Bruce L. Gary

**Introduction**

This web page describes "calibration" analyses for the Geophysica aircraft outside air temperature, OAT, readings and the Microwave Temperature Profiler, MTP, "window corrections" for a flight series in CY'2003. EUPLEX flights occurred between 2003.01.15 and 2003.02.11, and ENVISAT flights were from 2003.02.28 to 2003.03.16.

I first describe the calibration of Geophysica's *in situ* temperature
instrument, which I shall refer to as TDC (Thermal Dynamic Complex), built
by Central Aerological Observatory, Moscow, Russia. I calibrated
the TDC OAT using RAOBs as a standard.

The MTP window correction table, WCT, corrects for reflections and absorptions by the high density polyethylene microwave window for each of the 10 elevation angles of the MTP scan. For the Geophysica flights the MTP operated at 3 frequencies, so the WCT consists of a 3x10 element table of empirical corrections that are to be added to measured antenna temperature. The underlying assumption for use of this WCT is that RAOBs are accurate, on average, and microwave absorption is well-enough understood to permit a calculation from RAOB temperature profiles of what the MTP should have measured as it flew by a RAOB site. Typically, a dozen such comparisons of RAOB-based predicted Ta elevation profiles versus measure profiles is required in order to achieve a WCT that is accurate to ~0.2 K, which is a reasonable goal.

The procedure for obtaining the OAT calibration begins with an inspection of a flight track to determine which RAOB sites are located closer than ~150 km. A list of times and RAOB site designations is produced, and the RAOBs are downloaded from the following web site: http://weather.uwyo.edu/upperair/sounding.html These RAOB profiles are converted to my standard format, which facilitates their use for both OAT calibartion and WCT determination. The RAOB files are imported to a spreadsheet where plots of T(z) can be viewed (and printed out) for conducting interpolations in altitude, time and space (if necessary). Spatial interpolation is used when the flight path is straddled by two RAOB sites at approximately equal distances. For each OAT/RAOB comparison event a RAOB interpolation fraction is calculated for temporal interpolation between the closest previous and following RAOBs. An average aircraft-based OAT is calculated by averaging ~2.5 minutes worth of OAT measurements. This constitutes one RAOB/OAT comparison, which typically exhibits a standard error uncertainty of 1.1 K. Therefore, many such RAOB/OAT coparisons are needed in order to achieve an average OAT correction that has an associated SE uncertainty of approximately 0.30 K, which is a desired goal.

The following flights were used for deriving OATtdcCOR and the WCT.

**
OAT WCT Analysis**
** Flight Date
# # Date**

2003.02.06
2 2 3316

2003.02.09
2 2 3315

2003.02.11
0 0 3317 Unuseable

2003.03.02
3 3 3314

2003.03.08
1 0 3416

2003.03.12
2 0 3415

2003.03.16
1 1 3416

The "#" symbol refers to the number of RAOB flybys that were useful for an OATtdcCOR and WCT analyses. The "Analysis Date" is coded as YMDD. Not all RAOB flybys that were used for OAT calibration were suitable for WCT analysis since the WCT analysis requries that RAOB temperatures extend well above flight level whereas for the OAT analysis it is merely necessary that the temperature data extend to at least flight level.

**OAT Calibaration**

The following graph shows RAOB-predicted and measured OAT for 12 flyby events.

**Figure 1.** *Measured OAT versus RAOB temperature for Geophysica
flybys of RAOB sites. The dotted line is a fit to the PTW data, and
is offset -0.39 K from RAOB temperatures.*

The population SE in the above figure is 0.68 K, which is better than average (~1.1 K). There does not appear to be a trend in the difference between Geophysica TDC OAT and RAOB "truth."

Under the assumption that the Geophysica TDC OAT exhibit a simple offset from RAOB temperatures, constant for the entire flight period, the following CORRECTION should be applied to Geophysica TDC temperatures to achieve agreement with RAOBs:

**OATtdcCOR_euplex&envisat = +0.34 +/- 0.20
K**.

**Window Correction Table**

The following 3 graphs show corrections that must be added to measured antenna temperature to yield RAOB-predicted brightness temperature.

