1
Supplementary Material for the article:
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3
Route optimisation and solving Zermelo’s navigation problem during long distance
4
migration in cross flows
5
Graeme C. Hays, Asbjørn Christensen, Sabrina Fossette, Gail Schofield, Julian Talbot, Patrizio
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Mariani
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Ecology Letters
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9
Track analyses: We assumed that turtle motion is regulated by two components: (i)
10
swimming behaviour, and (ii) local ocean circulation. By combining individual swimming
11
velocity vector (w) and ocean surface velocity vector (u), we calculated the total velocity of
12
swimming turtles as:
13
v(x, t) = w + u
14
(Eq. S1)
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16
17
The observed longitude and latitude (x, y) values along the trajectories of a turtle,
x(t) , were used to calculate the velocity vector v using the equation:
18
v(x, t) =
19
R(x, t)
Dt
(Eq. S2)
20
21
where R is the vector for the distance covered between two consecutive points in time,
22
Dt = 4h .
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1
We calculated u(x, t) through the linear interpolation of the tracks superimposed on
2
sea surface daily data at a spatial resolution of 1/16o using the Mediterranean Ocean
3
Forecasting System database (Pinardi et al. 2003; Tonani et al. 2009).
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The swimming vector was then calculated as w(x, t) = v - u . Speed [ w º w ] and
5
direction [ q ] were used to analyse turtle swimming behaviour. To determine the relative
6
importance of local current circulation and turtle swimming behaviour in the observed path,
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we projected u in the direction of w, using the equation:
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9
Pu,w =
u×w
w
2
w (Eq. S3)
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11
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The above current speed is as defined negative if the angle uÙw > p .
The vectors for the observed trajectories (v) were mainly directed southwest (Fig.
13
S1); however, other components following a north-northeast-northwest direction were also
14
present. The surface ocean currents interpolated on the tracks (u) were uniformly
15
distributed in all the directions while w peaked at around -110 oN (Fig. S1).
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Turtle swimming speed has a Gaussian distribution with median w = 0.46 ms-1 , and
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reached maximum values of w =1.0 ms-1 (Fig. S2). The projected ocean velocities ( Pu,w )
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appeared to follow a Poisson distribution, with a zero mean and a maximum magnitude of
19
0.2 m s-1 both with and against the swimming velocity (Fig. S2).
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Simulations: We simulated individual trajectories using the modelled local surface ocean
22
circulation conditions and simple models for the behaviour of adult turtles.
23
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For a single numerical particle, we integrated Eq. (1) with a fixed time step Dt = 4h
using a Runge Kutta second order method.
2
1
Turtles are assumed to swim at a constant speed, w, toward the direction of an
2
assigned final destination, Xend º ( xend , yend , tend ) , matching the final position of the observed
3
trajectories, i.e. Xend Î x(t). Specifically, for each observed x(t) , we release numerical
4
particles in (xstart , ystart ,tstart ) Î x(t) , assuming a constant swimming speed and final target
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Xend . Then the trajectory x S (t)was simulated until t = tend .
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Computation of the travel distances: To assess the similarity between the observed track,
8
x(t) , and the simulated trajectories, x S (t), we calculated a set of distance metrics: (1)
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Euclidean distance x ts - x t , (2) Hausdorff distance, and (3) straightness index distance. In
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addition, we calculated (4) the theoretical migration time for each model (in days), and
11
compared those to the observed values.
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All of the calculated distance metrics above, (1 – 3), were obtained after matching
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the migration time for the observed and simulated tracks, using linear interpolation. For the
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values of metric (3), we first calculated the straightness index (D/L, where D is the distance
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between the final and starting point Xend - Xstart and L is the total path length travelled) for
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x(t) and x S (t), then we computed the absolute difference between these two values.
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Finally, the distances of metrics (1) and (2) were normalised by the total path length L.
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Those geometric distance metrics are calculated for all the observed tracks (Fig. S3 a – h)
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and the behavioural models considered in the study.
