Bayesian Learning of Spatiotemporal Source Distribution for Beached Microplastic in the United States Gulf Coast

Bayesian Learning of Spatiotemporal Source Distribution for Beached Microplastic in the United States Gulf Coast David A. Pojunas1Avishek Chakraborty2 1Demand Side Analytics LLC, Atlanta, GA 30312 2University of Arkansas, Fayetteville, AR 72701 Abstract Over the last several decades, plastic waste has gradually accumulated while slowly degrading in terrestrial and oceanic environments. Recently, there has been an increased effort to identify the possible sources of plastic to understand how they affect vulnerable beaches. This is of particular concern in the Gulf Coast of the United States due to the presence of oil, natural gas, and plastic production. In this study, we expand upon existing Bayesian plastic attribution models and develop a rigorous statistical framework to map observed beached microplastics to their sources. Within this framework, we combine Lagrangian backtracking simulations of floating particles using nurdle beaching data with estimates of plastic input from coastlines, rivers, and fisheries. This allows us to build a spatiotemporal source-to-sink distribution for microplastic beached in the Gulf states. We infer that the main sources of microplastics found on beaches throughout the region are centered around New Orleans, Galveston Bay, Corpus Christi, M´erida, the Grijalva and Pearl Rivers, as well as from fishing activities around the Mississippi River Delta. We also find strong seasonal effects of microplastic transport in the Gulf caused by the time-varying ocean currents and tourism. Key Words: Backtracking simulations, Lagrangian ocean analysis framework, Nurdle, Virtual particles 1. Introduction Since the introduction of synthetic polymers in the 1950s, plastic waste has accumulated at increasing rates in terrestrial and oceanic environments. Global plastic production reached 390.7 million metric tons in 2021, a nearly 780% increase from 1975 (PlasticsEurope, 2015, 2022). In 2010, the estimated annual flux of plastic waste from land into the ocean was approximately 4.8-12.7 million metric tons, projected to increase nearly 10-fold by 2025 (Jambeck et al., 2015). In 2015, the global estimate of floating plastics at sea was 5.25 trillion plastic pieces weighing 268,940 metric tons (Eriksen et al., 2014) ranging in sizes from meters to nanometers. Only a small portion of the plastic waste that enters marine environments is positively buoyant and continues floating in the ocean (Lebreton and Andrady, 2019). The remaining plastic waste either sinks to the ocean floor, is consumed by marine organisms, or beached on coasts. Therefore, plastic debris poses a significant risk for marine and terrestrial ecosystems (Amobonye et al., 2021) and our health and well-being (Vethaak and Legler, 2021). 1 Over the past few decades, numerous studies have estimated the global distribution of plastics, focusing on microplastics (sizes < 5mm). Microplastics are of more significant environmental concern due to their size and variable distribution in marine and terrestrial environments making survey, collection, removal, and recycling difficult. Microplastic can come in a variety of shapes, sizes, and polymers. They are either purposely produced to create other plastic products (primary microplastics) or are the by-products of environmental degradation (secondary microplastics). Globally, microplastics account for only a small portion of floating plastic marine debris, about 13.2% of the total floating plastic in the ocean (Lebreton et al., 2019). Global estimates are highly uncertain due to the variation in sampling methods, in-situ measurement types, laboratory experiments, and data sources (Van Sebille et al., 2020), which makes comparing and combining data on microplastics a complex exercise and can result in high levels of uncertainty. It is important to note that most global studies estimating microplastic presence in the open ocean refer to secondary microplastics. Therefore, most models underestimate microplastics’ abundance (or mass) by potentially a significant margin (Boucher and Friot, 2017; Eriksen et al., 2014; Van Sebille et al., 2015). Quantifying the abundance of primary microplastics in isolation is challenging due to various sources and transport methods into marine environments. According to Boucher and Friot (2017), between 15 and 31% of all floating plastic debris in the ocean could originate from primary sources. In some countries, the abundance of microplastics outweighs larger plastic sizes ( > 5mm). Overall, understanding the distribution of plastic sizes and their sources is an ongoing research effort. While regional studies on microplastics are abundant in literature, many oceanic regions worldwide still need to be explored. In this study, we investigate the transport of plastic debris to the Gulf Coast beaches of the United States to identify the sources of microplastic in the region. The Gulf of Mexico, also known as the Gulf of America (GoA), is the world’s largest Gulf. It is connected to the Atlantic Ocean and the Caribbean Sea, with a surface area of 615,000 square miles, a width of 810 miles, and a maximum depth of 4,384 meters. The GoA is rich in natural resources: the coastal waters are full of seafood and shellfish, the offshore waters are an essential source for the fishing, oil, and natural gas industries, and the beautiful beaches attracting millions yearly provide an economic backbone for many coastal communities. The GoA is particularly subject to microplastic pollution because of the many plastic discharge locations by coastal regions through port areas, tourism activities, river systems, and industrial activities. It is also home to the most plastic manufacturers in the United States, predominately in and around Galveston Bay, Texas, with additional processing plants stretching through the beaches of