Abstract

The two-way Lagrangian particle-tracking model (PTM) is proposed for specifying sources of objects drifting with random-walk processes on the sea surface. First, to determine object source candidates, modeled particles are released from the point (hereafter, “receptor”) where an observer finds the objects using a backward-in-time PTM with modeled ocean currents of which directions are reversed in sign. Second, the modeled particles are released from these source candidates in a forward-in-time PTM using ocean currents originally computed in hydrographic models. Third, the source candidates are considered to be reliable at a 5% significance level if the observed receptor is located inside the ellipse whose center is the mean position of the modeled particles at the time when the observer found the objects and whose axis length is twice the standard deviation computed using all modeled particle positions. The two-way PTM experiments are carried out in a realistic hydrographic model over the East China Sea shelf for the period from June through August 2004. Statistically significant sources are well specified close to the true source because 58%–90% of source candidates are rejected in the experiments.

1. Introduction

Recent Lagrangian particle-tracking model (PTM) experiments using high-performance computers provide us with reliable numerical solutions with respect to the fate of drifting objects such as fish eggs, spilled oil, and marine litter if the point from which the objects originate (hereafter referred to as the “source”) is given before the experiments. For instance, in a current field computed using the Ocean General Circulation Model for the Earth Simulator (Masumoto et al. 2004), Kim et al. (2007) carried out a series of PTM experiments to investigate the migration of Japanese eels (Anguilla japonica) from their spawning area in the western North Pacific. Spilled-oil behavior after tanker incidents is another example for which PTMs are available (e.g., Varlamov et al. 1999; Ohshima and Simizu, 2008).

However, we have encountered difficulties in adopting PTMs to specify drifting-object sources because object motion always includes irreversible random-walk processes in the actual ocean. Let us consider drifting objects found at an observation station (hereafter, “receptor”). To seek the object source(s), one may adopt the PTM in which modeled particles are released from all plausible source candidates in a modeled current field. Thereafter, particles reaching the receptor at the time when the observer found the objects are chosen, and their release point(s) is regarded as the object source(s) in the actual ocean. Apparently, in these forward-in-time PTMs, difficulty arises from the fact that source candidates required for the PTMs are likely to become too numerous, so experiments for seeking object sources must be computationally inefficient. In addition, it is difficult to specify the most reliable source among all sources detected using forward-in-time PTMs. In backward-in-time PTMs, particles initially placed at the receptor are carried by modeled ambient currents of which directions are reversed in sign for both horizontal current components. Thereafter, the area(s) where the particles reach within the window of time of interest is regarded as the object source(s) in the actual ocean. However, a question arising naturally is whether or not object motion, including irreversible random-walk processes, is able to be computed “backward in time.”

Even if irreversible random-walk processes are included in drifting-object motion, backward-in-time PTMs adopted on relatively short time scales may provide us with valuable information on object sources. In fact, the backward-in-time PTMs have been frequently used especially for air-quality modeling (e.g., Lin et al. 2003; Seibert and Frank 2004). These models compute particle motion backward on a time scale of less than 10 days because of the advection time typically seen for atmospheric phenomena. In the case of an oceanographic application, although the modeled ocean was quite simple with an idealized topography, Batchelder (2006) computed trajectories of particles representing meroplankton larvae in the course of one week using a backward-in-time PTM with random-walk processes. It is indeed unreasonable to include irreversible random-walk processes in modeled particle motion in the backward-in-time PTMs. However, it is likely that these processes were useful in his experiments for providing a provability or likelihood map of object sources. In addition, the reliability of the sources was examined using a forward-in-time PTM in which the particles were released again from the sources found in the backward-in-time PTM. The computed sources were considered reliable because the distances were relatively short between the receptor and positions at which the particles reach in the forward-in-time PTM.

An advantage of the present study over the previous PTM studies is the establishment of a PTM approach available for specifying “statistically significant sources” of drifting objects carried by ocean currents. In particular, ocean currents computed realistically on the Yellow Sea and East China Sea shelves (Fig. 1) are used in the present application. Another advantage is that the PTM in the present application is available for computation in the course of several months required for objects to cross the areas by ambient currents. A backward-in-time PTM is combined with a forward-in-time PTM (i.e., a two-way PTM; see section 2 for its concept) to overcome the difficulty arising from particle motion with random-walk processes.

