Introduction

It is difficult to measure the true elevation angle of the antenna beam for any scanning location for the MTP instruements.  Measurements taken in the hangar, using a protractor, level, etc, have been made with a great deal of difficulty and with unsatisfactory uncertainties.  Although the half-power full-width beamwidth for the MTP antenna pattern is approximately 7.5 degrees, it is nevertheless a reasonable goal to know the elevation angle for sky views near the horizon with an accuracy of 1 degree (for views close to the nadir and zenith it is not necessary to achieve such accuracy).  I estimate that the hangar measurements are accurate to approximately 1 or 2 degrees, SE.

This web page describes an attempt to establish the horizon view's true elevation angle using the actual airborne measurements themselves.  The concept is based on the fact that errors in this viewing angle will produce errors in TB while observing the putative horizon, and that the errors will be proportional to the lapse rate near flight level. Thus, by correlating the "horizon" TB error versus lapse rate, it should be possible to infer the true elevation angle error, and hence E = error in assumed horizon pointing direction.

Impact of Pointing Offset Errors

Over the years we have grown to rely upon a method for determining "system gain" that is based on comparisons of an in situ measurement of "outside air temperature" OAT and the measured counts of the MTP system as it views the horizon and when it views a calibration target of known temperature.  Allowance is made for slight absorptions and reflections of the microwave "transparent" radome.  In order to calculate the impact of a pointing offset error upon gain it is necessary to review the procedure for calculating gain.  The equation for gain is based on concepts that begin with the equation for antenna temperature when viewing the atmosphere at the horizon through a partially absorbing and reflecting radome:

    Ta = TBsky * (1 - L - R) + L * tW + R * tMXR                                                (1)

            where TBsky equals OAT when E = 0,
            L = fractional loss of the window (~0.004),
            R = fractional reflection of the window (~0.006),
            tW is the window temperature [K],
            tMXR is the mixer temperature [K], used to represent the level of thermal radiation reflected by the window back to the horn antenna.

When viewing the calibration target, antenna temperature Ta = tTGT, which assumes that the entire antenna pattern is intercepted by the target.  For any antenna view the counts measured is simply C = a + G * Ta, where "a" is an unimportant constant, and G is the system gain (referenced to the horn antenna aperture).  The equation for G works out to:

    G = (Cs - Cb) / [TBsky * (1 - L - R) + L * tW + R * tMXR - tTGT]                (2)

            where Cs is the horizon view sky counts,
            Cb is the calibration target (base) counts, and
            tTGT is calbration target temperature [K].

We may substitute OAT for TBsky provided that the OAT used is the actual sky brigthness temperature in the direction viewed by the antenna, i.e., when E = 0.  But when E <> 0, we must consider that instead of OAT we should use a TBsky calculated from the following equation:

    TBsky = OAT + LR * Le * sine(E)                                                                 (3)

        where LR = lapse rate [K/km],
        Le = applicable range for the channel in question (Le = 2.1 km at FL = 20 km, averaging Channel 1 and 2; LeCh2/LeCh1 = 0.8, which is close enough to 1.00 to permit use of an average Le).

Uaing equation (3), when E = 0, TBsky = OAT for all LR and Le values.  It is unnecessary to evaluate Le for determining E, provided LR is determined from the slope of "TBsky versus elevation angle" - which may be done using the viewing directions that neighbor the putative horizon view.  This removal of Le as a parameter to be evaluated can be illustrated by rearranging equation (3), associating TBsky with TB,Loc6, and substituting LR = (TBloc5 - TBloc7) / [Le * (sine(+8.6 deg) + sine(-8.6 deg))].  When this is done we get the simple relationship:

    E [degrees] = 17.2 * (d(TB,Loc6 - OAT) / d(TB,Loc5 - TB,Loc7)

    where the "d" represents a derivative,
    TB,Loc6 is TB measured at the putative horizon location 6 (channels 1 and 2 may be averaged),
    OAT is the in situ measurement, and
    TB,Loc5 is TB measured at scan location 5 (+8.6 degrees elevation), and TB,Loc7 is TB measured at scan location 7 (-8.6 degrees elevation).