Texas, Louisiana, and Florida (BeyondPlastics, 2021). Over 47% of the U.S. petroleum refining capacity is located along the Gulf and contains 51% of the U.S. natural gas processing plant capacity (U.S. Energy Information Administration, 2023). The presence of oil, natural gas, and plastic waste makes the GoA a primary ecotoxicological concern for potentially high concentrations of toxins entering the Gulf. Due to the severity of the problem, many organizations have been developed to identify and minimize the impact of plastic debris in the GoA. The Gulf of Mexico Alliance (GOMA) is a regional ocean partnership focused on enhancing the environmental and economic health of the Gulf. GOMA’s Marine Debris Cross-team Initiative’s (MDCTI) goal is to assess, prevent, remove, and eliminate marine debris and litter in the GoA by supporting community science groups such as the EPA’s Escaped Trash Assessment Protocol, the Marine Debris Tracker, the Nurdle Patrol, and NOAA’s Marine Debris Monitoring and Assessment Project tool (Grace et al., 2022). Barring the Nurdle Patrol, most of these groups focus on large debris and do not directly assess microplastics. Still, their data and work have been essential to understanding plastic waste in the GoA. In addition to these efforts, several studies have been conducted on microplastics in the 2 GoA. A recent survey by Shruti et al. (2021) studied the 14 research articles published on microplastics in the GoA. They concluded that extensive research is needed to determine the impacts of microplastics in the GoA due to non-standardized methodologies making it difficult to unify the work already done. Even with ongoing efforts, there is still a shortage of knowledge about the transport of microplastics in the GoA from source to sink. The research done to understand plastic in the GoA is unique because of the focus on beached microplastics. More specifically, the collection of surveys conducted by the Nurdle Patrol, the most extensive publicly available microplastic data, provides a spatiotemporal distribution of beached primary microplastic pellets. We propose to analyze this sink data to map beached microplastic in the GoA to its source, using a Bayesian framework in combination with Lagrangian backtracking simulations, expanding upon the approach presented in Van Duinen et al. (2022). This article is organized as follows. In Section 2, we describe the features of study region as well as the beaching data collected from Nurdle Patrol surveys in this region. A robust description of the Bayesian framework including the numerical technique for likelihood calculation is included in Section 3. Application of this method to the data from GoA is described in Section 4. Section 5 contains posterior inference on the spatial and temporal aspects of microplastic source attribution in GoA. Finally, in Section 6, we summarize the contribution of this article and identify important directions for future research. 2. Data Collection and Study Region The Nurdle Patrol has been compiling surveys on beached primary microplastic pellets (nurdles) found in North America since 2018. Each survey records the date, location (latitude and longitude), and number of nurdles found. In this study, we consider 2,102 nurdle surveys conducted within the extended GoA region inside the rectangle [97 . 98 ◦W, 76 . 46 ◦W ] × [18 . 14 ◦N, 31 . 9 ◦N ] from July 2019 to December 2021. Throughout this article, we shall refer to this region as the GoA. Figure 1 displays a general-purpose domain map of this region. Figure 1: A map of the Gulf of America (GoA). For a rigorous definition of each state, country, and oceanic region, see Figure 2. Note that some names therein are abbreviated for convenience. The data is collected using citizen science, where volunteers survey beaches by handpicking nurdles. The survey is carried out in 10-minute intervals, where each interval is done at the following water line, starting at the current water line during low tide and ending at the vegetation line. Citizen scientists are encouraged to choose a beach or multiple beaches to 3 survey once a month, providing temporal sampling at regular intervals. Once the survey is complete, it is submitted to the Nurdle Patrol website and added to their database. The data is then standardized concerning the collection method and other factors that could artificially inflate the nurdle count. See Tunnell et al. (2020) for further details regarding this data or collection method. The datasets can be accessed on their website, see Appendix A.2. This study region consists of important geographic and oceanographic divisions as displayed in the left and right panels, respectively, of Figure 2. In Section 5, we are going to explore microplastic beaching events and source attribution aggregated at the scale of these regions. We will refer to these two groups as the coastal and oceanic regions, respectively. Figure 2: (left) Coastal labels are assigned by country or state in the domain. (right) Oceanic labels are assigned using Lagrangian dynamical geographic regions defined by Miron et al. (2017). These regions were chosen manually and not determined directly from their work. For this article, we assume that the data provided by the Nurdle Patrol reasonably represents the spatiotemporal beaching pattern of all microplastic types of similar size and shape in the US Gulf Coast. The spatial distribution of the empirical beaching pattern, averaged over the study period, is displayed in Figure 3. The beaching data is preprocessed to keep the computational load feasible, see Section 4.2. Figure 3: Spatial distribution of the number of nurdles collected within a 10-30 minute period at different coastal locations, between 2019-07-01 and 2021-12-31. The actual nurdle survey locations are binned to the closest coastal locations, see Section 4.1 for more details. Finally, Figure 4 provides a comprehensive summary of the