Bruce L. Gary

September 29, 2000

A proposed warning algorithm for Clear Air Turbulence, CAT, has been evaluated using existing ER-2 and DC-8 flight data.  The algorithm relies on a method for inferring vertical wind shear, VWS, from in situ temperature and wind measurements.  Lapse rate measurements by JPL's Microwave Temperature Profiler, MTP, were combined with VWS to calculate the reciprocal of Richardson Number, RRi.  A CAT "warning" threshold of RRi = 2.5 (i.e., Ri = 0.4) was used to successfully warn of CAT on 12 of 14 occasions, corresponding to a "hit rate" of 86 +/- 8 %.

A theory for CAT generation, the vertical compression hypothesis, was also evaluated.  Only 4 of the 9 cases for which an evaluation could be made support this hypothesis using enhanced static stability (i.e., lapse rate) as an indicator of vertical  compression.  Three of the remaining 5 cases exhibited enhanced VWS, which may also indicate vertical compression.  None of the 13 CAT encounters were consistent with the "breaking wave" mechanism for CAT generation.

Mountain waves appear to have played a role in generating most of these CAT cases since 11 of the 13 were embedded in mountain waves.  One of the non-mountain wave cases was associated with the subtropical jet.

A shortcoming of this study is that the VWS algorithm optimized for use at ER-2 altitudes does not work at DC-8 altitudes.  A major goal of future work will be to optimize a VWS algorithm using DC-8 data, so that RRi-based CAT warning studies can be performed at commercial flight altitudes.

Links in This Web Page
    Underlying Theory
    MTP Accelerometer Data
    MTP Flight Data
    Case Selection
    Case #1: ER911102
    Case #2: ER940528
    Case ER950515
    Case ER970710
    Case #3: ER911014
    Case #4: DC960208
    Case DC971013, Adiabatic CAT Layer
    Case #5: ER890120
    DD, SR and MR Methods for Inferring VWS
    Summary of Cases
    Compression Versus Expansion
    Hit & Miss Rates
    Provisional Warning Algorithm
    Lessons Learned
    Recommendations for Phase II Study


This web site describes work performed for Dryden Flight Research Center, DFRC, having for its objective the evaluation of a CAT warning algorithm.  The algorithm is based on a hypothesis for CAT generation, which I refer to as the "vertical compression CAT generation hypothesis."  The algorithm uses vertical gradient information from two instruments:  1) the Microwave Temperature Profiler, MTP, an airborne instrument built and operated by personnel at the Jet Propulsion Laboratory, and 2) any in situ system for measuring the horizontal components of wind, such as the Meteorology Measurement System, MMS, built and operated by personnel at the NASA Ames Research Center.  The MTP is used for determining the vertical gradient of potential temperature, and the MMS is used to derive the vertical gradient of both wind components.  These two vertical gradients are used to calculate Reciprocal Richardson Number, RRi, and the temporal pattern of RRi is used to issue warnings of upcoming Clear Air Turbulence, CAT.  The algorithm is described in Patent #5,117,689, "Microwave Temperature Profiler for Clear Air Turbulence Prediction," June 2, 1992.

The MTP instrument was developed by JPL for NASA starting in 1978.  To date, MTP instruments have flown on six research aircraft (CV-990, C-141, ER-2, DC-8, WB-57F and the NCAR Electra).  All MTP data taken prior to 1999, as well as the MMS data which accompanied almost all MTP flights, are in the public domain.  Since I have been the principal investigator for the MTP instrument for all NASA sponsored deployments before 1998, I am in an ideal position to make use of it for the present evaluative task.  Indeed, it would be nearly impossible for anyone else to use this archive of MTP data since 1) there have been many format changes over the years, 2) many of the accelerometer files are not included in the public archive, and 3) most "test" flight data, including some CAT encounters, have not been archived and are therefore not in the public domain.

Underlying Theory for Measuring RRi

The warning algorithm is described on the web page Patent#3.  Briefly, it predicts that CAT will be generated when the Reciprocal Richardson Number, RRi, of an atmospheric layer reaches or exceeds the critical value 4, and is maintained in this state for an unspecified time.  The algorithm assumes that CAT patches are surrounded by an annulus of air having RRi => 4, and outside this annulus Ri << 4.  RRi is obtained by combining remote sensing measurements of dT/dZ, using the MTP instrument, and an in situ measurement of dU/dZ and dV/dZ.  The vertical gradients of U and V are derived by a technique of correlating wind components U and V with potential temperature, producing dU/dtheta and dV/dtheta, which are multiplied by dtheta/dZ (available from the MTP's dT/dZ).