Fig. 1.

Modeled current vectors depicted every fourth grid at 4-m depth in the study area. Stippling indicates the area with currents faster than 10 cm s−1. The current field is averaged over the period from May through August in the 2004 computation. The open boundaries, a–b (Taiwan Strait), c–d (east of Taiwan), e–f (Tokara Strait), and g–h (Tsushima/Korea Strait) are indicated by the bold broken lines with open circles at both ends. The modeled areas between 20° and 23°N and between 131° and 135°E are omitted. Also shown by thin broken lines with labels are isobaths in meters.

Fig. 1.

Modeled current vectors depicted every fourth grid at 4-m depth in the study area. Stippling indicates the area with currents faster than 10 cm s−1. The current field is averaged over the period from May through August in the 2004 computation. The open boundaries, a–b (Taiwan Strait), c–d (east of Taiwan), e–f (Tokara Strait), and g–h (Tsushima/Korea Strait) are indicated by the bold broken lines with open circles at both ends. The modeled areas between 20° and 23°N and between 131° and 135°E are omitted. Also shown by thin broken lines with labels are isobaths in meters.

2. Concept of two-way PTMs for specifying object sources

We consider a thought experiment in which objects released from a source (S0 in Fig. 2) are carried by ambient currents. All objects can reach a single receptor if object motion is governed only by ambient currents. However, given random-walk processes in object motion, multiple receptors must appear, as in Fig. 2a (Ri: i = 1–4), at a time of interest.

Fig. 2.

Schematic diagram of two-way PTMs. (a) Objects released from the true source S0 reach the four receptors R1, R2, R3, and R4. (b) An observer finding the objects at receptor R4 attempts to specify the true source, and finds the source candidates S0, S1, S2, and S3, using a backward-in-time PTM. (c) Thereafter, the source candidates except the true source S0 are rejected in a forward-in-time PTM. See the text for detail.

Fig. 2.

Schematic diagram of two-way PTMs. (a) Objects released from the true source S0 reach the four receptors R1, R2, R3, and R4. (b) An observer finding the objects at receptor R4 attempts to specify the true source, and finds the source candidates S0, S1, S2, and S3, using a backward-in-time PTM. (c) Thereafter, the source candidates except the true source S0 are rejected in a forward-in-time PTM. See the text for detail.

First, an observer finding the objects at a receptor (e.g., R4 in Fig. 2b) at an arbitrary time attempts to specify their source using a backward-in-time PTM. Using PTMs without random-walk processes, the source to which particles return may not be the true source S0 but rather other sources (e.g., S2 along the broken line in Fig. 2b), because the path from S0 to R4 is determined partly by stochastic motion. Particles released from receptor R4 cannot return to a single source in the backward-in-time PTM with random-walk processes, so multiple sources (e.g., Si: i = 1–4; hereafter, “source candidates”) must appear, as in Fig. 2b. The number of particles at each source suggests the probability of the object sources. However, the probability map may not be the true one, unless the number of objects at receptor R4 is greatest among all receptors in Fig. 2a.

Second, using a forward-in-time PTM, particles are again released from all source candidates detected in the above backward-in-time PTM to specify statistically significant object sources in the manner mentioned below. Figure 2c shows that the particles released from S1 are widely dispersed at the time when the observer finds the objects at R4. The particle distribution is likely to be elongated in a particular direction because stochastic motion is not always isotropic in the actual ocean. The major-axis length of ellipses in Fig. 2c represents twice the standard deviation (σ) of particle distances from their spatially averaged position in a certain direction (θ). This direction measured counterclockwise from the positive x axis is computed so that the largest value of σ is obtained (appendix A) as

