IEEE Robotics & Automation Magazine - September 2013 - 45

The temperature measurement of each glider at a given location (x i, y i, z i) is assumed to result from the true value of the
temperature field at this location, T (x i, y i, z i) plus an independent Gaussian noise with standard deviation
v (x i, y i, z i) . Measurements are assumed synoptic, and thus
no time dependence is considered in the analysis. The representation error associated with the gliders and Scanfish measurements and accounting for the potential departure from
synopticity is assumed to be 0.1 °C, as previously estimated
in [16] for the same region, time period, and depth range.
Under these assumptions, the probability to get the set of
measurements " d (x i, y i, z i) ,i = 1fN for a given realization of
the field T (x, y, z) is provided by the likelihood density [17]:
p ^d T (x, y, z)h \ e - i /= 1
N

(Ti - d i) 2
2v i2

(1)

,

where Ti refers to the value of the field at the ith sampling
location, Ti = T ^x i, y i, z ih . According to Bayes' rule, the posterior probability to have the field T (x, y, z) given the set of
observations " d (x i, y i, z i) ,i = 1 " N is
P ^T (x, y, z) d h =

p (d T (x, y, z)) p (T (x, y, z))
,
p (d)

(2)

where p (d) is the probability density of the observations,
a
P ^T (x, y, z)h \ e - 2 F (T) is the a priori probability of the temperature field, and a is a smoothing parameter determined
from the data. Only the thin-plate model, F (T) =
###V d 2 T (x, y, z) dxdydz,is considered in this work, as it
provides significantly better performance than membrane

models defined by F (T) = ### dT (x, y, z) 2 dxdydz [12].
V
Under this consideration, the posterior probability is
P ^T (x, y, z) d h \ e - i /= 1
N

(Ti - d i) 2 a
- F (T)
2
2v 2i

,

(3)

and the maximum a posterior (MAP) estimate is defined by
the field T MAP (x, y, z) that satisfies:

N

T MAP (x, y, z) = arg min e /
T

i=1

(Ti - d i) 2 a
+ F (T)o . (4)
2
2v 2i

MAP

Thus T (x, y, z) is the most probable field compatible
with our level of knowledge described by the smoothness
constraint and the data collected by the fleet of gliders. The
field T MAP (x, y, z) can be calculated using a variational
approach, where satellite data constrain the boundary values
T (x, y, 0) . This procedure optimizes the exploitation of
information available from remote sensors and gliders.
Equation (4) is solved using a 3-D finite element approach.
The total ocean volume under consideration V is discretized
as an unstructured mesh constituted by prismatic elements
defined by 15 nodes [18]. In the present case, a 3-D grid of
1,319 nodes and 387 prismatic elements was generated from
0- to 85-m depth in the region of interest (Figure 5). This grid
corresponds to a segmentation of the volume with ten layers
of prismatic elements of 8.5-m depth and triangular faces
with approximating 12-km edges. This guarantees a minimum vertical and horizontal resolution of 4 m and 6 km,
respectively. At this resolution, the Rossby radius of deformation (representing a fundamental horizontal scale of mesoscale eddies and of the order of 12 km in this region [19]) is
resolved. Thus, the present discretization is appropriate to
estimate the main spatial variability in the region with a limited computational demand.
Following the standard finite element methodology, the
value of the temperature field T (x, y, z) inside an eth prismatic unit of this grid is encoded by the value of the field at
each node and a set of interpolation functions:
T (x, y, z) =

15

/ N k (r, s, t) Tk,

k=1

where Tk is the temperature at the kth node of the eth prismatic element and N k (r, s, t) are the interpolation functions
expressed in a local coordinate system {r, s, t} [13]. Notice
that the local coordinates in the interpolation functions are
functions of the global coordinate system (x, y, z). Confining
(4) to the eth prismatic element and substituting (5) into (4)
results in [20], [21]

Depth

^K eij + A eij h W j = g ie,
0
-10
-20
-30
-40
-50
-60
-70
-80
8.8

9

9.2 9.4
9.6
Longitude

9.8

44.2
44
43.8
43.6
43.4
Latitude

Figure 5. The prismatic elements used to discretize the volume
of the restricted area (only the first layer is fully displayed). The
coastline is represented by the black line in the upper right corner.

(5)

(6)

with matrices given by
p 2
t
2
2
2
q 2 Ni 2 N j + 2 Ni 2 N j u
q 2x 2 2x 2
2y 2 2y 2 u
q
u
2
q 22 Ni 2 N j
2
N
e
i 2N j u
+2
dxdydz
K ij = a ### q+
2x 2y u
2z 2 2z 2
Ve q
u
q
2N i 2N j
2N i 2N j u
q+ 2 2x 2z + 2 2y 2z u
v
Ne
N r(x k) d (x k)
g ei = / i
2
vk
k=1
Ne
N i (x k) d (x k)
e
gi = /
,
(7)
v 2k
k=1
september 2013

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

45



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