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The values of (iv) are only showed for the simulations in which the behavioural
21
model reached the final destination, i.e. when the turtle headed directly toward target and
22
the Zermelo optimisation.
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All of the algorithms for the analyses were developed in MATLAB© R2012.
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3
1
2
Fig. S1. Distribution of the direction, q , for the three motion components: (i) ocean surface
3
velocity (u), and (ii) observed velocity (v) and (iii) turtle swimming velocity (w = v-u).
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Fig. S2.Distribution of swimming speed (w) and surface velocity along the direction that
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turtles swam ( Pu,w ). Note that the speed Pu,w has both positive and negative values,
9
which indicate whether the ocean surface velocity is with or against the swimming velocity.
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Distance score
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2
Fig. S3.Non-dimensional distance score for the eight observed turtle migratory tracks (a – h)
3
using the four different behavioural models: (i) heading to the goal, (ii) single heading (iii),
4
optimal Zermelo route, (iv) course correction. The three metrics shown are: distance in
5
straightness index (black bar), Euclidean average distance (grey bar), Hausdorff distance
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(white bar), with the latter two normalised by the observed total length of the track.
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1
2
Optimal swimming behaviour: A complementary approach for simulating behavioural rules
3
to explain the observed animal migration paths is the use of optimal behaviour models.
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Such models shift the focus from the behavioural response mechanisms to the pay-off. The
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major weakness of optimal behaviour models is the assumptions of (i) perfect knowledge
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about the ambient conditions that influence the pay-off and (ii) perfectly adapted response
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patterns to the behavioural problem. The major advantage of such models is that one avoids
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the overt parameterisation of the behavioural rules. In this way, optimal behaviour models
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indicate the upper limits of the pay-offs that might be achieved by adapting behaviour.
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In the present study, the behavioural problem was formulated as a horizontal (i.e.
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2D) travel problem between an observed initial point 𝒙0 and an observed final point 𝒙𝐸 . The
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subsequent sections present the mathematical and computational details of the
13
implementation of the models. In an optimal behaviour model the turtle chooses a travel
14
path between 𝒙0 and 𝐱𝐄 that minimises a hypothesised cost functional. A functional is the
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mathematical term for a function that takes a function as argument - in this case the
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swimming path x(t). The cost functional is then the price the turtle pays when choosing x(t).
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Examples of cost functionals include swimming time (corresponding to the original Zermelo
18
problem) or swimming energy expenditure. The behavioural scope of the turtle is to select
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its swimming velocity vector (both speed and direction), 𝐰(𝑡), as a function of time along
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the swimming path. Swimming velocity is understood with respect to water, so that the
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effective velocity v in the earth coordinate system is v = u + w, where u is the local water
22
current vector. The challenge for the turtle is to use prevailing hydrographic current
23
patterns and variability optimally, by swimming along the current where possible and
24
avoiding swimming against the current, if not necessary. The minimisation of the cost
25
function is formulated in the time domain, and involves solving the corresponding Euler-
26
Lagrange equations, which are a set of coupled ordinary differential equations for 𝐰(𝑡). The
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initial conditions of 𝐰(𝑡) must be reverse engineered so that the path ends at 𝒙𝑬 , given that
28
it starts at 𝒙0 . If only the swimming direction is variable or currents u are stationary,
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additional mathematical transformations are possible to reduce the problem (Comincioli,
30
2004). This approach is technically challenging with time-varying currents, especially when
31
the behavioural scope has more than one dimension (e.g., swimming speed and direction).