distributions of conducted surveys and collected nurdles across different calendar months and Gulf Coast states. Survey sites of beaching were located in Texas (81%), Louisiana (5%), Mississippi (9%), Alabama (2%), and Florida (3%). Since the study period is 1.5 years long, the months January to June 4 occur only once, and the months July to December occur twice. Hence, the members of latter group are expected to have taller bars in the panels representing monthwise counts for surveys or nurdles. Surveys have been conducted at higher rates in Texas throughout the year, potentially due the Nurdle Patrol’s project origin at Harte Research Institute at Texas A&M University–Corpus Christi. However, higher survey rates do not necessarily imply higher nurdle counts. For the months of August and October, Texas had a decrease in the number of nurdles found, whereas Mississippi saw a significant increase in the number of nurdles given relatively low survey rates. Another example is the comparison between Louisiana and Mississippi. In April, they had an equal number of surveys (= 11) but more nurdles were collected in Mississippi. Going to August, again they have an equal (but larger) number of surveys (= 18) but Mississippi actually deceased nurdle counts from April while Louisiana saw the huge increase in nurdle counts. Therefore, the number of nurdles do not show any systematic pattern of dependence on the number of surveys. Furthermore, we examined nurdle collection rate per survey as a function of month, shown in the bottom right panel of Figure 4. We glean that the highest density of nurdles found for a given survey are collected in Louisiana during the Fall and Winter months with an additional peak in February, suggesting seasonal periods of higher plastic debris. In Texas, we see clear seasonality in plastic debris with higher nurdle density in the Spring and Summer months and lower counts during Fall and Winter, agreeing with the peak accumulation seasons identified by Wessel et al. (2019). Overall, the dominance of nurdles found on Texas coastlines and density found in Louisiana indicate that these regions are major sinks for plastic debris (Grace et al., 2022; Tunnell et al., 2020; Wessel et al., 2019). State # % Nurdles # Mean Surveys Surveys collected Nurdles Nurdle/Survey Texas 1,711 81.399 200,977 57.557 117.462 Louisiana 104 4.948 115,107 32.965 1,106.798 Mississippi 177 8.421 30,725 8.799 173.588 Alabama 44 2.093 317 0.091 7.205 Florida 66 3.14 2,051 0.587 31.076 Total 2,102 100.0 349,177 100.0 1,436.128 Figure 4: Important summary statistics from Nurdle Patrol Surveys conducted between 201907-01 and 2021-12-31 (Tunnell et al., 2020): (top left) State-wise numbers and proportions of surveys conducted and nurdles collected, (top right) Monthand State-wise distribution of number of surveys, (bottom left) Monthand State-wise distribution of collected nurdles, (bottom right) Monthand State-wise distribution of nurdle collection rate per survey. 5 3. Methods Let A denote the spatial domain and W = [ tL,tR ] denote the time interval for this study. Let ξ denote a generic microplastic, also referred to as a particle, that gets released in the ocean at time to and returns to the beach at time tb . We shall refer to to and tb as the source and beaching times, respectively. We assume a microplastic can float in the ocean for a maximum time length ν , so tb−ν < to< tb . For this article, we only consider microplastics that beached in A during W , which implies tb∈W and to∈˜ W = [ tL−ν,tL ) ∪W . Let X ( t ) be the function that represents location of ξ at any time t∈ [ to,tb ]. Hence, X ( to ) represents the location where ξ got released in the ocean and X ( tb ) represents the location where it was beached. We shall refer to X ( to ) and X ( tb ) as source and beaching locations, respectively. The set { ( t,X ( t )) : to≤t≤tb} represents the entire spatiotemporal trajectory of ξ in the ocean. For this study, we assume that the spatial trajectory of ξlies entirely inside A. Our dataset consists of information on where and when a collection of n particles, denoted by Ξ n = {ξ1,ξ2,...,ξn} , were found on the beach across all surveys. To denote their individual beaching locations and times, we append a particle-specific subscript to the notations introduced in the previous paragraph. So, we can represent the available information on beaching events as { ( ti,b,X ( ti,b )) : i = 1 , 2 ,...,n} . The interest lies in inferring about unknown source locations and times for these particles that can be similarly represented as {(ti,o,X(ti,o)) : i= 1,2,...,n}. We introduce partitions for A and W into disjoint bins {A1,A2,...,AS} and {W0,W1,W2, ...,WT} , respectively. We refer to the spatial bins as gridcells and temporal bins as windows. If we assume d additional windows are needed to cover the time span [ tL−ν,tL ), the augmented collection {W−d,W−d+1,...W−1,W0,W1,...,WT} represents the corresponding partition of ˜ W . All subsequent probability computations are done at the level of these bins. Hence, if we want to infer about source time ti,o and location X ( ti,o ) for the i -th particle ξi , it will be in terms of which window and gridcell they lie inside. In a Bayesian paradigm, the answer will be expressed in the form of a discrete posterior probability distribution over members of {As : 1 ≤s≤S}×{Wη : −d≤η≤T} . In Section 3.4, we provide reasons behind carrying out inference at the bin-level which is a common practice in ocean microplastic modeling literature (Pierard et al., 2022; Van Duinen et al., 2022; Van Sebille et al., 2018). We develop this approach below. It follows from Bayes theorem that, PosteriorSource Event  Beaching Data ∝LikelihoodBeaching Data  Source Event ×PriorSource Event . Using available data on the beaching time ti,b and location X ( ti,b ) for the particle ξi , we easily determine the window and gridcell for its beaching event and, denote them as