Geoscience Reference
In-Depth Information
a
b
c
d
e
f
0-30
m
in
0-60
m
in
0-90
m
in
0-120
m
in
0-150
m
in
0-180
m
in
4.0
3.0
2.0
1.0
0.0
4.0
3.0
2.0
1.0
0.0
4.0
3.0
2.0
1.0
0.0
4.0
3.0
2.0
1.0
0.0
4.0
3.0
2.0
1.0
0.0
4.0
3.0
2.0
1.0
0.0
-1
0
1
2
-1
0
1
2
-1
0
1
2
-1
0
1
2
-1
0
1
2
-1
0
1
2
N
0r
N
0r
N
0r
N
0r
N
0r
N
0r
g
h
i
j
k
l
1.0
1.0
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.8
0.8
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0
500 1000
0
500 1000
0
500 1000
0
500 1000
0
500 1000
0
500 1000
a
g
a
g
a
g
a
g
a
g
a
g
Fig. 3.8
Posterior two-dimensional marginal parameter PDFs for increasing numbers of obser-
vations assimilated in an MCMC parameter estimation experiment. Each row corresponds to a
different pair of parameters, while each column represents a different set of assimilated observation
times. (
a
)-(
f
) are 2D marginal PDFs of the slope intercept of the rain particle size distribution (N
0r
)
and threshold cloud mixing ratio for autoconversion to rain (q
c0
). (
g
)-(
l
) are 2D marginal PDFs of
the coefficient and exponent in the graupel diameter - fall velocity relationship. True parameter
values are indicated in the
red cross-hairs
.The
thin dash-dot
,
dashed
,
solid
,and
dotted black
contours
enclose the 99.7, 95, 68.3, and 38.3 % probability contours, respectively
any parameter values for which snow falls faster than graupel. The result is shown
in Fig.
3.7
m-p: the solution reduces to a single mode centered on the true parameter
values for the snow and graupel fall speed parameters. Graupel density (Fig.
3.7
p)
continues to exhibit a mode that stretches from its true value to the maximum
allowable value, indicating a loss of sensitivity to changes in this parameter above
a certain value. The 2D marginals contained in Fig.
3.7
demonstrate the utility of
MCMC for exploring the relationship (e.g., covariance) between parameters, the
effect on parameter estimates of adding additional and/or more accurate observa-
tions, and the need for physical constraints in the data assimilation procedure.
Now, the MCMC algorithm is a static inverse method; control variables are not
updated sequentially but are instead assumed fixed over the length of the simulation.
We may address the question of when non-uniqueness in the posterior parameter
PDF arises by running several MCMC experiments, and changing the number of
observation times included in each one. The results of such an experiment, in
which we included observations from 30, 30-60, 30-90, 30-120, 30-150, and
30-180 min in the MCMC algorithm, are presented in Fig.
3.8
. Posterior PDFs for
the cloud-rain autoconversion threshold and the slope intercept of the rain particle
size distribution (Fig.
3.8
a-f), and the graupel fall speed parameters (Fig.
3.8
g-l)
are shown. Several conclusions can be drawn from this figure. First, changes in the
warm rain parameters affect the solution most during the convective phase of the
simulation (0-90 min) with influence that wanes as the system is forced to make
the transition to stratiform. The opposite is true of the graupel fall speeds, which
strongly influence the solution at stratiform times, but have limited effect early in
the simulation. In addition, it is clear that multiple modes do not arise in the solution
Search WWH ::
Custom Search