6
1
Alternatively, the problem may be formulated in the space domain. In this case, the
2
problem is simplified by local optimality, which eliminates 𝐰(𝑡) from the problem (this is
3
elaborated on later in this text). In this approach, the optimal swimming path is obtained by
4
minimising the general functional:
5
6
𝑑𝒙
(Eq. S4)
𝐸[𝑥] = ∫ 𝜀 (𝒙(𝑡), , 𝑡) 𝑑𝑥
𝑑𝑡
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directly over all (valid) paths between 𝒙0 and 𝒙𝐸 . Here, E is the overall cost functional and ε
9
is the corresponding local cost function (with unit cost per length) that depends on the
𝑑𝒙
10
path𝒙(𝑡), which is parameterised by time t and the effective velocity vector
11
vector with respect to a stationary coordinate system). The local cost function ε has clearly
12
delineated time dependence, because water currents vary with time. Computationally, this
13
approach involves relaxing a guessed initial path between 𝒙0 and 𝒙𝐸 using an iterative
14
minimisation technique, in which the swim path is used as an independent variable.
𝑑𝑡
(i.e. velocity
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Examples of cost functions: When the local cost function is:
17
1
𝑣
=
1
(Eq. S5)
18
𝜀=
19
we resolve the Zermelo problem, where v is the effective velocity in the earth coordinate
20
system. Informally, since v=dx/dt, this means the integrand in Eq. S4 εdx = dx/v = dt is the
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time used to travel a small piece dx of the travel path, so that the integral Eq. S4 sums up
22
the total travel time. To have a well-posed problem, the swimming speed w must be a
23
constant, so the turtle only has the option of swimming direction, otherwise minimisation
24
obviously leads to infinitely fast swimming speeds to achieve the minimal travel time. The
25
construction of the effective velocity v is indicated in Fig. S4a, where the swimming
26
direction is constrained by the preferred swimming direction t in the earth coordinate
27
system (corresponding to the local path tangent vector). Therefore, the positive solution w
𝑑𝒙
| |
𝑑𝑡
7
1
shown in Figure S4a represents the locally optimal swimming direction. In this way, by
2
applying local optimality, the cost functional does not depend on 𝐰(𝑡), but only the path
3
x(t). Figure S4b indicates the minimal cruising speed required to (i) cancel the transversal
4
current component, and (ii) acquire a positive definite effective velocity v along the path
5
tangent. This requirement is met by:
6
7
𝑤 ≥ √𝑢2 − 𝑢2𝑡
𝑤 ≥ |𝑢|
𝑢𝑡 > 0
𝑢𝑡 < 0
(Eq. S6)
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9
If this inequality is not satisfied, the cruising speed w is too small to allow the turtle to swim
10
in all directions locally, because the current is too strong. In this instance, a forbidden cone
11
of directions is present locally, which constrains how the travel paths may cross this area of
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strong currents. Fig. S4c illustrates the construction of the effective velocity v when an
13
energetic cost function ε(w) with a positive slope is applied. This would normally be the
14
case, because travel cost is expected to increase with swimming speed, e.g. the water drag
15
force. In this case, to have a well-posed problem, a certain minimal effective velocity 𝑣𝑚𝑖𝑛 is
16
required; otherwise, the minimisation of travel costs would obviously lead to an infinitely
17
slow swimming speed at path segments that oppose the current, which is biologically
18
irrelevant. In the case of the energetic cost functions, there is no imposed upper limit on
19
swimming speed w (only a high cost). In this case, there are no forbidden cones, as in the
20
Zermelo problem, with a constant cruising speed. Fig. S4d indicates the minimal cruising
21
speeds required to compensate the transversal current component, and to acquire a
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positive definite effective velocity 𝑣𝑚𝑖𝑛 along the path tangent. If a biological limit is imposed
23
on the swimming speed, this would be represented as a diverging cost at the limiting
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swimming speed. In this case, there would also be forbidden cones in areas of very strong
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opposing currents. In this work, we only focus on the Zermelo problem, Eq. S5, when
26
calculating the optimal travel path.
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1
2
3
Fig. S4: Construction of the local optimal effective velocity v and minimal swimming velocity
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w to realise swimming in the preferred direction t, when the current vector is u for the
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Zermelo problem (a + b), where the turtle controls swimming direction, and the energetic
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problem (c + d), where the turtle controls both swimming direction and speed. (a) The
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decision circle for cruising speed w, with the arrow indicating optimal swimming direction.