Wη′(i) and As′(i) , respectively. We use Bayes theorem to stochastically connect the bins for its source and beaching events as: P(ξisourced in As×Wη|ξibeached in As′(i)×Wη′(i)) =P(ξibeached in As′(i)×Wη′(i)|ξisourced in As×Wη)× P(ξisourced in As×Wη) P(ξibeached in As′(i)×Wη′(i)),1≤s≤S, −d≤η≤T(3.1) where the expression in the left hand side represents the joint posterior probability mass for the particle’s source location and time being in bins As and Wη , respectively, given the gridcell and window for its beaching location and time. Clearly, this probability would be 6 nonzero only if Wη is at most d bins to the left of Wη′(i) . The first term in the right hand side is the bin-level likelihood of available beaching data for ξi assuming the bins for its source location and time. This quantity is estimated using Lagrangian backtracking simulations discussed in Section 3.1. The numerator in the second term, P ( ξisourced in As×Wη ), is the prior probability mass for the source location and time being in bins As and Wη , respectively. Our interpretation of this quantity is described in Section 3.2. The denominator is the marginal probability, for the observed beaching window and gridcell for ξi no matter where and when it has been released in the ocean. It is the normalizing constant for the posterior distribution that is discussed in Section 3.3. 3.1 Likelihood Calculation Using Lagrangian Framework To calculate the bin-level likelihood in Equation (3.1) , we employ a Lagrangian ocean analysis framework using OceanParcels (Delandmeter and Sebille, 2019; Lange and Van Sebille, 2017) to backtrack a particle’s trajectory in time. We follow the framework presented by Van Sebille et al. (2018) and Van Duinen et al. (2022) to provide an overview of the simulation tool. The Lagrangian kinematic approach represents particles floating in the open ocean in a fluid reference frame where particles are defined as infinitesimal small fluid particles called fluid parcels. The continuous accumulation of these fluid parcels describes fluid motion. The Lagrangian approach leverages fluid descriptions (i.e., velocity fields) from Eulerian kinematics, which describes fluid motion in a reference frame fixed in space, to estimate the trajectories of virtual fluid particles. The OceanParcels simulator allows us to represent microplastics as virtual particles advected by velocity field data. The location X ( t ) of a particle at time tis updated by time stepping the surface flow velocity  v(X(t),t) as: X(t−∆t) = X(t)− t Z t−∆t  v(X(τ),τ)dτ +R√2K∆t, (3.2) where a stochastic noise term is added to represent the effects of horizontal diffusion on sub-scale grid eddies. This term includes (i) R∼N (0 , 1), (ii) the integration time step ∆ t chosen by user, and (iii) the eddy diffusivity K . In Section 4.1, we discuss how  v ( X ( t ) ,t ) is calculated for the current application using data on local ocean characteristics. Value of K is determined locally using the Smagorinsky method (Smagorinsky, 1963) as: K=Cs∆x∆ys(∂u ∂x)2+ (∂v ∂y )2+1 2(∂u ∂y +∂v ∂x)2,(3.3) where 0 < Cs< 1 is the Smagorinsky constant, a dimensionless tuning parameter chosen by the user that determines the size of the sub-scale grid eddy bounded by the cell area ∆ x ∆ y . The velocity gradients under the square root term are derived from Eulerian ocean velocity field data where the components of the velocities  v ( X ( t ) ,t ) are split into their eastward ( u ) and northward ( v ) velocities and calculated with respect to their x and y coordinates. For additional information about the interpretation and computation of the Smagorinsky method, see Van Sebille et al. (2018) and the OceanParcels documentation webpage in Appendix A.2. For each particle in Ξ n , we release one virtual particle exactly at the beaching time and location of that particle and simulate its trajectory backwards in time using Equation (3.2) . Hence, for the virtual particle corresponding to ξi , the simulation starts at time ti,b and location X ( ti,b ). Since we have assumed that a particle can not float in the ocean beyond a time span ν , this backtracking simulation runs up to time ti,b −ν . Every trajectory 7 constructed in this way is essentially a sequence of location-time pairs where two successive time points are ∆ t apart. Once trajectories for all virtual particles are simulated backwards in time as above, we use them to calculate the likelihood, at the bin-level, as outlined below. Consider all virtual particles that are released inside a gridcell As′ within a window Wη′ . As we previously assumed that d windows are needed to cover a time span of length ν , their trajectories go up to d windows preceding the initial window Wη′ . Now we find the gridcells these trajectories pass through only during the first window (i.e., the beaching window Wη′ ). For each s∈{ 1 , 2 ,...,S} , we count the number of location-time pairs from these trajectories that lie inside As during Wη′ and denote it using Ns′,η′ ( s,η′ ). Hence, the longer a trajectory spends inside a specific gridcell (during that window), it contributes more location-time pairs there, pushing the corresponding count higher. We repeat this counting step separately for each of the d preceding time windows Wη′−1,Wη′−2,...,Wη′−d . Consequently, we obtain a set of counts Ns′,η′ = {{Ns′,η′ ( s,η ) : 1 ≤s≤S, η′−d≤η≤η′} where Ns′,η′ ( s,η ) represents the number of location-time pairs inside As×Wη from the trajectories of all virtual particles released in As′×Wη′. Next, we shift to another spatiotemporal bin (by changing s′ or η′ or both), consider virtual particles released inside that bin, and repeat the above steps of computation. We keep doing this until we cover all virtual particles released in A×W to obtain the superset of counts ∪S s′=1 ∪T η′=1 Ns′,η′ . Regrouping these counts according to unique ( s,η ) pairs, we can view them as observations from multinomial distributions depending on source bins. Hence, the maximum likelihood estimate of the bin-wise probability for the beaching events given the source bin can be computed using the corresponding sample proportion as: P(Beaching Event in As′×Wη′|Source Event in As×Wη) =Ns′,η′(s,η)S X a=1 η+d X b=η Na,b(s,η).