 
formula

where N is the total number of particles released from a source candidate, and xi(yi) represents a particle distance in the x(y) direction from the average position. Likewise, twice the standard deviation is computed in the direction perpendicular to θ and is used for the minor-axis length of the ellipses; these ellipses are referred to as “2σ ellipses” for ease of reference in the following description. If the objects are assumed to disperse with a normal distribution in all directions, as is likely, we are able to consider that receptor R4 located outside the 2σ ellipse should be rejected as a member of positions of modeled particles released from S1 at the 5% significance level. In fact, we have to reject S1, not R4, in this procedure because we cannot reject the actual observation of the objects at the receptor R4. In addition, S0 should be accepted because the receptor for observed particles is located within the 2σ ellipse computed using all particles released from S0. Thus, S0 is referred to as a “statistically significant source” for convenience in the present application. In appendix B, this two-way PTM is applied to a simple flow field to investigate the distances between the true and significant sources without the spatiotemporal current variation present in the actual ocean.

3. Model descriptions

a. Hydrographic model

The numerical ocean circulation model used in the present study was presented by Chang and Isobe (2003), who gave the model details. The model boundary conditions and forcing are briefly described below. The Princeton Ocean Model (POM; Blumberg and Mellor 1987) is adopted to compute the current field covering the Yellow Sea, the East China Sea, and part of the North Pacific (Fig. 1). The grid spacing is (∼9 km) in both the zonal (x) and meridional (y) directions. The model domain is divided into 15 σ-layers vertically. We use the topographic dataset with resolution [5-minute gridded elevations/bathymetry for the world ocean (ETOPO5)] provided by the U.S. National Geophysical Data Center. First, the model is spun up using climatological datasets for the volume, temperature, and salinity transports at the open boundaries (a–b, c–d, e–f, and g–h in Fig. 1), the dataset for sea surface heat, freshwater and momentum fluxes, and the river discharge (Chang and Isobe 2003). In addition, the tidal elevation and currents of four major tidal constituents (M2, S2, K1, and O1) are added along the open boundaries using Matsumoto et al.’s (2000) dataset. The above boundary conditions and forcing vary consistently with the observed annual cycle [see Chang and Isobe (2003) for details]. Time integration is continued until the end of year 8, when year-to-year variability disappears. Chang and Isobe (2003) concluded that this model reproduces the surface currents over the Yellow and East China Seas accurately because the Changjiang-derived freshwater plume moves in the model domain as observed in the actual ocean.

In particular, forcing conditions in specific years are given so that modeled surface currents are realistic because the PTMs in the present study are developed to reproduce drifting-object behavior in the surface layer. Hence, data for wind directly driving the surface currents should be replaced with data for the specific years. Quick Scatterometer (QuikSCAT) Level 3 wind data (Jet Propulsion Laboratory 2003) from 2002 through 2004 are found for each grid using spline interpolation in space and linear interpolation in time. In the thin surface layer the currents may be influenced by river water spreading over the East China Sea, especially in the summer (Isobe et al. 2002). Changjiang discharge data (International Research and Training Center on Erosion and Sedimentation 2003, 2004, 2005) for the years specified above are used in the model after 2002. All boundary conditions, except those for wind, and Changjiang discharge are kept as in the climatological computation.

b. PTM

The procedure to determine particle locations is as follows. The particle location [X = (x, y)] at time t + Δt, where Δt is the time increment in the hydrographic model (360 s), is calculated as

 
formula

where U[=(u, υ)] and Kh are current vectors and diffusivity computed at a specific depth in the hydrographic model; i and j denote unit vectors in the x and y directions, respectively; and R represents a random number generated at each time step with an average and standard deviation of 0.0 and 1.0, respectively; otherwise, the notation is standard. In general, PTMs with random-walk processes use a random number to determine the direction of particle motion [e.g., the oil-spill model of Proctor et al. (1994)]. Instead, the present model applies a random walk to particle motion independently in the x and y directions. The diffusivity in the hydrographic model is used again in the two-way PTM, although the diffusivity in backward-in-time PTM experiments represents negative diffusion or aggregation. Nevertheless, this artificial procedure is useful in generating multiple source candidates, among which the “true” source is chosen in the following forward-in-time PTM experiments. Objects drifting near coasts over the course of one tidal cycle are unlikely to return to offshore waters in the actual ocean because they are likely to be washed ashore by tides. Hence, particles in grid boxes next to land boundaries for more than 12 h are removed from the model domain.

