MTP Spatial Resolution: Horizontal
The question of "what's the MTP resoultion, both in
altitude and horizontally?" occasionally arises. The last chapter
describes altitude resolution; this chapter describes horizontal
We shall adopt the same MTP and observing setting to illustrate the
concepts of calculating horizontal resolution; namely, an MTP/ER2
flying at 20 km in a standard atmosphere. Channel #1 (LO = 56.65 GHz)
observes an atmosphere with a microwave absorption coefficient, Kv = 0.39
Nepers/km, corresponding to an applicable range of ~2.6 km. Channel #2 (LO = 58.8 GHz) observes an atmosphere with a microwave absorption coefficient, Kv = 0.9
Nepers/km, corresponding to an applicable range of ~1.1 km. An observing cycle consists of switching between each frequency at each of 10-elevation
angles. The elevation sequence is +60.0, +44.4, +30.0, +47.5, +8.6, 0.0, -8.6, -20.5, -36.9 and -58.2 degrees. The next three graphs are similar to those ones in Chapter 6, but they have
been calculated for the specific case under discussion in this chapter.
If the ER-2 were not moving through the air the applicable ranges for
each channel, at each of the 10 scan angles, could be depicted by the
Figure 13.01. Applicable range locations for a 2-channel MTP that is
not moving through the air as it completes an elevation scan. The
elevation angle sequence and applicable ranges are given in the text.
Since the ER-2 is moving at 210 meters/second the applicable locations of a scan are offset as shown in the next figure.
Figure 13.02. Applicable range locations for
a 2-channel MTP that is not moving through the air as it completes an
elevation scan. The scan direction is downward.
Elevation scan sequence are 10 seconds apart so the air sampled by one scan cycle is 2.1 km offset
from the air sampled by the previous scan cycle - as shown in the next figure.
Figure 13.03. Applicable locations for two MTP scan cycles (for cycle spacing of 10 seconds).
Remember that each applicable location symbol is just the 1/e location
of an exponential weighting function. For the horizon views (scan
location 6 for each channel) the weighting functions for two scan
sequences is shown in the next figure.
Figure 13.04. Horizon view weighting functions for two scans, for Channel #1 (red) and #2 (blue).
Now we're ready to give a first approximation answer to the question
"What's the horizontal resolution of MTP temperature profiles?" Our
answer will apply to the MTP-based OAT at flight level, called OATmtp. Assuming Ch
#1 and Ch #2 horizon view brightness temperatures, TB1 and TB2, are weighted equally for the calculation of
the parameter OATmtp (outside air temperature based on MTP), each elevation angle scan
cycle's OATmtp comes from a quasi-exponential weighting function
shown in the next figure.
Figure 13.05. OATmtp weighting function (the average of horizon
view weighting functions for Ch #1 and Ch #2). The next cycle's
weighting function is shown by the dotted trace.
The 1/e distance of the OATmtp weighting function =
1.8 km. This is approximately the same as the spacing of neighboring
OATmtp weighting functions (2.1 km). It is tempting to say that horizontal
resolution is ~2 km. But there's the matter of undersampling. If TB
measurements of the horizon were made twice as often then we could be
confident of the 2 km resolution answer. Let's look at the matter of
horizontal resolution differently.
Fourier Representation of T(x)
The term "resolution" will have different meanings for observing
systems having different weighting functions. For example, an in situ
sensor reading is actually the weighted average of the source function,
T(x), multiplied by an exponential function that tails off behind the
aircraft with a distance of tens of meters. The e-folding distance for
the MMS in its low resolution mode is ~200 meters.
An MTP reading of brightness temperature, on the other hand, is the
product of T(x) multiplied by an exponential weighting function that
tails off in the opposite direction (assuming the MTP views the horizon
in the direction of flight). Even if the OAT sensor and the MTP had the
same e-folding distance, and same measuring interval, they would have
different observable sequences. In theory, however, it should be
possible to reconstruct the same source function, T(x). This would be
done using a Fourier component representation of the observations and
reconstructing the source function by shifting component phases in an
An observing system can be imagined that does not have an exponential
weighting function. For example, consider representing T(x) by the
function 2 * TB2 - TB1, where TB1 and TB2 are the MTP Ch#1 and Ch#2
brightness temperatures for the horizon view. This is a simple Backus-Gilbert procedure for
improving horizontal resolution; it is the horizontal analogue of the
vertical resolution improving procedure described in the previous
chapter (see Fig. 12.10). The location x where this temperature applies
is ahead of the aircraft's position, and the distance correction is
merely the applicable range of the averaging kernel times airspeed.