The possibility of deriving the vertical gradients of U and V by correlating these wind components with potential temperature, theta, is a crucial concept for this CAT warning procedure.  It rests upon the 1987 MTP finding that isentropes are almost always "wrinkled" at ER-2 altitudes, and the later 1991 MTP finding that isentrope wrinkles are also present at DC-8 altitudes, although with a lower amplitude.  An analysis of the the wrinkle amplitude, or "mesoscale fluctuation amplitude," MFA, shows that there is a dependence of MFA upon altitude, season, latitude and underlying topography.  These results are presented at MFA.  The empirical model fit shows, for example, that at DC-8 altitudes (11.4 km, average) MFA has 61% the value of MFA at ER-2 altitudes (19.4 km, average).   The topography effect is also pronounced, with the ratio of mountainous MFA to ocean MFA being 1.5 at mid-latitudes, and 2.7 at polar latitudes (after averaging out seasonal effects).

It is reasonable to treat isentrope surfaces as proxy indicators of the altitudes of "streamlines" (where clouds are not present, and when flying parallel to the wind) - provided the analyses are restricted to mesoscale distances (where diabatic heating and cooling effects are negligible).  With this interpretation, the omnipresent wrinkled feature of the atmosphere testifies to the existence of an ever-present field of up-and-down motions.  Thus, an airplane that flies at a constant altitude in a wrinkled atmosphere is flying the equivalent of an up-and-down altitude pattern through an unwrinkled atmosphere.  In either scenario the airplane is capable of sampling vertical gradients of any quasi-conserved property.

Potential temperature, theta, is conserved (on mesoscale timescales), and I make the crucial assumption that to first order the components of wind speed are also conserved (on mesoscale spatial scales).  In other words, I assume that the isentrope and isotach surfaces move up-and-down together to produce the wrinkled features noted by the MTP.  If the isotach surfaces do not do this, then the scheme of inferring VWS from an algorithm that correlates U and V with theta would be faulty, and the RRi derived by combining VWS with dT/dZ would also be erroneous.  There seems to be no simple way to determine if the isotachs behave in the assumed manner.  One approach to validating the assumption is to adopt it and see if inferred RRi behaves according to the Kelvin-Helmholtz theory for CAT generation; namely, proceed as if the isotachs move with the isentropes, and determine if RRi always approaches 4 before CAT encounters.  If it does, then the isotach surfaces probably behave in the assumed manner.  In any case, if RRi always predicts CAT, it would represent a pragmatic solution to the long-standing quest to warn of upcoming CAT.

MTP Accelerometer Data

CAT intensity is to be derived from pilot reports and a vertical accelerometer mounted in the various MTP instruments.  The MTP vertical accelerometer was measured at approximately 1 second intervals (the interval ranged from 1.4 second to 0.4 seconds, depending on the mission).  Some of the archived data includes only the peak-to-peak excursion for the vertical accelerometer during the MTP data cycle (for some missions the MTP data cycle was 9 seconds, for others it was as long as 16.5 seconds).  For other missions the entire set of accelerometer readings were recorded in separate files.  A few flights lack accelerometer data, due usually to a loss of data diskettes.

Although 10 to 20 Hz sampling of an accelerometer is generally accepted as essential for inferring CAT severity, some use can be made of the approximately 1 Hz data recorded by the MTP instrument (my funding did not specify CAT studies).  Empirical multiplication factors have been found acceptable for converting the under-sampled accelerometer excursions to corresponding pilot-reported CAT intensity.  Obviously, the flights with records of raw accelerometer counts at 2.6 Hz will require a smaller multiplication factor than the ones having only peak-to-peak excursions of data taken at 0.7 Hz.

MTP Flight Data

The first task for Phase I work was to compile a summary of basic information about the MTP flight data that is available for evaluation using the RRi algorithm.  The MTP data to be surveyed comes from two aircraft:  1) the NASA ER-2 aircraft (the MTP flew on two of them), and 2) NASA's DC-8 aircraft.  Starting in 1987, the Jet Propulsion Laboratory provided scientific support for many atmospheric experiments using JPL-constructed MTP instruments in these aircraft.  The survey includes all flight data before mid-1999 (data after mid-1999 is still not in the public domain).

The following missions were used for selecting CAT events to serve as case studies evaluating the Ri-trend algorithm for predicting CAT encounters.  Another web page includes 6 columns of information for each flight (for all missions):  flight number within the mission, flight date, flight duration, maximum CAT (either reported by the pilot or inferred from the MTP accelerometer data), flight location, and pilot's description of the CAT.  All data are in the "public domain."  Since I was the Principal Investigator for the MTP instrument for all these missions, I am familiar with all aspects of the MTP data to be used in this analysis.