The current velocities are computed using the “offline” velocities saved for each time step over the period in which the hydrographic model is carried out. These current velocities at a specific depth are linearly interpolated using the values for the four horizontally neighboring grid boxes and the two vertically nearest layers [the particle tracking code can be downloaded from the POM Web site (http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/)]. As mentioned by Ådlandsvik et al. (2004), higher-order schemes (e.g., the fourth-order Runge–Kutta scheme) with accuracy greater than that of the linear interpolation are less important for the horizontal advection when random-walk processes are added to particle motion. In the present application, model particles are set to move at 4-m depth.

4. Results

The applicability of two-way PTMs in finding object sources was investigated using the modeled surface (4-m depth) currents. Two sources (A and B in Fig. 1) were chosen for experiments such that particles were carried far from the sources by relatively intense northeastward currents around the sources. Two experiments for each source (hereafter, source A and B experiments) were carried out using the same current field. On 15 June 2004 each source released 10 000 particles that were traced until the end of the August 2004, when most particles have been able to cross the East China Sea shelf (shown in Fig. 3). Thereafter, given the particle locations on 31 August (i.e., receptors), we attempted to find the two true sources using the two-way PTM.

Fig. 3.

Modeled particle numbers in each grid box on 31 Aug 2004 for the (top) source A experiment and (bottom) source B experiment. The numbers of samples are represented by circle sizes in each panel. The open (solid) circles indicate the sources (receptors) used in the two-way PTM experiments. The bold broken line represents the particle trajectory in the experiment without random-walk processes.

Fig. 3.

Modeled particle numbers in each grid box on 31 Aug 2004 for the (top) source A experiment and (bottom) source B experiment. The numbers of samples are represented by circle sizes in each panel. The open (solid) circles indicate the sources (receptors) used in the two-way PTM experiments. The bold broken line represents the particle trajectory in the experiment without random-walk processes.

As suggested above, irreversible random-walk processes make particle motion complex and source specification using backward-in-time PTMs unreliable. Particle trajectories from two sources without random-walk processes are shown until 31 August by the bold broken lines in Fig. 3. Also indicated with the size of gray circles is the number of particles within each grid box in the experiments with random-walk processes. If the particles were carried only by ambient currents, they would reach position 1 in both experiments. However, the particles are dispersed widely over the whole East China Sea because of random-walk processes.

Given observers finding the particles at several representative receptors [1–5 in Fig. 3 (top) and 1–4 in Fig. 3 (bottom)] on 31 August 2004, the experiments are carried out to specify the true sources for the particles at these receptors. The receptors inside the 2σ ellipse of all 10 000 particles are chosen in each experiment; otherwise, the two-way PTM cannot find true sources. This restriction in the procedure is reemphasized in the conclusions (section 6). Before conducting the two-way PTM experiments, we carry out backward-in-time PTM experiments without random-walk processes from four receptors in the source A experiment for the period from 31 August through 15 June (Fig. 4). Four particles would return to the true source (A) if particle motion was determined only by ambient currents. However, the particles from all receptors except receptor 5 are located far from source A on 15 June; note that the backward-in-time PTM experiment for receptor 1 is not carried out because this experiment is a reversible process.

Fig. 4.

Particle trajectories in the source A experiment using the backward-in-time PTM without random-walk processes from receptors 2, 3, 4, and 5; the digits in the box represent the receptors of each trajectory. The backward-in-time trajectories during the period from 31 Aug through 15 Jun are depicted except for the particles from receptors 3 and 4 because these particles leave the model domain before the end of the experiments. The open and solid circles indicate the source and receptors. The line type and thickness differ for different trajectories for ease of reference. Also shown by thin dotted lines are 50-, 100-, and 200-m isobaths.

Fig. 4.

Particle trajectories in the source A experiment using the backward-in-time PTM without random-walk processes from receptors 2, 3, 4, and 5; the digits in the box represent the receptors of each trajectory. The backward-in-time trajectories during the period from 31 Aug through 15 Jun are depicted except for the particles from receptors 3 and 4 because these particles leave the model domain before the end of the experiments. The open and solid circles indicate the source and receptors. The line type and thickness differ for different trajectories for ease of reference. Also shown by thin dotted lines are 50-, 100-, and 200-m isobaths.

