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Simulation of an Algorithm for Determining Whether There Are MTP Pointing
Errors
MJ Mahoney
When examining Microwave Temperature Profiler (MTP) data from two recent
campaigns on a new platform, we noticed that there were unusually large window
corrections. These are corrections to the measured brightness temperatures
that account for sidelobe pickup that changes with the MTP scan position.
It was conceivable that these errors could be due to a pointing error, although
the pointing was carefully checked when the instrument was installed on the
aircraft. Nevertheless, it was still possible that there could have been a
change in the INS alignment from which the MTP receives pitch and roll in
real time to affect accurate pointing with respect to the horizon.
Bruce Gary has developed a technique to do this which involves comparing
differences of the measured brightness temperture at the horizon (T5)
with the outside air temperature (To) and the measured brightness temperature
at a nearby scan position (T7). That is, T5-To is plotted against T5-T7 for
an hour or so of level flight. As will be shown shortly, the slope of such
a plot is directly related to the pointing error. The slope would be zero
if there wasn't a pointing error. When such plots are made, they can appear
very noisy; this is shown in Figure 1. Therefore, it was decided
to run some simulations to understand what the limiting factors are in this
procedure.
Some Background
MTPs operate at two or three frequencies, and generally scan to 10 different
locations from near zenith to near nadir. We adopt the notation that Tij is
the brightness temperature measured at frequency i and scan position j. Now
imagine the MTP beam looking straight forward. In the absence of large
mesoscale temperature fluctuations, the brightness temperature that the MTP
measures is equal to the physical temperature at the e-folding distance, or
the reciprocal of the absorption coefficient in nepers/km. This is also approximately
try when there is a linear variation of temperature with altitude. Under
these assumptions, the temperature T measured at a scan angle y will be:
T = To + LR*R* sin(y) , where LR is the lapse rate and R is the e-folding
distance, or applicable range.
Since MTP channel 1 is by convention always the lowest frequency channel,
is will always have the largest applicable range, and therefore show the largest
temperature change (and therefore signal) relative to flight level. For the
measurements in questions, the horizon scan position was position 5, and
we will use as the second scan position , scan position 7, which is 21 degrees
below the horizon. Denoting the applicable range for channel one as R1, and
assuming that the horizon position pointing error is x, we can then write:
(1) T15 = To +LR*R1*sin(x), and T17 = To +LR*R1*sin(x+y)
Thus, the quantity (T15-To)/(T15-T17) = m
(2) m = LR*R1*sin(x)/LR*R1*(sin(x) - sin(x+y) = sin(x) / (sin(x)
- sin(x+y))
If the slope is measured in a scatter plot of measurements, Equation (2)
is easily inverted by subsituting sin(x+y) = sin(x)cos(y) + cos(x)sin(y),
and we get an expression for the pointing error:
(3) x = atan(-m*sin(y) / (1 - m*(1 - cos(y))
To determine how well we could determine a pointing error, x, we ran a simulation
to generate estimates of (T15-To) and (T15-T17). There are a number of noise
sources that must be considered in such a simulation.
- radiometric noise for each brightness temperature measured This is
nominally 0.6 Kelvin for one measurement.
- pitch error. On the aircraft in question, the pitch had an rms noise
of 0.43 degrees
- To accuracy. By radiosonde comparisons, this error is 0.2 K.
- Lapse rate range. It is intuitively obvious that the larger the the
range of lapse rate during measurements, the bigger the value of T15-T17 and
the better the R-squared on the least squares fit to determine m.
Figure 1. A simulation of 1000 estimates of (T15 - Tnav) and
(T15 - T17) with 0.6 K radiometric noise, 0.43 degrees aircraft pitch noise,
0.3 K outside air temperature noise, and random lapse rates varying between
-8 and -2 K/km. The actual pointing error is -2 degrees. The slope of this
least squares fit to the data was m = -0.0619, leading to -2.27 degrees
as the pointing error (an over-estimate of -0.27 degrees).
Initially this was done in a spreadsheet (see Figure 1). However,
it was quickly realized that hundreds of iterations for a given set of assumption
would need to be run in order to estimate the error in the pointing error
determination, since everytime the noise was recalculated a different pointing
error estimate was found. This was because the noise in the simulation can
have a big impact on how well the pointing error can be determined.
POINTerror.VB6 Simulations
As a result, a program (POINTerror.vb6) was written to carry out the simulations.
For a given set of assumptions, we would estimate the pointing error (Perr)
using 100 simulated (T15-To) and (T15-T17) pairs for an assumed lapse rate
range (LRR), and do a least squares fit to get an estimate of Perr. This will
be called the "ensemble" algorithm (developed by Bruce Gary). This process
is repeated 100 times to get a handle of the accuracy of the retrieved Perr
value.
In addition, another approach was taken. Rather than fit 100 points, we
calculated the slope m for each pair of points (the "instantaneous"
slope), calculated the average value of m, and then a value for Perr
using Equation (3). The results of both of these simulations are presented
in TABLE I, using blue lines and error bars
for the results based on "ensemble" algorithm, and green
lines and error bars for the "instantaneous" algorithm.
TABLE I. Results of simulations using 100 sample points repeated
100 times to estimate the standard deviation of the result. In these plots,
the blue items are the results based on the ensemble results in each LRR
bin, and the green items are the "instantaneous" results for each LLR bin..