Since we may assume that the true elevations for scan locations 5 and 7 are offset by same value, to first order we may assume that the above differences in TB values for scan locations 5 and 7 is unaffected by E.  The task for measuring E is straightforward, in concept:  merely correlate "TB,Loc6 - OAT" with "TB,Loc5 - TB,Loc7" and multiply the slope of this correlation by 17.2 to arrive at E [degrees].

This will be demonstrated shortly, but the purpose of this section is to assess the impact of non-zero E upon G.

Suppose E = +1 degree, and tTGT = OAT + 19 K.  When LR = 0, the there is no error in the derived gain (since TBsky = OAT in that case).  But when LR = - 7 [K/km], as is common in the troposphere, TBsky differs from OAT by 0.19 K when flying at 14 km, and 0.26 K at 20 km.  The "TBsky - OAT" error is proportional to E (for E << radian).  Referring to equation (2), flight at 14 km (which can be tropospheric at subtropical latitudes) will produce gain errors of 1.0 % (when tTGT exceeds OAT by 19 K).  It is important to note that the gain error will grow as OAT approaches tTGT.  Using typical values for tTGT, OAT, a flight altitude of 20 km, and an "OAT - tTGT" difference of 20 K, we get:

    Gain Error = -0.18 [%] * E [deg] * LR [K/km]

with a scaling for "OAT - tTGT" that goes as 20 [K] / (OAT - tTGT), and a flight altitude scaling that goes as 1.0 + 0.136 * (Zp - 20 km).

This error can become large when LR departs greatly from isothermal, and when "OAT - tTGT" gets small. For example, if E = +2 degrees, LR = +5 [K/km], Zp = 20 km, and "OAT - tTGT" = 10 K, then the gain error will be -3.6%!

If OAT-based gains have errors as large as 3.6%, which it concievably could at some times during a flight, based on the previous analysis, then what would be the impact on T(z) retrievals.  Consider the high elevation observables, where TB is the most different from tTGT.  If tTGT = 280 K and TB,Ch1,Loc1 = 190 K, there's a 90 K difference, and a 3.6% gain error will lead to a TB,Ch1,Loc1 error of 3.2 K.  The retrieval coefficient for high altitudes typically multiply this observable by a factor of order 0.5, meaning that the retrieved temperature would be in error by 1.6 K.  Since neighboring obserbles would share the gain error, and contribute to a retrieval temperaature error in the same direction (with retrieval coefficients of order 0.4, etc), the final error could be of order 2 or 3 K.

I conclude that it is important to estimate E, or to at least understand the effects that non-zero E values will have upon the final retrieval product.

Estimation of E

In this section I will illustrate some of the difficulties in estimating E for the ER-2.  Earlier work with DC-8 data showed that apparently accurate solutions could be obtained quickly, but I now believe this was because the DC-8 flies at lwoer altitudes where level flight data encounters lapse rates that are typically -7 [K/km], or much different from isothermal.  The ER-2, on the other hand, has level flight at altitdues that are typically isothermal, and any departures from isothermal are often confined to shallow layers, making the assumptions in my analysis suspect.

The first case for illustrating these problems is flight ER000114 (January 14, 2000).  In the following figure I plot the difference "TB,Ch1,Loc6 - OAT" (green trace) and a parameter that is proportional to lapse rate, "TB,Ch1,Loc5 - TB,Ch1,Loc7" (red trace).

Figure 1.  Example of changes in parameter "OATmtp - OATnav" caused by changes in lapse rate, as represented by MTP's "TB1,Loc5 - TB1,Loc7".

If the lapse rate changes occurred over thick layers then there wold be a good correlation between these two traces.  However, early in the flight the red trace undergoes rapid, large variations without any discernible changes in the green trace.  The simplest interpretation os this data is that E = 0.  However, in mid-flight there appears to be a slow, slightly anti-correlated change in both traces.  For both "correlations" to be true it would be necessary to invoke complex T(z) structure.  For example, if at 47.8 ks the ER-2 was flying above a warm layer, and above this warm layer the air was isothermal, then TB,C1,Loc7 could be warmer than TB,Ch1,Loc5 while TB,Ch1,Loc5 and TB,Ch1,Loc6 could have the same value, and small values of E would have negligible effects upon TB,Ch1,Loc6. I haven't checked for evidence of this, but I present the scenario to illustrate a potential pitfall of an incomplete analysis procedure.