The particle sources are placed close to the coast in the present experiments, so grid boxes where particles remain over the course of 12 h (approximately one tidal cycle of the prominent M2 tide in this area) are regarded as source candidates in the backward-in-time PTM experiments of the two-way PTM. This assumption is reasonable for seeking sources of objects such as anthropogenic marine pollutants and beach litter that are mostly released from the land, and fish larvae whose spawning grounds are located in shallow waters. To find the source candidates in the backward-in-time PTM experiments, multiple source candidates in all five neighboring grid boxes along the coasts are replaced with a single candidate where the number of particles is maximal among the multiple candidates. The particles are released midmonthly in the following forward-in-time PTM experiments, regardless of the date when the particles reach the candidate in a month. Explicitly, the present application of two-way PTMs has resolutions of about 50 km (five grid boxes) in space and one month in time. The choice of the 50-km distance is reasonable because the distance between significant and true sources exceeds 50 km even if the current velocities and diffusivity are spatiotemporally homogeneous over the East China Sea shelf (appendix B).

Figues 5 and 6 demonstrate to what extent the two-way PTM is able to find the particle sources A and B. The result for receptor 2 is chosen for the source A experiment (Fig. 5), and that for receptor 4 is chosen for the source B experiment (Fig. 6) as examples. In the source A experiment, 29 source candidates are suggested using the backward-in-time PTM, as shown by the gray circles in the top row of Fig. 5, which may lead readers to conclude that the particles were carried over a short distance during the period from July through August and that sources are located close to the receptor. The two-way PTM, however, rejects 18 (62%) sources, as shown in the bottom row of Fig. 5, suggesting that the sources are located close to the true source (the open circle in the June panel), although four sources are still accepted as significant sources close to the receptor in July. Likewise, 19 (76%) source candidates are rejected among 25 suggested by the backward-in-time PTM in the source B experiment (Fig. 6). The sources suggested by the two-way PTM are located close to the true source (open circle in the June panel) in May and June, although the sources in the Taiwan Strait are accepted in June. Ratios of the rejected sources are listed in Table 1 for all experiments.

Fig. 5.

(top) Source candidates detected in the backward-in-time PTM experiments and (bottom) statistically significant sources specified using the two-way PTM for receptor 2 in the source A experiment. The open (June) and solid (August) circles indicate the source and receptor. The sizes of gray circles in the top row represent the number of particles at each source candidate, while large (small) circles in the bottom row show significant sources specified using the two-way PTM in which the receptor is located inside the σ (2σ) ellipse computed using modeled particles. Also shown by thin dotted lines are 50-, 100-, and 200-m isobaths.

Fig. 5.

(top) Source candidates detected in the backward-in-time PTM experiments and (bottom) statistically significant sources specified using the two-way PTM for receptor 2 in the source A experiment. The open (June) and solid (August) circles indicate the source and receptor. The sizes of gray circles in the top row represent the number of particles at each source candidate, while large (small) circles in the bottom row show significant sources specified using the two-way PTM in which the receptor is located inside the σ (2σ) ellipse computed using modeled particles. Also shown by thin dotted lines are 50-, 100-, and 200-m isobaths.

Fig. 6.

As in Fig. 5, but for receptor 4 in the source B experiment.

Fig. 6.

As in Fig. 5, but for receptor 4 in the source B experiment.

Table 1.

Number of sources detected in the backward- and forward-in-time PTM processes of the two-way PTMs. See Fig. 3 for locations of sources and receptors for each experiment.

Number of sources detected in the backward- and forward-in-time PTM processes of the two-way PTMs. See Fig. 3 for locations of sources and receptors for each experiment.
Number of sources detected in the backward- and forward-in-time PTM processes of the two-way PTMs. See Fig. 3 for locations of sources and receptors for each experiment.

5. Discussion

The efficiency of two-way PTMs for specifying object sources is next demonstrated by synthesizing all experimental results mentioned above. The number of particles at each source candidate is summed every 100 km from the true source in each month (Fig. 7). In addition, the number of significant sources specified by the two-way PTM is also summed inside these 100-km-wide boxes. Pins with a solid circle represent the result in the backward-in-time PTM experiments (i.e., possible source candidates), while those with an open circle are used for the two-way PTM.