STEP (ER2)             14 flights,   75 hours
AAOE (ER2)             18 flights,  110 hours
AASE (ER2)             19 flights,  124 hours
AASE II (ER2)          30 flights,  182 hours
AASE II (DC8)          18 flights,  135 hours
ASHOE (ER2)            46 flights,  282 hours
STRAT (ER2)            46 flights,  228 hours
AVSAR (DC8)            10 flights,   57 hours
TOTE/VOTE (DC8)        20 flights,  141 hours
SUCCESS (DC8)          18 flights,  100 hours
MACAWS (DC8)           20 flights,   93 hours
PEM Tropics B (DC8)    29 flights,  199 hours
POLARIS (ER2)          31 flights,   83 hours
SONEX (DC8)            16 flights,  111 hours

TOTALS                332 flights  2026 hours

There are a total of 332 flights and 2026 flight hours available for analysis using the MTP-based Ri-trend CAT warning algorithm.  A detailed listing of flights, including each flight's maximum CAT intensity, is presented in the following web page: Flight List Details.

Missing CAT Intensity Reports

At the present time not all flights have entries for estimated maximum CAT intensity.  Of the 204 ER-2 flights, 147 have pilot-reported CAT intensity.  Of the 128 DC-8 flights, only 19 have MTP operator reported CAT intensity (and another 18 have entries for peak-to-peak vertical accelerometer).  Many of the missing ER-2 CAT entries are for ferry flights, which often have mountain wave induced CAT, and this provides some incentive to determine CAT intensity by analyzing accelerometer data.  Since 85% of the DC-8 flights are missing CAT intensities, there is an even greater incentive to analyze accelerometer data for them as well during Phase II.

The procedure for converting vertical accelerometer peak-to-peak excursions (hereafter referred to as Akk) to a "subjective" CAT intensity consists of 4 steps: 1) determine Akk for all flights, 2) create a scatter plot of Akk versus pilot (or observer) reported CAT intensity, 3) fit a line through this scatter plot, and 4) use the equation for the fitted line to convert Akk to subjective CAT intensity.  Because of FY'00 resource limitations this sub task was not performed during the Phase I study being reported here.

Case Selection

Considering only the flights for which pilot reports of CAT are available, moderate or greater CAT was encountered by the ER-2 on 10 flights, and by the DC-8 on 6 flights.  Six of the 10 ER-2 CAT encounters occurred  during altitude changes, and are therefore NOT suitable for determining whether or not an annulus of increasing RRi exists around a CAT patch (at the same altitude as the CAT).  These cases, indicated with a parentheses around the CAT intensity entry, are included because they CAN be used to determine whether or not decreasing Ri exists above and below the CAT patch, since high RRi might surround CAT patches above and below, as well as horizontally.  Knowing this could be useful for predicting encounters with CAT based on the same kind of mesoscale Ri trends that are the subject of study to be addressed by the present work, but is has not been studied during this Phase I investigation.  The 18 CAT encounter flights are listed below:

 #  Inten-   A/C       Location   Pilot report (or observer notes)
     sity  & date

 1    3    ER890120    Norway    "CAT at 67 N; a lot; mod at N end"
 2   (3)   ER910820    CA        "Heavy CAT climbing thru 50 kft.
 3    3    ER911014    AK/CA     "Moderate CAT ~200 mi east of Anchorage."          Case #3
 4    3    ER911102    CA/ME     "Rough air fr UT to NY, sometimes mod."            Case #1
 5    5    ER931022    CA/Can    "Lots of CAT; severe, const 10 min" (No MMS data)
 6   (4)   ER940321    HI        "Real gd turb 10m > TO; mod turb dive"
 7    3    ER940528    NZ/n      "Mod CAT & lt/mod..."                              Case #2
 8   (3)   ER940608    NZ        "Mod CAT 15-20 kft, some..."
 9   (3)   ER950515    CA        "Lt/mod <39N; lt FL 610; FL260 mod."
10   (3)   ER960201    CA/Can    "Mod CAT mid 30s on climb; smooth elsewhere."
11   (3)   ER961204    CA/l      "Mod CAT during dsc mid 30s, 5 kft thick."
12    2    ER970710    Ak/l      "lt/moderate; lt chopFL 510, 410."

The following are DC-8 flights that are known to have encountered moderate or greater CAT.  The CAT intensity entries are either g-units (peak-to-peak) or observer noted subjective intensity.

 1   0.37  DC920212    CA/AK     CAT at 50.95 ks, 31 kft, brief
 2   0.41  DC920320    ME/CA    38 kft & 20 kft, level flt
 3    3    DC951222    HI/CA     "Mod CAT 0958"
 4    3    DC960130    AK/e      "Mod, 37 kft 0300; mod 0332"
 5    4    DC960208    AK/HI     "Mod 3 min 1255; CAT v strong 1500"                Case #4
 6    3    DC960217    HI/s      "Mod 1055, thunderheads below"

The FY'00 work statement calls for the analysis of 3 ER-2 encounters and 1 DC-8 encounter.  My original choice was to study ER-2 entries numbered 4, 7, and 9, and DC-8 entry 5, above.  However, I had to abandon ER-2 flight 9 (for reasons given below), so I chose in their stead ER-2 flight 12.  This flight also had to be abandoned, so I chose ER-2 flight 3, which was satisfactory.