I'm not suggesting that this procedure be used for converting MTP
measurements to T(x); rather, I'm using it to illustrate that an
exponential weighting function is not the only one to describe
observations that are to be used in deriving T(x). When someone asks
"What's the horizontal resolution of your measurement?" you have to ask
"How do you want to define 'resolution'?"
Let's work-up to a suggested definition in steps. Consider that the observations
can be used to derive a most-likely source function, T'(x). A first
approximation to T'(x) is merely:
T(x+Ra) ~ T'(x+Ra) = TB(x)
where TB is measured brightness temperature while flying at location x, and
Ra is the applicable range corresponding to the TB observable.
Four equally-spaced measurements (thanks, MJ) are needed to measure a sinsoidal
funtion's amplitude and phase (at, for example, phase = 0, 90, 180 and 270
degrees). Therefore, the shortest wavelength that can be represented by
TB(x) is ~8.4 km (4 x 2.1 km), assuming TB measurements are made
every 2.1 km along the flight path. Longer wavelngths can be
represented, the longest being set by the length of the TB sequence.
It will be useful to now consider the response of an MTP to Fourier
components of the source function T(x+Ra) with the weighting function
geometries for our case study MTP.
First, recall that the MTP's estimate of OAT is obtained
by averaging the horizon brightness temperatures for two (or three)
channels. The weighting does not have to be equal, and in the past the
weighting has been influenced by the RMS performance of each channel.
Whenever a weighting is performed the effective weigting function
versus range is no longer an exponential. The following figure is a
plot of the weighting functions for a set of 6 TB measurements of the
horizon view (using the same shape as in the previous figure).
Figure 13.06. Weighting functions for a set of 6 MTP OAT estimates.
The set of 6 measured TBs are obtained by multiplying these 6 weighting functions with the
source function T(x).
Now, think of the true source function, T(x), as consisting of Fourier
sine and cosine components with wavelengths corresponding to all
relevant spatial frequencies. Let's start with a T(x) sinusoidal
component having a wavelength of 12 km.
Figure 13.07. 12 km wavelength Fourier component of T(x), air temperature along the flight path, T(x). The y-axis can be thought of a a Kelvin scale for the Fourier component.
Using this graph it can be seen that the MTP's OAT response to a 12 km
wavelength component of variation in T(x+Ra) appears to be close to 100%.
A detailed calculation (involving sliding the sequence of weighting
functions in phase to get the maximum response) shows that the maximum
observed TB variation is 68% of the true source variation. Thus, the
function TB(x) has a 68% response to Fourier components with a 12 km
wavelength. Similar calculations for other wavelengths have
been made, and are plotted in the next figure.
Figure 13.08. Response of MTP's OAT versus Fourier wavelength of T(x) function.
What is the meaning of the non-zero response shortward of 8.4 km
(which is the shortest wavelength for which we can derive Fourier
components for the desired source function)? The non-zero response
to short wavelengths should be viewed as "noise" cased by real T(x+Ra)
variations whose Fourier components cause TB fluctuations but whose
amplitudes and phases we cannot recover. This is a problem related to
having weighting functions with an abrupt "turn-on" near the aircraft.
Since the weighting function width for each averaged horizon view TB is
slightly shorter than the sampling distance there will be additional
"noise" for Fourier components near 8.4 km. If the Ch#1 TB were used by
itself to create a TB(x) sequence then there would be less noise
near 6.3 km due to real T(x+Ra) structure with wavelengths shorter than 8.4 km.
Lest we forget, I should mention that there's another component of
noise that will limit horizontal resolution, and that's radiometer
noise. Any analysis done with the observed TB sequence for the purpose
of recovering T(x+Ra) will have to take into consideration the stochastic
noise on each TB measurement, which is ~0.2 K.
If no attempt is made to recover T(x+Ra) structure from TB(x) then it
can be said that the MTP's horizontal spatial resolution is ~2 km,
corresponding to a highest spatial frequency
that can be represented by the function "true air temperature versus
distance along the flight path" of ~ 8 km. Due to undersampling this
resolution is subject to some additional "noise" caused by T(x+Ra)
structure with wavelengths shorter than 8 km.
This web page's objective is to answer the question "What's the MTP's
horizontal resolution?" Nevertheless, this is a good place for
describing how the observed set of TB(x) measurements (actually, the
average of Ch#1 and Ch#2 TB measurements) can be used to produce
improved T(x+Ra) solutions. We will conclude that this is an unworthy goal.