(3.4) Setting η′ = η′ ( i ) and s′ = s′ ( i ) in above equation gives us the bin-level likelihood for the beaching data of particle ξi that appears in Equation (3.1) . See Figure A.1 of the Supplementary Information for a visual illustration of likelihood computation. 3.2 Prior Specification The prior probability mass P ( ξisourced in As×Wη ), mentioned in Equation (3.1) , is assigned based on the knowledge about possible microplastic origin locations for every gridcell in A at different time windows within ˜ W . This can be done utilizing available data on the different sources that contribute microplastics to the ocean (Kaandorp et al., 2020; Van Duinen et al., 2022) such as plastic waste generated in land near coastal areas, plastic carried by the rivers that run into the ocean and human activities in the ocean including shipping, fishing and aquaculture. In this article, we consider three source types: land (L), river (R), and fishing activity (F) so that, for any generic particle ξ, P(ξsourced in As×Wη):=πL(s,η) + πR(s,η)+πF(s,η).(3.5) The quantities in the right-hand side can be interpreted as follows. Out of all microplastics that got released in the ocean within A during ˜ W from all source types, πL ( s,η ) is the a priori fraction that got released in gridcell As during the window Wη from land. Similar interpretation follows for each of πR ( s,η ) and πF ( s,η ) with change in source type. One can marginalize these quantities in different ways (across gridcells, windows) to compute prior probabilities that show how the relative prevalence of three source types varies across seasons 8 or by geography. For example, to learn a priori contribution of a source type B∈{L,R,F} in the microplastic beached somewhere in Aanytime during W, one can compute P(ξhas Source type B ) = S X s=1 T X η=−d πB(s,η).(3.6) In Section 4.2, we discuss how the prior mass is assigned across gridcells, windows and source types for the current application. 3.3 Posterior Inference The remaining term in the formula for the posterior probability in Equation (3.1) is the denominator which represents its normalizing constant and can be calculated as: P(ξibeached in As′(i)×Wη′(i)) = S X s=1 η′(i) X η=η′(i)−d P(ξibeached in As′(i)×Wη′(i)|ξisourced in As×Wη) ×P(ξisourced in As×Wη), (3.7) where the first term inside the summation is the bin-level likelihood calculated using Equation (3.4) and the second term is the prior probability given by Equation (3.5). It may be of interest to compute posterior probabilities for the three source types discussed in Section (3.2). For example, we may want to compute the posterior probability for ξi being released in the Ocean from a land source. Note that since the numerical simulation that determines the likelihood of beaching events is invariant to the source type, only the prior terms need modification. In general, for any source type B∈{L,R,F }, we can compute P(ξihas Source type B |ξibeached in As′(i)×Wη′(i)) =1 C× S X s=1 η′(i) X η=η′(i)−d P(ξibeached in As′(i)×Wη′(i)|ξisourced in As×Wη) ×πB(s,η), (3.8) where Cis the normalizing constant from Equation (3.7). If our interest lies in the source event for any generic microplastic ξ (that is beached somewhere in Aanytime during W), that can be computed as: P(ξsourced in As×Wη|Ξn) =1 n n X i=1 P(ξisourced in As×Wη|ξibeached in As′(i)×Wη′(i)).(3.9) This expression is the weighted average of probabilities from Equation (3.1) with weights determined by empirical distribution of beached particles. Since the likelihood and prior probabilities are determined at bin levels (gridcell and window), this particle-wise sum can also be grouped according to how many particles were beached at any gridcell-window combination. Note that the terms inside the above summation will be positive only for particles with beaching window index ∈{η,η + 1,...,η +d}. Inference focussing solely on source location for ξ requires marginalizing Equation (3.9) over 9 Figure 7: Map of gridcell-wise posterior source probabilities for particles from fishing, land, and river sources, aggregated over the duration of study (2019-2021). River sources that contributed less than 0.001% were excluded so as not to overcrowd the graph. in the prior. The Caribbean has been identified as the possible source region, as was also seen in prior. This suggests that regions southeast of the GoA may impact the amount of microplastic beaching on northern coastlines, further supporting a positive association between a source location and tourist destination. In Table 1, we present how overall contributions from different source types are updated in the posterior from their prior values. The regional fishing source proportions doubled from the prior estimates. The posterior suggests that fishing operations closest to shore on the nWFS side of the Mississippi Delta region may attribute more microplastic pollution than other regions. It is clear that fishing activities near the coastlines can greatly impact the amount of microplastics beaching on the Texas and Louisiana coastlines. On the other hand, river sources nearly halved and were mostly attributed to the Grijalva River near Frontera, the Pearl River near New Orleans, and the rivers that exit into Galveston Bay. We note that the Grijalva River has a 3.2% probability of being the source of microplastic for beaches where nurdles were found (see Figure 3). We note that, in the prior distribution, there is strong contribution for river input from the Grijalva River, which matches the results shown in the River Plastic Pollution Map created by the Ocean Clean Up (see Appendix A.2) for the Grijalva River. We recognize that the model did not