Case #1 Analysis:  ER911102

The description for this flight's data will be especially detailed, as it can serve to show how I settled upon a specific algorithm for deriving RRi.  During the course of this flight's analysis, I developed what I believe is an improved algorithm compared with the one described in the patent.

This flight features a total of 7 CAT encounters.  Three of these occurred during a 40 minute descent/ascent altitude "dip" and one of these was a "moderate" CAT encounter.  There were 4 level flight CAT events, ranging from light/moderate to moderate.  The presence of CAT during the altitude dip affords an opportunity to gain insight into the dynamics that underly the other level flight CAT.

Details of this case are at ER911102.  This web page is unusually long and detailed, as it introduces many concepts and procedures that will be used during the analysis of other flights.  Briefly, this web page describes three methods for correlating the U (and V) horizontal wind components with potential temperature, theta.  With dependencies of U and V upon theta, it is possible to calculate "vertical wind shear," VWS, which then permits the calculation of Reciprocal Richardson Number, RRi.

The three methods for determining the correlation of U (and V) with theta are referred to as the DD method, the SR method and the MR method.  The "single regression" method (SR) produced correlation values (i.e., VWS) that were 70% of those produced by the "difference span" method (DD).  The "multiple regression" method (MR) produced correlation values that were, in turn, 70% of those produced by the SR method (i.e., they were 48% of the DD correlations).  Whereas the SR method allowed theta to absorb the entirety of correlation, leaving none for horizontal gradients, the MR method allowed for the simultaneous correlation of U and V with both theta and horizontal gradient.  I do not completely understand these differences in correlation amplitude, but I suspect they are somehow related to the fact that small "bobbing motions" of mesoscale-sized chunks of air do not preserve their horizontal wind vector in a conserved manner.  Theta will be conserved, so any discrepancies with my simple model, which require that isotachs and isentropes follow the bobbing motions, should be attributable to a breakdown in the isotach assumption.

I have adopted the DD method for correlating U and V with theta because this method produces RRi results that are in very good agreement with theory.  I could just as well have adopted the SR (or MR) method.  If this were done, however, I would have to adopt RRi thresholds for CAT that were 1/2 (and 1/4) of the values I have adopted for use with the DD method.  There is nothing in the present flight's data to give preference to any of these three methods, since they all showed rising RRi before (most) CAT encounters, and they all showed RRi features at specific locations where CAT did not exist (i.e., they had the same specific "false alarms").  If future flights indicate that the DR, or MR, method is superior to the DD method, I may switch to the superior performing method.

Considering only the 4 level flight CAT encounters, which can be scored for forward flight as well as simulated backward flight, we have a "score" of 5 hits and 2 misses for the 7 CAT encounters that can be scored.  One of the 8 possible scoring situations is presently left unscored since it had a 6.7 minute warning time, which may be too long to qualify as a valid warning.  The median warning time for the 7 scored cases is 3.0 minutes, and the average is 2.3 minutes.

The issue of false alarms will be an important issue to address, but I prefer to do this at a later stage of this work.  Flights identified as "smooth" by the pilots should serve as the best data source for evaluating false alarm rates.

The "dip" data provide dramatic support for the "compression hypothesis" for CAT generation.  During the dip's ascent 2 CAT encounters occurred at altitudes that had undergone a dramatic switch from inversion layer during descent to adiabatic during ascent.  This is consistent with the descent flight data occurring during a compression phase, whereas the ascent flight data occurred after CAT had been triggered and converted the former inversion layer to an adiabatic layer.


      Flight Direction     CAT Event      Warning Time
      Forward     "1A"    3.3 Minutes
      Forward     "1B"    3.0 Minutes
      Forward      "2"    0.0 Minutes
      Forward      "7"    0.8 Minutes
      Backwards     "1A"    3.3 Minutes
      Backwards     "1B"    0.0 Minutes
      Backwards      "2"    1.0 Minutes
      Backwards      "7"    6.7 Minutes 

Case #2 Analysis:  ER940528

This flight had a pair of CAT encounters described as "moderate" by the ER-2 pilot.  The two CAT encounters occurred 5.1 minutes apart, and exhibited the same intensity, so they are treated as one "event" for warning purposes.  9177-foot Mount Ruapehu was nearby, and the isentrope surfaces were distorted in a way characteristic of mountain waves.  The CAT events occurred within the mountain wave.

The CAT events were embedded in a region of high RRi.  However, the high RRi was not due to increased static stability, since lapse rate rose only a small amount; the higher RRi was caused by increased vertical wind shear.  This flight's data does not support the compression hypothesis for CAT generation in the way I expect. Warning times for forward and backward flight were found to be 3.3 and 7.3 minutes.