A word of clarification about what is meant by the ordinate parameter "Lapse
Rate Range (LRR)," or Lapse Rate Variability. It represents how much the lapse
rate was allowed to vary during each simulation iteration. So if the Lapse
Rate Range is zero, there is no variability, and if it is 6, there is maximum
variability. A further clarification is needed as well. Although the LRR
is 0 to 6 K/km in each of the panels of TABLE I, the actual lapse
rate varied from -8 to -2 K/km in the left hand panels [(a) and (c)], and
from -3 to +3 K/km in the right hand panels [(b) and (d)]. The reason for
doing this is that the "instantaneous" algorithm becomes very noisy whenever
a lapse rate near 0 K/km is encountered. So, for example, consider panel (d).
The "instantaneous" algorithm does very well until the Lapse Rate Range reaches
~2 K/km, then it goes south. The reason is that the computations begin using
the full range (-3 to +3 K/km, LRR=6), then (-2.75 to +3 K/km, LRR=5.75),
and so on in 0.25 K/km steps to (+3 to +3 K/km, LRR=0). Therefore, beyond
the range (0 K/km to +3 K/km, LRR=3) zero lapse rate is no longer involved
and this algorithm improves. In panel (c) where where the smallest absolute
value of the lapse rate never gets smaller than 2 K/km (LRR<2 K/km), the
instantaneous algorithm works very well; in fact, much better than the single-fit
or ensemble algorithm.
The difference between the upper and lower panels of TABLE I
is simply the pointing error that we are trying to determine; it is 0 degrees
in the lower panels and -2 degrees in the upper panels. The presence of 0
K/km is a killer for the "instantaneous" algorithm, when left and right panels
are compared, but it doesn't seem to affect the "ensemble" algorithm. When
upper and lower panels are compared, it is clear that while error bars don't
change as a function of Lapse Rate Range, the ability to determine the pointing
error does. Under the assumed noise conditions in these simulations, a Lapse
Rate Range greater than 3 K/km is needed if there is to be any chance of retrieving
the pointing error. If the absolute value of the actual lapse rates encountered
are always greater that one, the "instantaneous" algorithm works well under
all Lapse Rate Ranges.
Having completed these simulations, it was realized that 100 sample points
was not realistic since we typically use the longest flight legs available
to estimate the pointing error. So for example a 12 ks flight leg with 12
s scans would yield 1000 samples. Therefore, the simulations were repeated
using 1000 sample points for each of the 100 iterations in each LRR bin. The
results are shown in TABLE II.
TABLE II. Results of simulations using 1000 sample points repeated
100 times to estimate the standard deviation of the result in each lapse rate
range bin. In these plots, the blue items are the results based on the ensemble
results in each LRR bin, and the green items are the "instantaneous" results
for each LRR Range bin.
The most obvious difference between TABLEs I and II is that the size
of the error bars on the mean estimate of Perr has been greatly reduced, but
the mean estimate of Perr has not changed significantly. Certainly, the variation
with LRR is much smoother in TABLE II simply because there were 10
times as many samples going into the simulation. The other thing to note
is that as LRR goes to zero, the error in the ensemble algorithm approaches
the actual pointing error.
Figure 2. Standard deviation of Perr estimates as a function of LLR
for sample sizes from 10 to 10,000 using the ensemble algorithm.
We did a simulation under the assumptions of panel (a) of TABLE
II, that is, -8 to -2 K/km and -2 degree Perr to see how the accuracy
of the Perr estimate improved with N, the number of samples. This is shown
in Figure 2. Obviously, 0.1 degree accuracy requires LLR>3 K/km
and N>1000 if the ensemble algorithm is to be useful. We did a sanity check
of the improvement in the standard deviation from the mean of the Perr accuracy,
and it does improve as one over the square root of N.
Figure 3. The error in determining Perr as a function LRR for six
values of the radiometric noise between 0 and 1 K using the ensemble algorithm.
The number of samples was 1000.
So how important are the various noise sources in affecting the accuracty
of the Perr determination by the ensemble algorithm? This is shown in Figure
3 under the assumption that 1000 samples are used. If we assume that a
LRR = 3 is available, then a radiometric noise of 0.6 K (representative of
a single MTP integration cycle) would result in a Perr standard deviation
of 0.2 degrees, and this improves with larger LRR. The same result could
be achieved with ) 0.4 K radiometric noise at LRR=2, and the accuracy improves
to 0.1 K by LLR=3.
TBD Charts of instantaneous algorithm, Two Frequency approach.
-----------------------------------------------
STILL WORKING ON THIS
An issue with the previous approach is that is requires the outside air
temperature To to be known accurately, because if it wasn't this wouldn't
be distinguishable from a pointing error. If we use two MTP frequency channels,
then we can write:
(1) T15 = To +LR*R1*sin(x), T25 = To +LR*R2*sin(x),
T17 = To +LR*R1*sin(x+y), and T27 = To +LR*R2*sin(x+y)
Thus, the quantity (T15-T25)/(T17-T27) = m
(2) m = LR*(R1-R2)*sin(x)/LR*(R1-R2)*(sin(x) - sin(x+y) = sin(x)
/ sin(x+y)
If the slope is measured in a scatter plot of measurements, Equation (2)
is easily inverted by subsituting sin(x+y) = sin(x)cos(y) + cos(x)sin(y),
and we get an expression for the pointing error:
(3) x = atan(m*sin(y) / (1 - m*cos(y))
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