**Figure 2.** *Channel #1 window corrections (add to MTP
measured antenna temperatures). The 8 red line traces correspond
to individual comparisons with RAOB-based calculations for passage close
to a single RAOB site (or between a pair of sites). The thick black
trace is an average.*

**Figure 3.** *Channel #2 window corrections.*

**Figure 4.** *Channel #3 window corrections.*

As with all previous WCT analyses there is "shape" agreement among all individual profiles. Departures increase with distance from the horizon because of the way individual profiles are subjected to an offset normalization (in which the middle three values are forced equal to zero).

The corresponding table of corrections is given below:

**
EL Ch#1 Ch#2 Ch#3**

**
55 5.20 1.04 1.33**
**
38 3.85 1.47 0.90**
**
24 4.91 1.83 0.52**
**
12 4.10 0.72 0.05**
**
0 0.00 0.00 0.00**
**
-10 -0.64 1.35 0.75**
**
-21 0.56 1.82 0.58**
**
-34 1.53 1.13 0.14**
**
-49 1.40 0.85 0.45**
**
-70 1.56 0.57 0.60**

The table entries are to be ADDED to measured antenna temperature.

Each WCT entry is the average of 11 numbers. The RMS for each WCT entry is presented in the following graphs.

**Figure 5**. *RMS scatter for Channel #1's WCT, with 0.3
K orthogonally added. The dashed trace is a 3rd-order polynomial
fit.*

**Figure 6**. *RMS scatter for Channel #2's WCT, with 0.3
K orthogonally added. The dashed trace is a 3rd-order polynomial
fit.*

**Figure 7**. *RMS scatter for Channel #3's WCT, with 0.3
K orthogonally added. The dashed trace is a 3rd-order polynomial
fit.*

A listing of the RMS variation of the data for each WCT cell is presented below (with orthogonally added 0.3 K and based on a 3-rd order polynomial fit to achieve smoothness).

55
1.81 0.84 0.82

38
1.40 0.55 0.51

24
1.17 0.44 0.39

12
1.03 0.41 0.36

0 0.94 0.43 0.37

-10
0.90 0.48 0.42

-21
0.88 0.56 0.49

-34
0.89 0.66 0.58

-49
0.94 0.79 0.68

-70
1.01 0.92 0.77

I recommend using these values for the *a priori* uncertainty when
calculating retrieval coefficients.

**Scan Location Offset**

Is the horizon scan location, #5 for the Geohphysica, really at the horizon? It was adjusted in the hangar to be horizon-viewing, but is it close to the horizon during flight?

There's a straightforward way to determine, empirically, the scan offset and it involves the measured horizon-viewing brigthness temperature, TBh, and in situ air temperature, OAT. A scatter plot is made of "TBh - OAT" versus lapse rate. If there's no correlation then the horizon view is really pointed at the horison. But if, for example, there's a positive correlation of "TBh - OAT" with dT/dZ, then the horizon view is above the true horizon. For a given scan offset, such as one degree, the slope of "TBh-OAT" versus dT/dZ is proportional to applicable range, Ra.

Note that Ra is the 1/e distance for the microwave weighting function, which equates to 1/Kv when viewing the horizon, where Kv is the absorption coefficient [Nepers/km]. Since each MTP channel consists of a finite RF passband function, ith upper and lower IF sidebands, it is necessary to calculate a passband-weighted Ra. This is done by calculating 1/Kv for each RF increment and performing a weighted average Ra.

I will use the GE030206 flight (51.2 to 55.8 ks) to illustrate how to determine an MTP horizon view scan offset error, E. When the MTP/Geophysica is flying at 18.46 km (pressure altitude) in air having a temperature of 197 K (appropriate to the first half of GE030206), Ra has the following Ra values:

Ra (Ch#1) = 8.29 km

Ra (Ch#2) = 2.16 km

Ra (Ch#3) = 1.10 km

The follwoing graph shows the MTP/Geophysica RF weighting function for Ch#1 and the Ra values for RF passband increments.