predict significant source attribution to the Mississippi River watershed discharge contrary to findings from several authors (Mauro et al., 2017; Shruti et al., 2021; S´anchez-Hern´andez et al., 2021; Wessel et al., 2019) on significant microplastic presence in the area. This discrepancy potentially comes from the averaging done on a global scale for the river MPW input data provided by Lebreton et al. (2017) that does not necessarily match the regional studies. Therefore, more localized information could be used to inform the prior distribution about the amount of microplastic discharged from the Mississippi River. We explore how probabilities of different source regions vary as a function of calendar month of the beaching event and display the values in Figure 8. Strong seasonal variation is 16 Table 1: Prior and posterior microplastic contributions by source types Source type Land River Fishing activities Prior contribution (%) 66.6 13.4 20 Posterior contribution (%) 52.3 7.2 40.5 clear, and source probabilities follow the general pattern of microplastics being attributed to southern rivers and coastlines during the Summer months, while microplastic source contributions from the northern coasts, rivers, and fisheries are strongest the rest of the year. The spike in source probabilities during the Summer is attributed to coastlines and rivers in Mexico, which coincides with peak tourist seasons. Fishing sources in the LaTex and nWFS shelf regions peak in the months of May and November. This can be explained by the temporal variability in the prior πF ( s,η ), matching the peak fishing activity in the region (see the Global Fishing Watch Map in Appendix A.2). Figure 8: Posterior source probabilities for different coastal and Gulf regions as function of beaching months of the year. Note that source probabilities for some Gulf regions may be too small to be visible. 5.1 Regional Particle Crossings From the posterior distributions of beached nurdles, one can infer on the likely transport crossings, i.e., chance of microplastic traveling between source and beaching regions. We use the regions defined in Figure 2 to display probabilities for such interand intraregional transport in Figure 9. We find that, for any beaching region, the coastal region with highest source probability is usually the beaching region itself (except Texas, which had the highest source probability from Mexico), and the oceanic region with highest source probability usually borders the beaching region. This is consistent with the results of multiple ocean current and tracer studies and microplastic surveys done in the region (Gough et al., 2019; Miron et al., 2017; Shruti et al., 2021; Wessel et al., 2019). The regional oceanic geography is a driving factor for beaching locations in the GoA. We see this in Figure 3 where higher densities of nurdle beaching sites are found on the western Gulf coast. This beaching distribution matches the regions of attraction found by Miron et al. (2017) and can be explained by the Lagrangian dynamical geography. These spatial patterns 17 Figure 9: Posterior source probabilities for different coastal and Gulf regions as function of assigned nurdle coastal beaching regions. Note that this represents the crossing matrix for regions with the most significant source contributions. can also be explained by the large seasonal loop current responsible for many oceanic events in the GoA, most notably contributing to the intensity of hurricanes during the summer. When the Loop Current is fully extended (see left panel in Figure 10) it will eventually shed a large eddy which drifts westwards (see right panel in Figure 10) while the Loop Current retracts to the south (NOAA, 2019). Combining this with the strong seasonal westward winds and surface currents, microplastic dispersion can stay regionally locked in the northern shelves and is less likely to be dispersed to other regions in the GoA. This is even less likely given that microplastics are often found near sources. Figure 10: (left) Loop Current visible from Sea Surface Temperature observed by satellite (source: NOAA/AOML OceanViewer). (right) Track of Hurricane Katrina (2005) overlaid on top Tropical Cyclone Heat Potential (TCHP) condition on August 20, 2005. TCHP is the quantity of heat available in the ocean’s upper layer. Both images are sourced from NOAA (2019). For the GoA, we find a 33% possible source contribution of microplastics from Mexico beaching on Texas coasts. A Lagrangian transport study by Gough et al. (2019) was done for the LaTex shelf region and found that particles sourced from Texas had a 19% chance of breaking from the shelf after 28 days, and particles sourced from further south had the same chance of breaking into shelf after 28 days. This supports our results given for the Texas coastline and LaTex shelf region since the majority of all particles released for backtracking 18 originated along this coastline, providing evidence for these escape events. Therefore, it is possible for particles to escape their regional shelf and disperse across the Gulf. 6. Discussion In this article, we propose a probabilistic method for microplastic source attribution in the GoA using data on nurdle beaching events. Treating the nurdle source locations and times as unknown parameters, we use a Bayesian framework to infer about them from posterior distributions. Inference related to behavior of a generic microplastic particle can subsequently be drawn by weighted aggregation of nurdle-specific posterior probabilities based on empirical beaching distribution. This study uses input from real microplastic beaching data in a region that has not yet been studied to this degree. Our application creates a spatial map of dominant microplastic source regions for the years between 2019 and 2021. Our results show that microplastic sources in the GoA can be attributed to direct land input, fishing, and major rivers. We