Details of this flight's analysis can be found at ER940528.


    Flight Direction    CAT Event    Warning Time
   Forward    76.37 ks    3.3 minutes
   Backward    76.69 ks    7.3 minutes

Case Analysis:  ER950515 (Not Used)

The pilot reported "light to moderate CAT south of 39 degrees latitude; light at flight level 610 [18.6 km]; moderate at flight level 260 [7.9 km]."  In retrospect, this report is ambiguous, since the only feature in the accelerometer record (during level flight) is at 39.9 degrees latitude.  The "moderate" encounter was during descent, when the method for predicting CAT using RRi trends cannot be evaluated.  Due to these ambiguities this flight is judged unsuitable for the present task.  Details, for anyone wanting to verify my reasons for rejecting the flight, can be found at ER950515.

After performing a few hours of analysis that brought me to the above conclusion I became curious about the mysterious, large accelerometer feature that had un-CAT-like properties, so I spent a couple more hours with the data and eventually determined that I had been fooled by a "porpoise" maneuver performed for the purpose of calibrating the MMS.  I shall leave the above link to that flight data on this web site, as it serves to illustrate the need to remain ever-vigilant about data that doesn't "look right."  This one almost "fooled me"!

Case Analysis:  ER970710 (Not Used)

This flight isn't suitable for this study because the only turbulence during level flight is "light chop at FL510."

Case #3 Analysis:  ER911014

This flight was the subject of an RRi analysis in 1992 (Original ER911014).  That analysis employed the "single regression" (SR) method for deriving vertical wind shear, VWS.  Since the Difference Data (DD) method is used throughout the present study, it will be applied to this flight.  Appropriate portions of the original analysis, plus new analyses that rely upon the DD method for deriving VWS and RRi, are presented in the web page ER911014.

The results of this analysis provide strong support to the vertical compression hypothesis for CAT generation.  The warning times for forward and backward flight of the moderate CAT encounter are 19.8 minutes and 12.3 minutes.  It is possible that such long warning times are useless to pilots, and should instead be categorized as "false" alarms.  This question will be resolved during a Phase II study.

As a side study for this flight, I determined how quickly lapse rate changed at a CAT patch edge.  At precisely the end of a moderate CAT event dT/dZ changed from the sub-adiabatic value -6.1 to an inversion layer value +3.6 [K/km] in a distance of 1.4 +/- 0.4 km.  This corresponds to 6.7 +/- 2.0 seconds of ER-2 flight time.  This may be the first measurement of such an abrupt transition distance for lapse rate changes, and it might be usefully employed as a constraint upon mesoscale models for CAT generation.  For graphs of this event go to Figure 6a and 6b at the ER911014 link, above.


   Flight Direction    CAT Onset    Warning Time
   Forward    69.4 ks    19.8 minutes
   Backward    69.6 ks    12.3 minutes

Case #4 Analysis:  DC960208

The DC-8 flight of February 8, 1996 has two CAT encounters, one "moderate" and the other "very strong" - which I take to mean "moderate to severe."  Details of the RRi analysis are at DC960208.

The moderate/severe CAT encounter is accompanied by a perfectly correlated rise of static stability (from -8 [K/km] to -2 [K/km]), indicating that vertical compression played a role in generating the CAT.

However, when RRi is calculated using the DD method for inferring VWS, high values are produced throughout the tropospheric portion of the flight, and the pattern of high RRi preceding the main CAT event was not present.

After spending many hours with this flight data I have discovered a possible explanation for this unexpected RRi behavior.  This flight does not have the good quality MMS wind and temperature data that I apparently need for inferring VWS and RRi; instead, the wind and temperature data come from the DADS facility instrument, and the data exhibit obvious quantization and time response problems.  I have concluded that an RRi analysis is not possible using the DADS data (for this flight).

The fact that vertical compression is clearly occurring for this DC-8 flight, within tropospheric air, augurs well for finding a method for inferring VWS from the in situ data since enhanced VWS must be present where it is expected.

Case Analysis DC971013 (not used)

This flight exhbits what I believe is "light to moderate" CAT in a region where the DC-8 made small altitude changes that allowed the lapse rate within the CAT layer to be measured with very good accuracy.  Throughout a 400-meter turbulent layer the measured lapse rate (-9.74 +/- 0.01 [K/km]) was equal to the adiabatic value (-9.728 [K/km], latitude 39 degrees, altitude 11.0 km) to within approximately one part in 800!  Moreover, the horizontal wind components also indicated thorough vertical mixing.  Thus, measurements confirm to high accuracy that CAT is able to produce a complete mixing for layers as thick as 400 meters.  Details of this analysis are found at DC971013.

This flight data did not provide an opportunity to evaluate the RRi technique for predicting CAT, nor did the altitdue changes offer a test of the "compression hypothesis for CAT generation," so data from this flight cannot be used to contribute to RRi-based warning statistics.