**Figure 8**. *RF passband (green) for the MTP/Geophysica
and the applicable range for each passband increment (reciprocal of absorption
coefficient, i.e., 1/Kv) for the conditions T = 197 K and air pressure
69.8 mb (corresponding to a flight pressure altitude of 18.46 km). The
weighted average Ra is 8.29 km.*

This figure illustrates something that should be avoided in designing an MTP. Notice that the various RF increments sample a large range of Ra, from 5 km to 24 km. It's true that the 24 km Ra is associated with the edge of just one sideband, and therefore does not contribute greatly to the passband-averaged TB that's measured. However, the centers of the upper and lower sidebands have Ra values that are in the ratio of 2 to 1. The meaning of this is that Ch#1 samples a wide range of altitudes when looking above or below the horizon, and its information content is "diluted" more than the other two channels, which have more nearly uniform Ra plots. All of this information will be implicitly "allowed for" when retrieval coefficients are computed. Nevertheless, Ch#1 can be expected to exhibit "special needs" during calibrations. This situation may account in some way for the unusual WCT for Ch#1.

I refer to the vertical gradient of temperature, dT/dZ, as lapse rate, LR (strictly speaking, this is the negative of the the traditional meaning of "lapse rate"). Note that we really want to use dT/dZg, which is the vertical gradient of temperature with geometric altitude (as opposed to pressure altitude). RAOBs can't be used to derive LR for one major reason: a RAOB T(z) applies to only one location at one epoch and we need LR versus time along the aircraft's flight path in order to construct a scatter plot of "TBh-OAT" verus LR. We'll have to estimate LR(t) from the MTP data itself. The ideal MTP LR would come from the viewing directions that neighbor the horizon view, but for this particular mission there's a large window correction for Ch#1's +12 degree elevation view. This is the largest WCT feature I've encountered in my decades of MTP calibrations, adn I'm reluctant to use the +12 degree Ch#1 data for estiamting LR. Therefore, I have chosen to use scan elevation angles 0 and -21 degrees (scan locations 5 and 7) for estiamting LR. In making this change I'm assuming LR over this lower range of altitudes will be the same as for the layer between flight level and the altitudes probed by the horizon scan location (which will turn out to be below flight level).

Given that the effective applicable range for Ch#1 is 8.29 km, the -21 degree view has an altitude weighted 1/e altitude (with respect to the aircraft altitude) of -2.97 km (i.e., 8.29 km * sine (-21 degree)). Thus, dT/dZ = (TB1(5) - TB1(7)) / 2.97. This assumes that scan locations 5 and 7 are really at 0 and -21 degrees, so if a different result is determined from this analysis another iteration might be required (but note that even for a 1 degree error the correction will be small, since both scan locations will be corrected in the same direction, meaning that any corrections of this type are second-order). The following graph is a scatter plot of "TB1(5) - OAT" verus "TB1(5) - TB1(7)".

**Figure 9.** *Scatter plot of horizon view's TB (Ch#1) discrepancy
with OAT versus a paramter proportional to LR.*

The slope of this scatter plot fit is -0.211 +/- 0.038 [K per K difference b etween scan location 5 and 7].

Before proceding let's think about the significance of WCT on "TB1(5) - TB1(7)." Assuming the WCT is correct, we should add 0.66 K to TB1(5) and add 0.12 K to TB1(7). In other words, every point in the above figure should be adjusted to the right by 0.54 K. If this were done the fitted line slope would not be altered, and as will be seen it is only the slope that enters into the following derivation.

If the scan location error were zero, then dT/dZ could be derived from "TB(5) - TB(7)" by noting that TB(5) is the temperature at flight level and TB(7) is the temperature at an altitude that is below the aircraft a distance equal to Ra * sine (-21). Then, dT/dZ = (TB(5) - TB(7)) / (Ra * sine (21 degrees)). The problem with this derivation is that it assumes that the scan error E = 0.

A slightly modified method will be used here for deriving E, and it will be an exact solution (no approximations).

Consider the equations for each of the following measured TBs:

TB(5) = OAT + Ra * dT/dZ * sine (E)

TB(7) = OAT + Ra * dT/dZ * sine (E-21 degrees)

Let's express the slope fit in the above figure in terms of the two TBs just described (abbreviating the sine terms in an obvious way and replacing dT/dZ with L):

Slope = d(TB(5)-OAT) / d(TB(5) - TB(7))

Substituting,

Slope = d(OAT + Ra*L*sinE - OAT) / d(OAT + Ra*L*sinE - OAT - Ra*L*sin(E-21))

Which can be simplified,

Slope = Ra*sinE * d(L) / Ra*(sinE - sin(E-21)) * d(L)

Note, d(L) is really d(dT/dZ), or a change in "lapse rate" - which is the thing that causes a pattern in the scatter of points in the above figure.