Fig. 7.

Space–time plot showing the efficiency of the two-way PTMs for the (top) source A and (bottom) source B experiments. The sum of the number of particles at each source candidate, computed every 100 km from the true source in each month, is shown by the length of pins with an open circle in these 100 km wide boxes. The length of pins with a solid circle indicates the sum of significant sources in these 100 km wide boxes. The pin length is normalized by the total number of particles or sources over the space–time plot. The box including the true source (i.e., the 0-km point on 15 Jun) is emphasized by stippling.

Fig. 7.

Space–time plot showing the efficiency of the two-way PTMs for the (top) source A and (bottom) source B experiments. The sum of the number of particles at each source candidate, computed every 100 km from the true source in each month, is shown by the length of pins with an open circle in these 100 km wide boxes. The length of pins with a solid circle indicates the sum of significant sources in these 100 km wide boxes. The pin length is normalized by the total number of particles or sources over the space–time plot. The box including the true source (i.e., the 0-km point on 15 Jun) is emphasized by stippling.

Doubtful sources are considerably rejected by the two-way PTM, as shown in Fig. 7. A total of 15 (22) boxes are chosen for source candidates by the backward-in-time PTM in the source A (B) experiment, while only 10 (7) boxes include significant sources after conducting the forward-in-time PTM experiments. The results of the backward-in-time PTM experiments suggest that the most plausible source is located in the 900–1000-km box in August in the source A experiment. Likewise, the backward-in-time PTM in the source B experiment shows that the source candidates are located over broad areas from July through August. However, the two-way PTM suggests that the long pin in the 900–1000-km box in August is erroneous in the source A experiment and that the pins in July and August are all doubtful in the source B experiment. As shown in Table 1, 58%–90% of the source candidates are rejected using the two-way PTM. As a result, the true source is suggested to exist in 0–400-km boxes in May and June. The distance between the true and significant sources is larger than that estimated in appendix B because of the spatiotemporal variability of the ocean currents over the shelf.

Both experiments suggest that an advantage of two-way PTMs over one-way (i.e., forward- or backward-in-time) PTMs is the efficient specification of object sources even in relatively weak current fields. It is noted that the number of rejected sources in the source B experiment is larger than that in the source A experiment (Fig. 7 and Table 1). Source A is located on the coast close to intense ocean currents (Fig. 1) originating from Taiwan Strait. On the other hand, as shown in Fig. 1, the ocean currents near source B are weaker than those near source A. Explicitly, the ratio of short-term fluctuations to long-term averaged currents around source B must be larger than that for source A.

6. Conclusions

In contrast to the case for objects in the air, it is difficult to trace objects drifting in the ocean along their entire paths. The location and time at which an observer finds an object is the only information available for specifying the object source. However, in general, numerical model approaches used so far have difficulty in specifying reliable sources; plausible source candidates required for forward-in-time PTMs are likely to be too numerous in space and time, while backward-in-time PTMs are unlikely to give reliable sources, especially in long-term computations because irreversible stochastic motion is included in object behavior. The present study has demonstrated that two-way PTMs are available for specifying statistically significant sources in the actual ocean. The two-way PTM experiments are carried out in a realistic hydrographic model over the East China Sea shelf for the period from June through August 2004. Statistically significant sources are well specified close to the true source because 58%–90% of source candidates are rejected in the experiments.

Nevertheless, we have to mention a restriction of two-way PTMs in concluding the present study. As mentioned above, the true source is specified when an observed receptor is located inside the 2σ ellipse computed using modeled particles released from a source candidate. This procedure requires the assumption that an object found by an observer is located inside the 2σ ellipse computed using all drifting objects originating from the actual source; otherwise, modeled particles from the true source cannot make a 2σ ellipse including the observed object. A straightforward explanation of this restriction is provided in an old Japanese poem in which a poet finding a single coconut at a beach imagines a far southern island where the coconut dropped into the ocean. Unfortunately, the presence of a single coconut was insufficient for specifying the source. Using a two-way PTM, the statistically significant island far from the beach can be specified only when the poet walking on the beach found a large number of coconuts sufficient for justifying the assumption that the beach was located inside the 2σ ellipse computed using all coconuts originating from the island.