identified that the most likely river sources in the GoA were attributed to microplastic inputs from the Grijalva and Pearl Rivers during the Summer. Land sources had higher probabilities near larger cities and tourist destinations, with the most likely locations being New Orleans, Galveston Bay, Corpus Christi, and M´erida. Fishing activities were stronger around the Mississippi River Delta with clear temporal patterns. Overall, our results match the predictions made by Shruti et al. (2021) and could be used for upstream prevention at the identified sources. As shown in Section 5, the present framework is suitable for learning about different aspects of the source distribution such as seasonal patterns, geographic distribution and source type. However, we reiterate that the applicability of our posterior inference to any microplastic particle beached anywhere along the GoA coastline is limited by the extent to which the Nurdle Patrol surveys represent the beaching events across the region. In case these surveys overor undersample beaching events that take place in a certain part of the domain and/or during a certain time of the year, posterior learning for source attribution will be affected by that sampling bias. If a more comprehensive data on beaching events is available, that can easily be incorporated in the present framework. We like to mention one aspect of the backtracking simulation and source attribution exercise that offers scope of further research. The simulated trajectories of the virtual particles can broadly be classified into two groups: (i) ones that stay in the ocean cells for entire six months period, and (ii) ones that move through the ocean cells to reach some coastal cell. Since all points on all trajectories are considered as potential source (in terms of location as well as time), this could potentially inflate the relative frequency of ocean cells as source locations resulting in higher posterior contribution from fishing activities that we have seen in Table 1. This limitation arises from the fact, whereas the ocean velocity field data is being used in Equation (3.2) to stochastically determine the spatial trajectory, no such auxiliary information exists that can be used to control the temporal trajectory, i.e., how long the simulation should be run for any given particle. This can be addressed by incorporating information on other features of the beached nurdles such as age, size, polymer type. Age represents the time spent by a particle in the ocean from its source to beaching destination. Microplastics in the open ocean can experience different forces during their lifetime that cause them to degrade. As Van Duinen et al. (2022) notes, age of a beached microplastic particle can be estimated from its state of degradation. Currently, the likelihood calculation in our method treats all points on any simulated trajectory as equally likely. If additional information is available on estimated age distribution of a nurdle, that can be used to reweigh these points (on the temporal scale) accordingly which will in turn change the likelihood calculation and improve the uncertainty associated with source attribution. 19 Another direction for potential improvement lies in how a virtual particle’s trajectory is simulated in Section 3.1. Here, particle transport was only modeled in 2D, neglecting upswelling and sinking. Also, windage, particle degradation, and resuspension of beached particles were not explicitly modeled. Different particle advection scenarios can be considered in future, conditional on availability of more detailed hydrodynamics information and feasibility of computation. The time window used for computing the likelihood can be adjusted according to the focus of the study. Computing the likelihood with smaller time windows can lead to an increase in variability. We assessed temporal variability in terms of monthly windows since choosing weeks or days would make the inference too unstable. However, any inference related to particle age necessitates working with daily windows since particles were only tracked for six months, and most particles were beached within 30 days. To reduce uncertainty of the posterior inference on source locations and times from our model, we would take the following approaches. First, we would release multiple virtual particle per nurdle leading to a proportional increase in simulated trajectories from the current beaching locations. This would allow us to quantify uncertainty associated with possible transport routes between the source and the beaching events. Second, we would extend the temporal domain beyond three years for both the beaching data and the hydrodynamic data. A longer temporal domain would lead to more consistent inference on the available paths between the source and the beaching locations. This would imply data from more beaching locations, again increasing the sample size for the number of virtual particles released. It would also allow us to analyze longer advection times in the GoA. Lastly, if more input data was provided for the GoA that included primary microplastics directly, the incongruity between primary and secondary source locations in our model could be resolved. However, this would require extensive computation on larger datasets (hydrodynamic, prior inputs or beaching data) over the temporal domain, which does not yet exist for the GoA at the resolution used in this article. The Bayesian approach has the advantage of easily being updated if new information becomes available. In this study, the data used for prior elicitation were taken from global plastic estimates to assess primary and secondary microplastic sources based on the beaching patterns of primary plastics. Often, microplastics are considered secondary microplastics since identifying the source amounts of primary plastics from surveys and models is difficult. In fact, there are no verified estimates for how many