The following graph is a redisplay of this data in another form.

Figure 2.  Correlation of OAT discrepancy (defined by "OATmtp,Loc6 - OATnav") with lapse rate (represented by channel-average difference in MTP TB for Loc 5 and Loc7).  The three lines are subjective "eye/hand fits."  The fits have negative slopes, indicating the Loc 6 is below the horizon.

This graph combines channel 1 and 2 data for the lapse rate parameter, TB,Loc5 - TB,Loc7".  This is justified because the two parameters are nealy equal; "TB,Ch2,Loc5 - TB,Ch2,Loc7" is always found to be 80% of "TB,Ch1,Loc5 - TB,Ch1,Loc7", based on correlations of the two parameters, which is in agreement with theory.  The slope of the fitted line (performed by hand) is -0.37 K of OAT difference per K of "TB,Loc5 - TB,Loc7."  Multiplying by 17.2 yields an estiamte for E of -6.4 degrees.  The estimated SE uncertainties range from -8.8 to -4.1 degrees.  This flight exhibits the most steeply sloped correlation, and the most negatively sloped correlation, of any ER-2 flight analyzed so far.

Te following figure is from another flight.

Figure 3.  Same type of plot as in previous figure, except it is for data from another flight.  For this data the slope is positive, instead of negative.

The slope for this correlation is positive, yielding E = +1.3 (range from -0.6 to +2.9) degrees.  This flight exhibits the most positively sloped correlation encountered in the ER-2 analyses, so far.

Analyses have been performed on 7 ER-2 SOLVE flights, and the results for E are shown in the following figure.

Figure 4.  Solutions for Loc 6 elevation angle, for several SOLVE flights (with flight dates presented in YYMMDD format).  The circled data have been carefully quality-checked, by editing-out roll periods, RFI-contaminated noisy data, data for non-level flight, and data at low altitudes.

This is unsatisfying.  Applying formal statistics to these 7 estimates produces the following weighted-average estimate:

        E = -0.96 +/- 0.37 degrees

I am not convinced that this is a good solution for E.  The inexplicable behavior exhibited in Fig.1 is troubling!

A Revealing Flight, ER000127

Figure 5.  Same type of graph as Fig. 1, except for ER000127.

Notice the extreme stability of dOAT (green trace) throughout a period when dTB1 (lapse rate) varies greatly - until 52.3 ks, when the final descent begins.  Early in the flight dOAT changes by no more than 0.15 K while dTB1 changes by 3.0 K; this corresponds to Loc6 Elevation Offset, E < 0.9 degrees.  During the initiation of the final descent, on the other hand, dOAT changes by 5.6 K while dTB1 changes by only 3.5 K, implying E = +27.5 degrees.  We know that such a high value for E is not true, so - whatever caused the big dOAT change starting at 52.3 ks is a factor that MUST be identified, if any of this analysis is going to make sense!

The next figure shows that temporal location of the "dOAT anomaly" in relation to ER-2 altitude.

Figure 6.  Altitude (blue trace) and dOAT and dTB1, during the last half of the flight.

Could the "dOAT anomaly" be explained by a pitch change, that presumably is not corrected by the real-time pointing code?

Figure 7.  Pitch (red trace) and dOAT and altitude for the last hour of flight (including landing).

Pitch changes canNOT account for the dOAT anomaly centered at 52.4 ks!

Could MTP instrument temperatures correlate with this well-defined "dOAT anomaly"?  No, there's nothing in the instrument temperatures that correlate with the dOAT anomaly.

As added confirmation that the proper elevation corrections were applied in real-time, consider the next figure.

Figure 8.  The parameter "ElCor" is recorded in the raw data A-line, and it is plotted here, along with the pitch that it is meant to compensate.  The blue trace is the sum of ElCor and pitch, and it should be zero.

