One of the prime equations for data assimlation is the Kalman Filter Equation. This equation is:
x(a)-x(f) = PH(T){HPH(T)+R}(-1)[d-Hx(f)]
Where
x(f) is the forecast, x(a) is the analysis, P is the background error
covariance matrix, and R is the observation error covariance matrix.
x(a) and x(f) are vectors, containing every model variable at every
model gridpoint.
P gives the information regarding errors of the model first guess field
R gives information about the errors of observations (instrument and representativeness errors).
This
equation, and the notion of 'covariance matrices', is often a little
overwhemling. It can be simpler to consider the case of trying to best
know the value of one variable.
For just one variable...
T(a) - T(f) = s(f)**2/(s(o)**2+s(f)**2)*[T(o)-T(f)]
Each
term in the equation for one variable has a match with the full blown
Kalman Filter equation. T(
a) is the
analysis value (best estimate), as
is x(a), T(
f) is the model
first guess, as is x(f). And so on.
Consider the temperature of the air at 500mb over Denver Colorado.
To estimate this value, T(
a) (
a for 'analysis') we have two estimates:
#1 is the value as reported from the DNR sounding.
I'll call that T(
o) (
o for '
observed') Using known history, the average error of these observations has an error variance s(o)**2
#2 is the value from the model first guess field
I'll call that "T(f)" (
f for '
first guess') Using known history, the average error of these observations has an error variance s(f)**2
Here is where the graphic comes in!
The graphic I am envisioning is rather similar to this one:
This graph would be re-worked such that instead of FCST A and FCST B, we have First Guess (T(f)) and Observation (T(o))
The applet I am envisioning is something that can be changed from the image on the left to the image on the right.
Sliders
(as well as text entires) will allow the user to adjust both the values
of T(f) and T(o) as well as the error variances s(f)**2 and s(o)**2.
Also there would be a box showing the value of T(a).
What
if our forecast model was always perfect? In that case, the error
variance of the observations, s(f)**2, would be zero. In that case, the
equation would collapse to T(a)-T(f) = [T(o)-T(f)]*(0), or T(a)-T(f)=0.
At this time, the image would show that T(a) and T(f) are the same
value, and that the spread (the tails of the Gaussian curves in the
figure above) would be zero.
Consider now the other extreme, where the
observations are perfect (but the forecast is not). In that case, the
equation becomes T(a)-T(f) = [T(o)-T(f)]*(s(f)**2/s(f)**2), or
T(a)-T(f)=T(o)-T(f), and thus, T(a)=T(o).
Users could adjust
the values of the error variances. The 'analysis value', T(a) would
move towards whichever value (T(a) or T(f)) had a smaller error
variance associated with it. As the error variances were increased, the
picture would look more like the right side of the above image. If
error variances were decreased, the image would look more like the
figure on the left of the above image.
Included with the estimate of T(a), the actual value, there would be a Gaussian shape around it with variance equal to [1/(s(o)**2+s(f)**2)](-1), showing that when the error variances of the two estimates are small, the error variance (uncertainty) of the analysis is also small.
This module will show the correlation between error statistics of the two estimates (first guess and observation) and the analysis value.