primary microplastics enter the marine environment (Boucher and Friot, 2017). However, these estimates could be provided if survey data was done near the watersheds of pellet factories in the GoA. Furthermore, if this type of research were continued for a years-long time span, this methodology could be used to monitor plastic input and identify possible plastic pollution anomalies in the case of extreme weather events (e.g., hurricanes or floods) or manufacturing/shipping accidents. This article highlights the need for more collection and survey studies on microplastics to understand the full scope of plastic debris. The amount of plastic that enters the marine environment globally is expected to nearly triple by 2060 (Lebreton and Andrady, 2019). Therefore, regions such as the GoA should be of significant interest for researchers to study and monitor microplastics in all environments. Beyond the GoA, other worldwide databases such as the Worldwide Nurdle Hunt (Fidra, 2023) provide similar data for beached nurdles. Hopefully, our framework could be used on current worldwide nurdle estimates to gain valuable knowledge on the potential impact of primary microplastic pollution on terrestrial and marine environments. Finally, the spatial and temporal nature of microplastic source, transport and beaching events provide ample opportunity for development of sophisticated hierarchical models. For example, one may utilize space-time smoothing on the observed nurdle counts to predict 20 the beaching events across the entire GoA coastline that would in turn result in more comprehensive source attribution. Similarly, use of dynamic models can be explored for a better description of the underlying physics of plastic transport in the ocean. Managing computational load that usually comes with a fine spatiotemporal grid is another notable challenge. We expect that the present work will draw more attention to this field leading to significant methodological and computational contributions. Acknowledgements The first author gratefully acknowledges utilization of computational resources from the Arkansas High Performance Computing Center ( https://hpc.uark.edu/hpc-about/index. php ) which is funded through multiple National Science Foundation grants and the Arkansas Economic Development Commission. 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Appendix A.1 An Illustrative Example of Likelihood Calculation We present an illustrative example for the method of likelihood calculation based on simulated trajectories. For this example, we consider a subdomain A consisting of 54 , 980 gridcells within [97 . 98 ◦W, 84 ◦W ] × [22 ◦N, 31 . 9 ◦N ] for the 12 temporal windows in W = { March 2019, April 2019,...,February 2020 } . Let Ξ n be the set of n = 1 , 765 particles contained within the subdomain and d= 6 months. Now, let us consider an event where a microplastic particle is beached at a bin near Galveston Bay, denoted as B = As′ , during September 2019. We want to find the likelihood of that beaching event given that particle was sourced at the Mississippi Delta River mouth bin, denoted as M = As , during Wη = August 2019. This probability can be computed using Equation (3.4) of the main article as: P[beaching event at As′during 09/19 |source event at Asin 08/19] =Ns′,η′(s,η)/ 54980 X a=1 02/20 X b:Wb=08/19 Na,b(s,η),(A.1) where the numerator is the number of location-time pairs inside M in August 2019 from trajectories that beach on B in September 2019, and the denominator is the total number of location-time pairs inside M in August 2019 from trajectories that beach anywhere in A between August 2019 and February 2020 (which is Wη+d ). Figure A.1 provides a visual representation of the trajectories relevant to this computation. 24 Figure A.1: Visualization of trajectories relevant for likelihood calculation using Equation (A.1) . There are 146 red trajectories and 13,081 black trajectories. The red trajectories, representing simulated paths where a virtual particle sourced at M (the blue □ , not to scale) during August 2019 beaches at B (the blue × , not to scale) during September 2019, account for the numerator of the fraction. The red and black trajectories together, representing all simulated paths where a virtual particle sourced at M during August 2019 beaches anywhere in A between August 2019 and February 2020, account for the denominator. Hence, the numeric value of the expression in Equation (A.1) will be 146/(13081+146). A.2 Data and Code Availability • All code for running and post-processing the GoA simulations are available at https:// github.com/davidavzP/BayesianLearning_Microplastics_GoM . Following Python packages have been utilized: xarray (Hoyer and Hamman, 2017), pandas (McKinney, 2010), scipy (Virtanen et al., 2020), matplotlib (Hunter, 2007) and cartopy (Met Office, UK, 2015). • The code for Parcels is available at https://www.github.com/OceanParcels/parcels . The Ocean-Parcels documentation can be accessed at https://docs.oceanparcels. org/en/latest/examples/tutorial_diffusion.html. • The Nurdle Patrol data are provided by Mission-Aransas National Estuarine Research Reserve, University of Texas Marine Science Institute (Tunnell et al., 2020), and can be downloaded at https://nurdlepatrol.org/map. • The HCOM + NCODA Gulf of Mexico Analysis data are provided by the HYCOM Consortium and can be downloaded at https://www.hycom.org/data/gomu0pt04/ expt-90pt1m000. • Global Wave Reanalysis WAVERYS data are provided by the Copernicus Marine Environment Monitoring Service (CMEMS) and can be downloaded at https://data. marine.copernicus.eu/product/GLOBAL_MULTIYEAR_WAV_001_032/description. • The data for fishing activities are provided by the Global Fishing Watch and can be found at https://globalfishingwatch.org/map. • The River Plastic Emissions to the World’s Oceans Map by the Ocean Clean Up (Meijer et al., 2021) can be found at https://theoceancleanup.com/sources/. 25