Figure 9.  OATnav, OATmms and OATmtp for the same flight segment.

For some reason the NAV's OAT differs from the MMS OAT (and the MTP OAT) during this "dOAT anomaly" centered at 52.4 ks!  Thus, I shall assume that there is some problem with OATnav shortly after a pitch-down maneuver, that lasts about 220 seconds.  Maybe it's a carburetor heater effect that's not compensated for in the NAV analysis.  If that's the explanation, then the rest of the dOAT versus dTB1 and TB2 correlations are OK.  I will assumes this is the "situation" and proceed with use of the rest of the flight data to obtain a solution for E for this flight.  Doing this yields E = -1.09 +/- 1.44,1.77 degrees for ER000127.

Figure 10.  "Latest" plot of E-solutions for SOLVE.

So far, there are 8 solutions for E from SOLVE flights, 4 of which have been quality checked.  The quality checking doesn't appear to matter, based on the fact that they give the same solution.  Using all 8 E-solutions yields E = -0.96 +/- 0.36 degrees.

Re-Assessment of Importance of Evaluating E

Based on the 8 flights that have already provided solutions for E, it is likely that E is approximately -1 degree.  It is unlikely, though possible, that E is outside the range -2 to 0 degrees.  The most "serious" of these two possibilities is that E = -2.0 degrees.  Before spending more time refining the E-solution it will be prudent to re-assess the importance of determining E with greater accuracy.  In other words, is it sufficient to merely establish that E departs from zero by no more than 2 degrees?

The following analysis is based on two reductions of the same raw data.  First, I assumed E = 0 degrees, and used OATnav to calculate gains, then TBs and then retrieved T(z).  Second, I assumed E = -1.0 degrees, and repeated theabove procedure.  In assuming E = -1.0 degrees, I determined a "horizon sky counts" by interpolating between Loc6 and Loc7 in the appropriate way.  The following figure shows the differences in retrieved T(z) between these two reductions.

Figure 11.  Change in retrieved T(z) profile caused by a Loc6 pointing error of 1 degree.  The four traces correspond to times of with a range of lapse rates: -1.4, +0.4, -4.4, and +0.4 [K/km].

The maximum error occurs near flight level, and it is greatest for conditions of the greatest lapse rate.  Both results are unsurprising.  However, what is surprising, and gratifying, is that the magnitude of the T(z) errors is <0.15 K in all cases.  This is acceptable, as it is less than the uncertainty assigned to the retrievals.

Since it is likely that E does not exceed 2 degrees, it is also unlikely that T(z) will exhibit errors (near flight level) that exceed 0.3 K - assuming lapse rates are less than 4.4 [K/km].  For |LR| = 7 [K/km], T(z) errors of 0.5 K are possible.  Errors of this magnitude are "borderline acceptable."  We should at least keep in mind that any anomalous behavior might be attributable to non-zero E when LR is also non-zero.

It is interesting that T(z) is least affected at the extreme high and low altitudes.  This must occur because at the altitude extremes the observbles have no T(z) information and the retrievals are returning the archive average.

Conclusion

I recommend that the MTP/ER2 for SOLVE be treated as if E = 0, but be mindful of the possibility that during the analysis phase in which gains are determined for each flight, with the goal of obtaining a mission average gain set, "unusual gain behavior" could be attributable to non-zero LR in combination with non-zero E.  The flight average gains will have errors that are greatly reduced from the amounts shown in Fig. 11, due to the range of lapse rates that will approximatley average close to zero over the course of the flight.  Of course, any flight that has LR persistently one side of isothermal is a candidate for producing gains that differ from the other flights.  It would be wise to keep a log of LR characterstics for each flight used in producing a set of gain values, for later mission averaging.

Since errors throughout a flight could amount to values as high as 0.5 K at times of large lapse rates, assuming that E is as large as 2 degrees, and since it is possible that E may differ for the different missions, I think it would be prudent to perform at least a quick study of E effects for each mission, and certainly for each aircraft configuration.  This will provide assurance that large E effects wil not be present in the final T(z) data.

This site opened:  November 12, 2000.  Last Update: November 21, 2000