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(b) The minimal cruising speed w required to compensate the transversal current
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component, and obtain the positive effective velocity v for swimming direction. (c) Local
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optimal effective velocity when there is a required minimal effective velocity 𝑣𝑚𝑖𝑛 . (d)
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Minimal swimming speed w required to compensate the transversal current component,
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and obtain the positive effective velocity 𝑣𝑚𝑖𝑛 for swimming direction.
13
14
Minimising cost functions: When using numerics, the path integral Eq. S4 is evaluated by
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sampling the travel path with nodes {xi}. In this study, around 20 nodes per travel path
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provided sufficiently meaningful representation. Hence, the functional derivative of Eq. S4
9
1
was mapped onto a finite dimensional gradient with respect to the nodal points {xi}. For the
2
Zermelo problem, the travel time becomes:
3
𝐸 = ∫ 𝑑𝑡 ~ ∑𝑀−2
𝑖=0 𝜏(𝒙𝒊 , 𝒙𝑖+1 )
(Eq. S7)
4
5
where M is the number of nodes in the path and τ the time to travel between points xi and
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xi+1. When the path sampling is sufficiently dense, it is assumed that the effective velocity v
7
varies linearly between 𝒙𝑖 and 𝒙𝑖+1 . Under this assumption we can analytically calculate τ:
8
9
| 𝒙𝒊+𝟏 − 𝒙𝒊 | 𝑑𝑠
𝑣(𝑠)
𝜏(𝒙𝑖 , 𝒙𝑖+1 ) = ∫0
=
|𝒙𝒊+𝟏 − 𝒙𝒊 |
𝑣
ln( 𝑣𝑖+1 )
𝑣𝑖+1 − 𝑣𝑖
𝑖
(Eq. S8)
10
𝑑𝒖
11
The spatial derivatives of τ also involve the current shear components
12
for v, constructed as indicated in Fig S4(a+c). A consistent choice for time points 𝑡𝑖 where the
13
turtle arrives to each node 𝑥𝑖 is:
𝑑𝒙
via the chain rule
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15
𝑡𝑖 = 𝑡0 + ∑𝑖−1
𝑗=0 𝜏(𝒙𝑗 , 𝒙𝑗+1 )
(S9)
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17
which is used to interpolate the time dependent current vectors 𝒖𝒊 = 𝒖(𝒙𝒊 , 𝑡𝑖 ) as elaborated
18
below. This update of the set of times {ti} also applies to the energetic cost functionals. In
19
the model implementations, we applied lazy update of the set of times {ti} above. This
20
means that 𝑡𝑖 is considered to be parametric when taking the derivative of τ in Eq. S9. Then,
21
the update of {ti} Eq. S9 is only performed after a minimisation step. This approach localises
22
the cost function, and simplifies the complexity and load of the gradient evaluation, because
23
the strict enforcement of Eq. S9 implies that a variation in 𝒙𝒊 affects all points on the path
24
after i in Eq. S9. The self- consistency of the set of times {ti} is obtained through iterative
25
updates using Eq.S9 during the minimisation of the cost functional E, so that it converges
10
1
with the proper minimum. This approach is similar to the Car and Parrinello scheme (1985),
2
which simplified molecular dynamics in computational materials science.