There are at least two methods for attempting to recover a better version of T'(x+Ra) than
by assuming it to be TB(x). The simplest of them is Bracewell's deconvolution method. You
begin with TB(x) as a first approximation of T(x+Ra) and then pretend
to observe it. The fictitious observation sequence will have less
structure than TB(x), and the difference between the two sequences
is the first iteration correction to TB(x). Each iteration
multiplies the effect of stochastic noise, so when the corrections
become comparable to, or smaller than, the noise level of the TB
observations you terminate the process. The Bracewell deconvolution is
almost equivalent to multiplying the high spatial frequencies in the
observations by their lowered response to true variations of high
Another method for recovering structure is to choose an x-region of
interest and decompose the series TB(x) into Fourier components.
This will be done using spatial frequencies (not wavelengths). The
minimum spatial frequency will be Fn = 1 / (2 * L), which is also the
spatial frequency interval, dF, and the maximum spatial frequency will
be (N-1) * Fn - where L is the length of the data set, and N is the
total number of evenly-spaced TB data. Windowing should be performed
first (my favorite is Welch). After calculating the entire set of N-1
sine and cosine amplitudes the amplitudes should be adjusted upward in
accordance to Fig. 13.08 (i.e., using the reciprocal of the response as
multipliers). The amplitude adjusted Fourier components should then be
added to produce a new T'(x+Ra). The new T'(x+Ra) will have the shorter
wavelengths of temperature variation restored. Unfortunately, the
"noise" produced by wavelengths that cannot be recovered (see previous
figure) will be amplified as they show themselves as spurious
fluctuations at the short wavelength end of what can in theory be
recovered. This problem originates in the weighting function having a
sharp "turn on" near the aircraft. The only way to overcome this
problem is to use a horizon view version of the Backus-Gilbert
averaging kernel shown in Fig. 12.10 and 12.11. These B-G averaging
kernels have "soft" shapes and they therefore will not have "noise"
produced by fluctuations shorter than can be used in a deconvolution
process. The only downside to this is radiometer noise. The B-G
procedure multiplies radiometer noise, as does the Fourier
deconvolution procedure. When each TB reading, for each channel, has a
stochastic uncertainty of ~0.2 K, the final deconvolved T'(x+Ra) will have
stochastic SE on the order of 1.0 K. I therefore do not recommend using
any of the structure recovery procedures for horizontal temperature.
Suggested New Equation for OATmtp
Considerations of the previous few paragraphs suggest that a new way be used for computing OAT from MTP measurements:
OATmtpNew = 2 * TB1 - TB2
where TB1 is
Ch#1's horizon view TB, and TB2 is Ch#2's horizon view TB.
The following figure shows the shape of the averaging kernel for "OATmtpNew":
Figure 13.09. Averaging kernel for OATmtpNew = 2 * TB1 - TB2 for two frequency switching choices.
The slight difference timing for the TB1 and TB2 measurements causes a
negative feature at the beginning of the OATmtpNew's averaging kernel.
The next figure shows the weighting functions for the frequency swtiching sequence "Ch#2 then Ch#1."
Figure 13.10. Weighting functions for Ch#2 before Ch#1 (horizon view). Ch#2 "turns on" at the origin while Ch#1 "turns on" ~0.1 km later.
Since the two measurements are made ~0.4 seconds apart (~80 meters of
difference along the flight path) only temperature fluctuations on this
scale will produce spurious, high spatial frequency artifacts in
OATmtp. These fluctuations are miniscule in comparison to those that
affect the old version of OATmtp. The only disadvantage in using
OATmtpNew is that the stochastic noise per 10-second observing cycle
be ~0.5 K instead of 0.2 K.
Notice that it is better to observe Ch#2 before Ch#1 at the horizon
view scan location since that frequency order produces a slightly
better averaging kernel shape. The FWHM for the "Ch#2 then Ch#1"
averaging kernel is 4.2 km. This is twice the interval between samples;
this means that using the new OATmtp equation there is no undersampling
problem. In fact, a sequence of OATmtpNew has the potential for a
proper deconvolution since its sampling meets the Nyquist requirement.
The MTP's horizontal resolution is ~2
km near flight level but the readings are undersampled by about a factor two. At 1 or 2 km above and below flight level the
horizontal resolution is about 2 to 3 km, and the same undersampling exists. For
all previous calculations of OATmtp a
component of high spatial frequency fluctuations is present because the
observables have exponential (non-Gaussian) weighting
functions with a sharp turn-on shape near the aircraft; this produces
observables that are influenced by temperature fluctuations having
wavelengths shorter than the stated resolution. A new equation for
calculating OATmtp is suggested which does not suffer from this
problem; it has a horizontal resolution of 4.2 km but because it meets
the Nyquist sampling requirement it may be possible to achieve a
somewhat better horizontal resolution, such as ~3 km.
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This site opened: January 5, 2006. Last Update: June 22, 2007