Geoscience Reference
In-Depth Information
Bowler et al. (2004) used the optical flow constraint (Horn & Schunck, 1981) approach that is
used for computer vision applications to derive the mean advection vector for tiles with
rain. Optical flow uses least squares to find the (u,v) that minimizes the two-dimensional
conservation equation
dR
R
R
R
u
v
0
(1)
dt
x
y
t
over a local neighbourhood.
Bowler et al. (2004) smoothed the field using a moving average over a (15 x 15) pixel mask
before calculating the partial derivatives using a finite difference scheme. The smoothed
image was then partitioned into 48 x 48 km
2
tiles and least squares used to estimate the
mean advection vector within the tile. The resulting vectors were then smoothed so as to
minimize
2
. Grecu and Krajewski (2000) used a similar approach over 40 x 40 km
2
tiles.
Foresti and Pozdnoukhov (2011) used optical flow to track areas with rain rates that
exceeded 10 mm/h. Essentially, this represents the application of optical flow to cell
tracking.
Germann and Zawadzki (2002) used the Variational Echo Tracking (VET) method of
Laroche and Zawadzki (1995) to derive the advection velocities. This technique partitions
the field into small tiles and then uses the conjugate gradient method to minimize a cost
function in one global minimization. The cost function includes a smoothness term. The
difference between this approach and optical flow is that optical flow applies the
smoothness constraint after the velocity field has been calculated for each tile, thereby
avoiding an expensive global minimization (Bowler et al. 2004). Ruzanski et al. (2011)
describe another approach using a linear least squares technique in the frequency domain.
Cell tracking algorithms assign a velocity to each object and this is advected with a constant
velocity during the forecast period. Such an approach is not optimal for field tracking
algorithms because it does not allow for changes in direction and speed of motion during
the forecast period. Germann and Zawadzki (2002) undertook a detailed analysis of several
advection algorithms and found that a modified semi-Lagrangian backward interpolation
scheme was optimal. Bowler et al. (2004, 2006) used the simpler semi-Lagrangian scheme
that is applied for each time step in the forecast time series. Semi-Lagrangian advection
requires a velocity at each pixel in the field and the optical flow technique does not provide
advection vectors for tiles that have no rainfall. Therefore the velocity at each pixel must
either be interpolated from the tiles with rain, or provided by a hierarchical approach that
progressively reduces the size of the tiles that are used in the analysis (e.g. Germann and
Zawadzki, 2002).
Kernel-based methods have been employed for advection by Ruzanski et al. (2011) and Fox
and Wikle (2005) using
y
(2)
y
t
1
t
where
[( ,),( ,),,,( ,)
T
(3)
y
y styst
yst
t
1
2
n