3
To evaluate Eq. S7–S9, a smooth current field interpolation was needed for arbitrary
4
time and space points (x,t) in the Mediterranean. The current fields obtained from the
5
circulation model described in Methods are stored as gridded data 𝒖𝑖𝑗𝑘 = 𝒖(𝒙𝑖𝑗 , 𝑡𝑘 ) on a
6
regular longitude/latitude grid 𝒙𝒊𝒋 at regular time intervals 𝑡𝑘 . Bilinear spatial interpolation
7
of 𝑢𝑖𝑗𝑘 at arbitrary positions x was found to cause kinks in the gradients of the cost
8
functional (via current shear terms 𝑑𝒙), which hindered the minimisation of the cost
9
functional close to a minimum. Therefore, each time frame 𝑡𝑘 , 𝒖𝑖𝑗𝑘 was Fourier transformed
𝑑𝒖
10
spatially into 𝑼𝑖𝑗𝑘 , and the corresponding wave representation was used to generate a
11
smooth spatial representation:
12
13
𝒖(𝒙, 𝑡𝑘 ) = ∑𝑖𝑗 𝑼𝑖𝑗𝑘 𝑒𝑖𝒌𝒊𝒋 𝒙
(Eq. S10)
14
15
where {𝒌𝒊𝒋 } is the set of wave vectors associated with {𝒙𝒊𝒋 } by the Fourier transformation.
16
Finally, a piece-wise linear interpolation in time was applied between adjacent space
17
interpolations, to generate time interpolation 𝒖(𝒙, 𝑡) for arbitrary time and space points
18
(𝒙, 𝑡), 𝑡𝑘 < 𝑡 < 𝑡𝑘+1 , in the Mediterranean:
19
𝒖(𝒙, 𝑡) = 𝒖(𝒙, 𝑡𝑘 ) +
𝑡− 𝑡𝑘
((𝒖(𝒙, 𝑡𝑘+1 ) − 𝒖(𝒙, 𝑡𝑘 ))
𝑡𝑘+1 −𝑡𝑘
(Eq. S11)
20
21
A pertinent problem generated by minimising the discretely sampled functional Eq. S7 is
22
that, under certain conditions, the value of the cost functional may decrease slightly due to
23
coalescence of certain sampling points along the travel path; thus, lowering the path
24
resolution. This is a problem that arises at the end phase of the minimisation process, where
25
weak modes are also relaxed. To overcome this problem, three parallel stabilizing measures
26
were implemented. First, close-lying points were removed from the initial path; specifically,
27
if adjacent points were separated by less than 25 km, the latter point was removed from the
11
1
initial path (unless it was the observed end point). Second, gradients of the cost functional
2
∇𝐸 supplied to the minimisation algorithm were projected onto path-normals (i.e. unit
3
vectors perpendicular to the swimming path):
4
5
𝛁𝑬 ← 𝐍 𝑫(𝑵𝒕 𝛁𝑬)
(Eq. S12)
6
7
where N is a matrix with path-normals in its columns, 𝛁𝑬 is a matrix with nodal gradients in
8
its columns and D() is a matrix function that sets the off-diagonal elements to zero. This
9
equation directly prevents sampling point coalescence. Third, to avoid certain conditions
10
causing zig-zag features in the final travel path, the highest modes were removed by Fourier
11
filtering. Here, the reference is a linear path L between 𝒙0 and 𝒙𝐸 . If the matrix B contains
12
included modes in its columns and c is a vector describing how much each mode is excited,
13
the transformation between the Cartesian path and the mode representation would be:
14
15
𝑿
𝒄
=
=
𝟏
𝑩𝒕 𝑩
𝑳 + 𝑩𝒄
𝑩𝒕 (𝑿 − 𝑳)
(Eq. S13)
16
17
and gradients would be transformed using:
18
19
𝛁𝒄 𝑬 = 𝑩𝒕 𝛁𝑬
(Eq. S14)
20
21
Minimisation is then performed in excitation space c. Consequently, the first measure
22
(pruning of initial path) would be applied once, whereas the second (normal projection of
23
gradients) and third (mode transformation) measures would be applied in this order for
24
each minimisation step.