This last expression for slope can be further simplified,

Slope = (sinE / (sinE - sin(E-21))

The simplest way to convert a measured slope to E is to consult a graph that plots slope versus E (or the reverse, which will be more user friendly).

**Figure 10**. *Dependence of "slope" (X-axis) on scan offset "pointing
error" (Y-axis), where "slope" is defined to be the rate of change of "TB(5)
- OAT" for changes in "TB(5) - TB(7)" caused by variations in lapse rate
dT/dZ. This plot is for scan locations 5 and 7 having their nominal
values of 0 and -21 degrees.*

In the above figure the slope is plotted as the X-axis as a convenience for the user wishing to convert a measured slope to an inplied pointing error, E. Based on the measured slope of -0.210 +/- 0.038, the pointing error E = -4.4 +/- 0.8 degrees.

Although Ch#1 is more affected by pointing error than the other two channels, and is thus a natural choice for estimating E (which presumably is the same for all 3 channels), similar concepts can be used to show that the slope of "TB2(5)-OAT" versus "TB1(5) - TB1(7)" is (Ra2/Ra1) * sinE / (sinE - sin(E-21)). The measured slope for Ch#2 is -0.077 +/- 0.031, implying that

E (Ch#2) = (8.29/2.163) * (-1.6+/-0.6 degrees) = -6.1 +/- 1.4 degrees.

This is statistically compatible with the Ch#1 solution for E, since the difference is 1.7 +/- 1.6 degrees.

For Ch#3 the measured slope is -0.098 +/- 0.032, implying E = (8.29/1.10) * (2.0 +/- 0.6) = 15.1 +/- 4.5 degrees. This result is discrepant with the other two values for E, and I thnk it reveals that the scatter plot of "TB3(5) - OAT" versus "TB1(5) - TB1(7)" is dominated by some factor other than the Ch#3 pointing error. Whatever that confounding factor is it will be present for the other two channels, and hopefully it is not as dominating for Ch#1. Thus, I recommend using only the Ch#1 value, E = -4.4 +/- 0.8 degrees.

A scan offset of 4.4 degrees will produce significant errors for some of the other analyses. For example, when dT/dZ = 1 [K/km], which will be typical for most stratospheric flight, the error in TB1(5) will be 0.64 K. For tropospheric flight, assuming a flight altitude of 10 km and dT/dZ of -7 [K/km], the error on TB1(5) will be ~1.2 K (the lower altitude causes Ra to be smaller, which compensates for dT/dZ being larger). If we adopt -4.4 degrees as the pointing offset error to be applied to all 3 MTP channel observations, then there is a simple procedure for incorporating this information into all subsequent analyses. The first step in this re-analysis procedure is to determine a new window correction table, WCT, using the new viewing elevations. Then, new instrument gains will have to be determined using a TB for each channel that is an interpolation between scan location 5 and 6 (44% of the way from 5 to 6). The TBs that result from this new set of gains will have to be used to refine the WCT, and perhaps a new iteration of gains and WCT will have to be performed until both WCT and gains are "stable."

This "recovery" from a scan pointing error represents a lot of extra work. Since the determinations of E, above, are based on a small range of variation of dT/dZ it will be prudent to verify the pointing error using a flight that encountered a greater variation of dT/dZ.

**Conclusion**

The OATtdcCOR and WCT calibrations are comparable in quality to most previous such calibrations. Since these data are from the first flight of an MTP aboard the Russian Geophysica aircraft it is noteworthy that good quality calibrations are achievable for an MTP on this platform. This means that it should be possible to produce MTP temperature profiles with a quality comparable to those produced on other platform aircraft, such as the ER-2 and WB-57 aircraft.

____________________________________________________________________

*This site opened: April
17, 2003. Last
Update: July 10, 2003*