Case #5 Analysis: ER890120 (Discovery of RRi Error for Other Flights)

During the analysis of this flight I discovered that I have been making an error in calculating the vertical gradient of theta for all the other flights described on this web site.  The ER-2 data have an error that underestimates RRi by a factor of approximately 2; for the DF-8 the error is about 1.5.  It now appears that the Multiple Regression (MR) method for inferring vertical wind shear, VWS, is better than the Data Difference (DD) method.  When the MR method is used for this flight, using the correct equation for dTH/dZg, acceptable CAT warning performance is obtained.  A longer discussion of the DD< SR and MR methods are presented in the next section.

Flight ER890120 had a "moderate" CAT encounter associated with the northern border of Norway, and also shows evidence of being mountian wave related.   Warning times for forward and backward flight are 8.6 and 4.7 minutes, using the MR-based RRi analysis.  Details for this flight, and the error in previous calculations of dTH/dZg, can be found at the link ER890120.


    Flight Direction    CAT Event    Warning Time
   Forward    33.9 ks    8.6 minutes
   Backward    35.4 ks    4.7 minutes

Thoughts About DD, SR and MR Methods for Inferring VWS (and RRi)

The crucial challenge of this study is to use in situ measurments of wind vector to infer the "scalar version of the wind vector's vertical gradient," which I refer to as Vertical Wind Shear (VWS).  I am convinced that the most important part of this challenge is the task of not being influenced by horizontal gradients.  This will be true for any algorithm that attempts to derive vertical gradients from data consisting of a time series in which both gradients exert their effects.

The simplest procedure to implement is to consider data spans, with a length DD seconds, and attribute the entirety of the wind vector change across such a time span to the vertical difference that is also associated with that time span (using theta as a tracer for vertical location).  By sliding the two ends of the time span many times, and averaging with some arbitrary weighting scheme (that rewards data having larger vertical differences, for example), it is possible to infer VWS.  This is the DD Method I have employed for most of the data reported at this web site.  The fact that it works quite well implies that vertical gradient information is indeed captured by this simple procedure.

Although the DD Method does capture VWS information, it also seems to produce values for VWS that are often too high.  By "too high" I mean that the RRi values calculated from the DD Method VWS are sometimes much higher than 4, the theoretical limit of what an atmosphere should be able to sustain.  These high values didn't bother me much, during most of my analyses, because it is possible for high values to be sustained for short times, and the high values usually were limited to short times.  However, this concern led to the derivation of two other methods for inferring VWS that promise to reduce the effect of horizontal gradients.

The Multiple Regression (MR) Method for inferring VWS consists of performing a multiple regression analysis on a "chunk" of the time series data in which a component of the horizontal wind (either U or V) is the dependent variable, and the two independent variables are time (the proxy parameter for horizontal location) and theta (the proxy parameter for vertical location).  This method will properly partition the horizontal and vertical correlations provided a sufficiently long chunk of data is used so that at least one cycle of theta fluctuations is present.  If, for example, only a half cycle of theta fluctuation is experienced, then there would be no way to distinguish between a change in wind component value caused by the varying horizontal location from a change caused by the varying vertical (theta) location.  I have made an empirical determination that the optimum chunk length is approximately 30 seconds (at ER-2 altitudes).

VWS inferred using the MR Method is always smaller than VWS inferred using the DD Method.  This is to be expected, since the MR method will "reject" some correlations of wind with theta as being better attributed to changes in horizontal location.  Assuming horizontal wind fluctuations are uncorrelated with vertical displacements of the air, DD-based VWS will consist of the true VWS plus a random component with some characteristic amplitude.  Therefore, any DD-based VWS, and the RRi that results, might be "useful" in predicting CAT.  However, the MR-based VWS, and its associated RRi, should exhibit greater CAT predictive value.

The Single Regression (SR) Method is intermediate between the DD and MR methods.  SR consists of subjecting a chunk of continuous time series data to a regression analysis in which theta is the only independent variable.  SR has an advantage over DD in that more data is used for each individual VWS determination, but since it involves more calculation there is a slight "burden of proof" that it has superior performance.  The same burden of proof, demonstrating benefits, should be applied to MR, as it requires even more calculation.

I have not spent enough time evaluating the three methods for inferring VWS to defend the MR Method over the other two.  However, given the poor performance of DD with ER890120, and the superior performance of MR, I am inclined to favor MR.  The only other flight data comparisons were made with VWS and RRi values that were too low by a factor two, and these DD values exhibited better agreement with theory than the MR values.  Until now this was the only basis for my preference of DD over MR.  Now that the error in calculating dTH/dZg has been discovered, and it is estimated that the calculated ER-2 values for RRi should be adjusted upward by about a factor two, I expect to prefer the MR Method for inferring VWS.

