Geoscience Reference
In-Depth Information
later study by Dong et al. (
2007
) simulated SWE in North America with and without the
assimilation of the SMMR SWE product. Assimilation of the SMMR product improved the
analysis where SWE \ 100 mm, provided the SMMR product was quality controlled.
However, this approach did not account for the changes in snow microstructure, which
affect the scattering, as the SMMR-based SWE product is based on a variant of the Chang
Algorithm. A more comprehensive approach is detailed in Durand and Margulis (
2006
),
who describe an Ensemble Kalman Filter (EnKF) approach of assimilating microwave
brightness temperatures.
The Kalman Filter approach consists of two steps to produce an analysis of the variables
of interest, which will be some vector x
a
whose components represent snow properties such
as the density and grain size of each snow layer. In the first step, the analysis x
k
1
from the
previous time step t
k
1
is propagated using a model M to produce a forecast x
f
k
:
x
f
t
ðÞ¼
M
k
1
x
a
½
ð
Þ
ð
4
Þ
t
k
1
This forecast is then updated with reference to observations:
x
a
t
ðÞ¼
x
f
t
ðÞþ
K
k
y
k
H
k
x
f
t
ðÞ
ð
5
Þ
where y
k
is the observation vector and H
k
is an operator, which converts the state vector
into an equivalent observation. In the case of snow remote sensing, it is some model of
snow's radiative transfer that converts the known snow properties from the state vector into
a vector of observable brightness temperatures or some combination thereof. K
k
is the
Kalman Gain, which acts as the weighting function and depends on the error covariances of
the forecast P
f
and the observations R
k
:
1
: ð
6
Þ
Here H
k
is the linearised approximation of the observation function H
k
. It can be seen
that as observational error decreases, the Kalman gain increases and greater weight is
placed on the observations. The forecast error covariance P
f
ð
t
k
Þ
consists of the model error
covariance Q
k
1
and the error covariance introduced due to errors in the previous step's
analysis, P
a
ð
t
k
1
Þ
:
K
k
¼
P
f
t
ðÞ
H
k
H
k
P
f
t
ðÞ
H
k
þ
R
k
P
f
t
ðÞ¼
M
k
1
P
a
Þ
M
k
1
þ
Q
k
1
ð
t
k
1
ð
7
Þ
where M
k
1
is the linearised approximation of the forecast operator M. Estimating this
component of the error covariance can be enormously computationally expensive, leading
to the attraction of the EnKF where a model ensemble allows the generation of statistics to
approximate M
k
1
P
a
t
k
ð Þ
M
k
1
and therefore allow the calculation of the Kalman Gain.
The Kalman Gain is also required to calculate the error covariance of new analysis
P
a
ð
t
k
Þ
, which is reduced by the assimilation of observations relative to the forecast:
P
a
t
ðÞ¼
I
K
k
H
ð Þ
P
f
ð
t
k
Þ ð
8
Þ
Durand and Margulis (
2006
) tested this approach with a synthetic experiment of
snowpack progression in the USA. The system truth was taken to be a single model run
with forcing perturbed by doubling the precipitation and adding autocorrelated noise to
mimic known issues of gauge undercatch. Synthetic passive microwave observations at
SSM/I or AMSR-E frequencies were simulated by the Microwave Emission Model of
Layered Snowpacks (MEMLs, Wiesmann and M
¨
tzler (
1999
)) corrupted with 2 K white