12
Many observed turtle tracks pass close to land; hence, to avoid artificial land
1
2
crossings during minimisation runs, a predictor-corrector scheme was applied after each
3
minimisation step, to retain the path in the sea, if it was displaced by the minimization
4
algorithm onto land. The geography of the Mediterranean region was represented by a land
5
mask:
6
0
𝑑𝑖𝑗 = {
1
𝑖𝑓 𝒙𝑖𝑗 𝑖𝑠 𝑤𝑒𝑡
𝑖𝑓 𝒙𝑖𝑗 𝑖𝑠 𝑑𝑟𝑦
}
(Eq. S15)
7
8
A continuous topography function d(x) was constructed through the bilinear interpolation of
9
𝑑𝑖𝑗 on the grid 𝒙𝒊𝒋 . The land areas were then defined by 𝑑(𝒙) > 1⁄2. This coastal
10
representation provides a softer coastline for the given spatial resolution 𝒙𝑖𝑗 compared to
11
other frequently used procedures, like looking on whether the closest grid point xij is wet or
12
dry.
13
14
Several standard algorithms are available to minimise functions over a continuous
15
space x. There are, however, a few features associated with the present problem that
16
complicate the application of standard minimisers. First, these features assume an
17
unrestricted (or at least convex/square) domain of the function; this assumption is not
18
satisfied, because many potential paths connecting observed end points are invalid, as the
19
paths artificially pass over land. Second, for the travel time functional, there are forbidden
20
cones in certain areas of strong currents for realistic cruising speeds. These two issues are
21
not easy to remove by suitable transformations to unrestricted domains. Third, the cost
22
functional is not strictly a unique functional of paths x, due to the lazy evaluation of the
23
current time evaluation points (Eq. S9), but depends slightly on search history during the
24
minimisation, even though the cost functional approximation converges with the proper
25
value and path at the end of the minimisation. This minor non-uniqueness of Eq. S4, due to
26
lazy time evaluation, is sufficient to confuse many standard algorithms based on iterative
27
line searches. Therefore, we applied a Newtonian annealing algorithm QuickMin (Stoltze
28
1997), which only depends on local conditions around the focal path; thus, it is insensitive to
13
1
the non-uniqueness issue. Furthermore, it is relatively easy to capture the two technical
2
issues (land crossing/forbidden swim directions and non-uniqueness) in this algorithm by
3
using a predictor-corrector amendment that ensures the path remains in the valid domain
4
following initialisation.
5
The landscape of 𝐸[𝒙] is generally governed by several local minima. For a given
6
initial path, 𝒙0 , the Quickmin algorithm (like other standard algorithms) only localises a
7
proximal minimum to 𝒙0 . To ensure that the global minimum was localized, an ensemble of
8
initial paths {𝒙0 } was produced by generating a random perturbation to the straight line
9
between 𝒙𝟎 and 𝒙𝑬 . In the presented runs, an ensemble size of 50 initial paths was applied,
10
which seemed sufficient for the travel time cost functional.
The presented cost functional minimisation is implemented as a class library in
11
12
Python. Automatic derivatives were found to be too slow numerically; hence, gradients
13
were coded by hand. Numerical consistency between the functional and (lazy) gradients was
14
tested.
15
16
17
References
18
1. Car, R., Parrinello, M. (1985). Unified approach for molecular dynamics and density-
19
20
functional theory. Phys. Rev. Lett. 55 (22), 2471-2474
2. Pinardi, N., Allen, I., Demirov, E., De Mey, P., Korres, G., Lascaratos, A., et al.(2003). The
21
Mediterranean ocean forecasting system: first phase of implementation (1998-2001).
22
Ann. Geophys.21(1), 3-20.
23
24
3. Comincioli, V (2004): Biomatematica (interazioni tra le. Scienze della Vita e la
Matematica),Universita' degli Studi di Pavia,
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1
4. Tonani, M., Pinardi, N., Fratianni, C., Pistoia, J., Dobricic, S., Pensieri, S., et al.(2009).
2
Mediterranean forecasting system: forecast and analysis assessment through skill
3
scores. Ocean Sci. 5, 649-660.
4
5
5. Stoltze, P. (1997). Simulation Methods in Atomic-scale Materials Physics. Polyteknisk
Forlag, Lyngby, Denmark.
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