A description of these methods for inferring VWS from the in situ data, and for converting VWS to RRi using MTP data, is found at RRi Derivation.  I expect to expand that web page in the near future.

I am anxious to continue study of this matter, and I recommend that it be performed as Task #1 for any future work.

Summary of Results to Date

The following table summarizes the results of analyses performed during this Phase I study.


CAT Case # Flight Date CAT Event# within flt. [ks] Time of peak CAT CAT severity Is there a donut of high static stability? Does data support "compression" or "expansion"? Was there RRi Warning? RRi Warning Time [minutes] Was CAT embedded in mtn wave? Comments
      1  ER911102   "1a"  55.9  2  Yes  Comp.  Yes, Yes  3.3, 3.3  Yes  
      2       "   "1b"  56.3  2  No  Neither  Yes, No  3.0, 0.0  Yes  
      3       "   "2"  57.0  1  ?  Neither  No, Yes  0.0, 1.0  Yes  
      4       "   "3"   57.7  1  n.a.  n.a.  n.a.  n.a.  Yes  Dip dsc 18.4 km
      5       "   "4"  58.6  2  n.a.  Comp.  n.a.  n.a.  Yes  Dip asc 16.0 km
      6       "   "5"  58.8  3  n.a  Comp.  n.a  n.a.  Yes  Dip asc 17.2 km
      7       "   "6"  59.2  1  n.a  Comp.  n.a.  n.a.  Yes  Dip asc 18.5 km
      8       "   "7"  61.5  2  No  Neither  Yes, Yes  0.8, 6.7  No (?)  
      9  ER940528   "1a"  76.38  3  Yes  Comp.  Yes, n.a.  3.6, n.a.  Yes  1a & 1b  close together
    10       "   "1b"  76.69  3  No  Comp.  n.a., Yes  n.a., 7.3  Yes  
    11  ER911014    "1"  69.5  4  Yes  Comp.  Yes, Yes  19.8, 12.3  Yes Extreme LR changes
    12  DC960208    "1"  54.2  4  Yes  Comp.       ?        ?  No  Sub-tropical jet related
    13  ER890120    "1"  33.9  3  No  Neither   Yes, Yes   8.6, 4.7  Yes

The CAT severity code is 0 = smooth, 1 = light, 2 = light/moderate, 3 = moderate, 4 = moderate/severe.

The "donut" of high static stability cannot be scored for the altitude dip data (CAT cases 4 through 7), explaining their "n.a." ("not applicable") entries.  If a level flight encounter has high static stability (dtheta/dZp) on one side of a CAT event, and additional CAT following shortly, I scored it as "yes."

I scored the data as supporting compression if it met either of the following conditions:  1) it had a pattern of rising static stability on at least nearby edge, 2) the temperature field went from inverted to sub-adiabatic from before to during CAT (applies to the altitude dip data, and also the ER911014 data), or 3) there was a pattern of high RRi surrounding the CAT.

In deciding whether to accept a warning as legitimate I have accepted all warning times less than 20 minutes.  This may have to be made more restrictive, and I seek guidance from pilots or other operational people on this protocol.

Most cases were associated with mountain waves (11 of 13 cases).  One flight dominates this result, so I don't place much significance on it.

Compression versus Expansion

None of the data support the "expansion" (i.e., "wave breaking") mechanism for CAT production.

Using my criteria for supporting the "compression" mechanism, 9 of of 12 events support the vertical compression mechanism.  However, it is noteworthy that of the 9 cases showing evidence for compression, only 4 show enhanced static stability.  Perhaps it can be argued that the other 5 cases, when VWS increased without a corresponding increase in static stability, are situations in which compression had not taken place.  If this position is taken, then the case for the "vertical compression hypothesis for CAT generation" is weakened, since only half of the 8 cases would then qualify as meeting the requirements of the hypothesis.  Alternatively, half the CAT cases of this study were generated by vertical compression, and the other half were generated by some other process.  This other process, however, is not "wave breaking," since the static stability data rules out this possibility.

Hit, Miss Rates

There are 14 CAT encounters that can be categorized as either "hit" or "miss" using RRi, and 12 of these are a "hit."  (I'm including both backwards and forward cases in this count.)  The hit rate is therefore 86 +/- 8 %, and the miss rate is 14 +/- 8% (the two numbers are complementary for this study due to the procedure for selecting case studies).  The two misses were "light/moderate" and "light."  The number of events is too small for making a final judgment of hit and miss rates, especially considering that only three flights contribute to these statistics.

False alarm rates were not addressed by this analysis.  It should be pursued using flight data that the pilots reported as being "smooth."  This is a recommended task for Phase II work.

Provisional Warning Algorithm

For ER-2 altitudes I propose using the DD-method for deriving vertical wind shear (and RRi).  The difference time of 10 seconds appears to work best (for the one flight where an optimization analysis was conducted).  I have adopted a RRi threshold for issuing a CAT warning of 2.5.  A description of the DD-method is contained at the web page DD Method.