Fig. B1. Schematic diagram of a two-way PTM experiment in a flow field with spatiotemporally homogeneous currents and diffusivity. (a) An object released from the source S0, reaches the receptor R1. The receptor is located a distance L (≫2σ) from the source. (b) The source candidate (S1) is detected in the backward-in-time PTM experiment, and (c) whether or not it is a statistically significant source is determined using the forward-in-time PTM.

Fig. B1. Schematic diagram of a two-way PTM experiment in a flow field with spatiotemporally homogeneous currents and diffusivity. (a) An object released from the source S0, reaches the receptor R1. The receptor is located a distance L (≫2σ) from the source. (b) The source candidate (S1) is detected in the backward-in-time PTM experiment, and (c) whether or not it is a statistically significant source is determined using the forward-in-time PTM.

Acknowledgments

The authors express their sincere thanks to Masahisa Kubota and two anonymous reviewers for their useful comments. This work was supported by the Global Environment Research Fund (D-071) of the Ministry of the Environment, Japan. TM is supported by the Japan Society for the Promotion of Science through Grant-in-Aid for Scientific Research (14204045 and 18340143).

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APPENDIX A

Least Squares Method for Finding the Major Axis of Scattered Particles

We consider the situation when the distribution of particles carried by ambient ocean currents gradually enlarges in a single direction. This direction can be regarded as the major-axis angle (θ) for an ellipse in which the particles are mostly included. The major-axis angle measured counterclockwise from the east is computed using a least squares method as follows.

First, the particle-position variance (σ2) along the major axis is obtained as

 
formula

where N denotes the number of particles, and xi = (xi, yi) is the position vector of the ith particle. As in the usual manner of the least squares method, differentiating Eq. (A.1) with respect to θ yields

 
formula

Replacing sinθ cosθ with ½(sin2θ), and replacing cos2θ − sin2θ with cos2θ, we obtain Eq. (2.1).

APPENDIX B

Two-Way PTM Application to a Simple Current Field

A simple current field (Fig. B1) is assumed in an investigation of how two-way PTMs are controlled by different combinations of advection and diffusion. We consider the situation in which drifting objects released from S0 are carried in a flow field [Fig. B1(a)] with spatiotemporally homogeneous current velocity (U) and diffusivity (K). The currents are directed only in the x direction in Fig. B1, so object motion in the y direction can only be caused by diffusion. In line with the definition of the diffusivity (K), the standard deviation (σ) of the drifting objects is computed as

 
formula

A thought experiment using a two-way PTM is carried out to specify the true source (S0) of the object found at the receptor (R1). The receptor should be chosen inside the 2σ ellipse (see section 2 for the definition) computed using all drifting objects originating from the true source; otherwise, the two-way PTM is unavailable for specifying the true source, as described in section 6. In this thought experiment, receptor R1 is chosen at the edge of the 2σ ellipse so that the contribution of irreversible random-walk processes is largest. The backward-in-time PTM experiment can provide the various source candidates. However, in the following forward-in-time PTM experiment, the modeled particles released only from source candidates, apart from the true source, at a distance of less than 4σ [e.g., S1 in Fig. B1(b)], can provide the 2σ ellipse including the receptor R1 [Fig. B1(c)]. Thereby, the largest computational error in the two-way PTM in this simple flow field is estimated as

 
formula

Substituting the East China Sea zonal size (∼700 km) for L, the typical current speed over this region (∼10 cm s−1; Isobe 2008) for U, and diffusivity of 105 ∼ 106 cm2 s−1 for K yields the maximum computational error ranging from 50 to 150 km in the course of about 3 (∼L/U) months.

Footnotes

Corresponding author address: Atsuhiko Isobe, Center for Marine Environmental Studies, Ehime University, 2-5 Bunkyo-cho, Matsuyama, 790-8577, Japan. Email: aisobe@sci.ehime-u.ac.jp