how probabilities vary 23 according to the location and age. We computed the source probability of beached par24 ticles, showing that plastic found at a given latitude is most likely to come from the clos25 est source. This framework lays the basis for source attribution of marine plastic. 26 Plain Language Summary 27 Plastic is commonly found floating near the surface of the ocean but it is difficult 28 to know where it was introduced into the environment. For some large plastic items, the 29 origin can be estimated by analysing the information printed on them, but for small par30 ticles, this information is typically missing. To estimate the origin of particles at sea, we 31 built a framework that assigns a probability indicating the chance of finding a particle 32 that came from a particular source, found at a specific location of the ocean. The frame33 work uses estimates of plastic emitted by rivers, in combination with a simulation of the 34 transport of particles at the ocean surface, to compute the probability that a particle, 35 found at a particular location in the South Atlantic, comes from a certain river. Sim36 ilarly, we computed the probability that a particle of a certain age (defined as the time 37 it has been drifting in the ocean) comes from a particular river, showing that the prob38 ability changes according to the particle age. Finally, we computed the probability for 39 particles stranded at the coasts of South America and Africa, showing that plastic found 40 on beaches is most likely to come from the closest river. 41

ity as a function of the particle age, at three locations, showing how probabilities vary 23 according to the location and age. We computed the source probability of beached par-24 ticles, showing that plastic found at a given latitude is most likely to come from the clos-25 est source. This framework lays the basis for source attribution of marine plastic. 26 Plain Language Summary 27 Plastic is commonly found floating near the surface of the ocean but it is difficult 28 to know where it was introduced into the environment. For some large plastic items, the 29 origin can be estimated by analysing the information printed on them, but for small par-30 ticles, this information is typically missing. To estimate the origin of particles at sea, we 31 built a framework that assigns a probability indicating the chance of finding a particle 32 that came from a particular source, found at a specific location of the ocean. The frame-33 work uses estimates of plastic emitted by rivers, in combination with a simulation of the 34 transport of particles at the ocean surface, to compute the probability that a particle, 35 found at a particular location in the South Atlantic, comes from a certain river. Sim-36 ilarly, we computed the probability that a particle of a certain age (defined as the time 37 it has been drifting in the ocean) comes from a particular river, showing that the prob-38 ability changes according to the particle age. Finally, we computed the probability for 39 particles stranded at the coasts of South America and Africa, showing that plastic found 40 on beaches is most likely to come from the closest river. 41 1 Introduction tics, the origin can be attributed by careful analysis of labels (e.g. Lebreton et al. (2018); small and nondescript for their origin to be identified this way. Nevertheless, it is im-48 portant to assess and possibly attribute the likely source for these smaller particles too, 49 as they are among the most harmful to marine ecosystems (Koelmans et al., 2019). 50 Here, we use numerical simulations to compute the pathways of virtual plastic par-

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By tracking particles, it is in principle possible to connect any source with any location. 53 However, the multitude of possible sources very quickly makes this a computationally 54 unwieldy approach. To overcome this computational challenge, we here propose using 55 a Bayesian inference approach to attribute sources in a probabilistic sense. 56 Such a probabilistic approach has been used before to locate objects lost at sea,  To develop this probabilistic framework for attribution of likely plastic sources, we 63 here focus on plastic emitted by rivers, as rivers are considered the principal pathway 64 for mismanaged plastic waste (MPW) into the ocean (Lebreton & Andrady, 2019). We a particle at a location S loc from a specific source R i .

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Bayes' theorem offers a way of estimating p(R i |S loc ), by combining prior knowl-79 edge with new observations. In our case, Bayes' theorem is where p(R i |S loc ) is the conditional probability that we aim to estimate, p(S loc |R i ) is the 81 opposite conditional probability that can be estimated by performing a numerical sim-82 ulation (see below), p(R i ) is the probability of a particle being released at a particular 83 source and p(S loc ) is the probability of sampling a plastic particle in a specific location, 84 regardless of the source. It is important to note that p(R i |S loc ) = p(S loc |R i ). The lat-85 ter term namely indicates the probability of a plastic particle found at a location to come 86 from a specific source, and the former indicates the probability of a particle coming from 87 a specific source being at a location. Each term is commonly referred to by it's inter-88 pretation. For instance, p(R i ) is denoted as 'the prior' because it represents the prior 89 knowledge of the problem, p(S loc |R i ) is 'the likelihood', which updates our prior knowl-90 edge from the problem, p(S loc ) is the 'normalizing constant', and p(R i |S loc ) is 'the pos-91 terior'.

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In eq. (1), computing the normalizing constant p(S loc ) requires observations for all 93 plastics in the ocean regardless of their source, which means that p(S loc ) also considers 94 plastic that comes from sources that are not taken into account in the numerator of eq. (1).

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Therefore, the posterior probabilities at each S loc would not add to one in each location 96 but instead will add to a fraction that corresponds only to the sources of plastic consid-97 ered in the study. This is inconvenient when the focus is only on plastic coming from spe-98 cific sources such as riverine plastic. To overcome this inconvenience, we can constrain 99 the sum of all posterior probabilities to be equal to one where the sum is defined for the N number of sources. Then, substituting p(R i |S loc ) for and by factorizing and solving for p(S loc ) we obtain a normalizing constant that only considers the sum of all our hypotheses (i.e.

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products of prior and likelihoods). Finally, by substituting p(S loc ) in eq. (1) we get  ing the total amount of plastic released by the rivers in the South Atlantic). 119 We then clustered the rivers in 10 groups that contained the top polluting rivers 120 and their neighboring rivers. These clusters are 2°by 2°square regions centered around 121 ten locations that coincide with important cities or river estuaries. We used the result- Salvador, and Recife; and three on the river estuaries of Rio de la Plata, Itajaí and Paraibá. 128 We defined the prior distribution p(R i ) to be the fraction of plastic emitted at each associated probability defined between 0 to 1, and the sum of the 10 probabilities is 1.

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The probability associated with each source is shown in Table 1.  Other rivers 12.1 - Table 1. The proportion of the total annual plastic released to the South Atlantic and the prior probability p(Ri) of a particle being released at a specific source Ri. The "Other rivers" row indicates the proportion of plastic from rivers outside the clusters and is therefore not considered in p(Ri).
-7-manuscript submitted to JGR: Oceans specific particle. We implemented a stochastic parametrization for beaching of buoyant

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The posterior age distributions yield the probable sources of a particle of a certain The panel for sampling location A in Figure 4, located in the western part of the 246 subtropical gyre (32.37°S, 37.64°W), shows for example that a particle sampled at that 247 location with age younger than 0.4 years is very unlikely to come from any of the con-  American coast, the nearest source to the bin S lat has the highest probability, which peaks 271 at the same latitude as the source or in its vicinity. This suggests that the plastic found 272 on beaches close to a source is most likely to come from that source. Santos is the only 273 -12-manuscript submitted to JGR: Oceans exception to this trend because its probability is overshadowed by its proximity to Rio 274 de Janeiro which emissions are six times larger.

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In the right panel of Figure 5,  We introduced a Bayesian probabilistic framework that allowed us to estimate p(R i |S loc ), 289 the probability that a plastic particle, sampled at the surface of the South Atlantic Ocean, 290 came from a particular source. The framework supports different types of analyses and 291 can be used, for example, to compute spatial probabilities, compute local probability as 292 a function of particle age, or analyse the probabilities once a physical process (such as 293 beaching) alters the particles' state.