For data at DC-8 flight altitudes I don't know what algorithm to use, since I have not yet shown that the DD method works at these flight altitudes.

Lessons Learned

During the past workmonth of this study I've learned a few lessons.  First, it's difficult to shift from mission to mission since data formats changed every mission.  I've had to write many data analysis programs that I hadn't anticipated for handling the 6 missions of this study - 28 programs for the ER-2 and 12 for the DC-8.  This has been the most time-consuming part of the entire study.  It would have been more efficient to select just one or two missions, and restrict the CAT flight selections to them.  I will be mindful of this in the future.

Second, the air at DC-8 altitudes appears to be different from the air at ER-2 altitudes, and a procedure for inferring VWS for one regime may not work for the other.  Switching from one to the other would be trivial, but determining which two procedures are optimum in their respective altitude regimes will be difficult.  A Doppler lidar would surely simplify this.

Thirdly, good quality wind and temperature data are needed for inferring VWS and RRi, and the DC-8 DADS winds and temperatures are not good enough for this task.

Recommendations for Phase II

In anticipation of completing Phase I studies soon, I shall present a tentative recommendation for a FY'01 Phase II study.  Specifically, I recommend that a Phase II study be conducted.

I recommend that for Task 1 VWS (and RRi) should be recalculated using the MR method.  New warning statistics should be generated, and new hit/miss rates should be determined for the entire data set.  I do not anticipate any significant changes in these statistics since even a wrongly calculated RRi performed well, indicating that "information" does indeed exist in the data for warning of CAT encounters.  This overall conclusion should not change.

More ER-2 cases of moderate CAT should be found by overcoming the current limitation of relying upon pilot reports of CAT and instead using MTP's vertical accelerometer to infer CAT intensity.  The ferry flights are more useful for this study for several reasons, yet most of the ferry flights did not produce pilot reports (that I was able to log) because of the logistics of travel (I had to provide MTP support at the launch site, and was not present at the landing site to attend the pilot's post-flight debriefing).  Therefore, I recommend for Phase II that the MTP vertical accelerometer be systematically "calibrated" against existing pilot reports so that the existence of CAT can be assessed for the remaining flights.  There are 57 ER-2 flights without pilot debrief notes (28%), and these were not considered for analysis during Phase I.  During Phase II they should be brought into the "search net" for CAT study candidates.

The DC-8 has an even smaller fraction of operator recorded CAT intensity.  Of the 128 DC-8 flights a whopping 85% are missing operator records of CAT.  The same procedure of "calibrating" the MTP/DC8 vertical accelerometer against existing operator recordings of CAT intensity should be conducted, so that the remaining 109 flights can become candidates for study.

Therefore, I recommend that for Task 2 (of Phase II) the MTP vertical accelerometer shall be "calibrated" against existing reports of CAT intensity, for both the ER-2 and DC-8, and the remaining flights shall be scored for CAT intensity.  Also, a prioritized list of best candidates for additional study shall be constructed - taking into account the greater efficiency of working with data from as few missions as possible.

It is important to assess whether any of the RRi procedures under evaluation here can be used at DC-8 altitudes.  Today, approximately half of commercial flying is done in the troposphere, so a stratospheric solution for issuing CAT warnings will not translate to a commercial solution.  Therefore, I recommend that for Task 3 (of Phase II) an emphasis be placed on DC-8 flight data, with an exploration of vertical wind shear algorithms, similar to the exploration performed for flight ER911102 of this Phase I study.  Specifically, the single-regression (SR) and multiple-regression (MR) methods shall be evaluated, using various chunk sizes and weighting schemes, in order to identify the optimum algorithm for use in the troposphere.

An interesting idea is to allow the MR technique identify times when horizontal gradients are large, and to discount the DD solutions for those times.  Such a scheme might be useable for all altitudes.  Exploring this MR-inhibiting method, and other "hybrid" procedures, should constitute Task 4 (of Phase II).

False alarm rates have not been studied in Phase I, and they should receive attention during Phase II.  Therefore, I recommend that for Task 5 (of Phase II) a systematic study be conducted of false alarm rates using the CAT warning algorithms found to be optimum for the stratosphere and troposphere.  The result of this study shall consist of "flight hours per false alarm" statistics for both the ER-2 and DC-8.

In consideration of the proposed funding for Phase II, forecast to be for 25% of a workyear, I recommend that for Task 6 (of Phase II) a total of 5 ER-2 flights and 10 DC-8 flights be analyzed for RRi patterns in relation to CAT.  In addition, a complete summary of hit, miss and false alarm rates shall be presented for Phase I and Phase II case studies.


This site opened:  June 21, 2000.  Last Update: February 26, 2002 (2012.04.12)