Remote Sensing of Turbulence:
Radar Activities
FY04 Year-End Report
Submitted by
The National Center For Atmospheric Research
Deliverable 04.7.3.1E4
1
Introduction
In FY04, NCAR was given Technical Direction by the FAA’s Aviation Weather Research
Program Office to perform research related to the detection of atmospheric turbulence by remote
sensing devices. This report covers work performed under task 04.7.3.1, which focused on the
development and testing of the NCAR Turbulence Detection Algorithm (NTDA) for use with
WSR-88D radars.
The report consists of two sections. The first is a draft journal article, “Remote Detection of
Turbulence using Ground-Based Doppler Radars with Application to Aviation Safety”. This
article describes the NCAR Turbulence Detection Algorithm and reports on a number of
verification studies that have demonstrated its skill. The second section is a report from NSSL
describing work performed by Ming Fang and Dick Doviak. The report is entitled, “Separation
of Shear and Turbulence Contributions to Spectrum Width Measured with Weather Radar”.
Remote Detection of Turbulence using Ground-Based Doppler Radars
with Application to Aviation Safety
John K. Williams, Larry Cornman, Danika Gilbert, Steven G. Carson, and Jaimi Yee
National Center for Atmospheric Research
DRAFT—last revised Thursday, October 14, 2004, 6:02 PM
Corresponding author address: Dr. John K. Williams, Research Applications Division, National Center
for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307. Email: [email protected]
Abstract
Real-time remote detection of in-cloud turbulence would provide a valuable new input to decision
support systems that help pilots and air traffic controllers assess weather-related aviation hazards, and in
particular offers the potential to improve safety and air traffic flow during convective events. This
capability may be provided, in part, by a new Doppler radar turbulence detection algorithm designed for
use on the operational NEXRAD and TDWR radars that effectively cover the continental US. The new
fuzzy logic algorithm, developed at NCAR under the auspices of the FAA's Aviation Weather Research
Program, makes use of the radar-measured reflectivity, radial velocity, and spectrum width to perform
data quality control and produce estimates of eddy dissipation rate (EDR), an aircraft-independent
turbulence metric. It is anticipated that the algorithm will eventually be installed on all NEXRAD and
TDWR radars, and that the radar-derived EDRs will be combined with satellite, in situ, and numerical
weather model data to produce a nationwide integrated turbulence detection product.
In addition to describing the new turbulence detection algorithm, the authors present highlights of the
verification process that has demonstrated the algorithm's skill and potential operational utility. This
process has included extensive comparisons of radar-derived EDR estimates with in situ EDR values
obtained from aircraft winds data, including both post-processed data from the NASA Boeing 757's
spring 2002 flight tests and EDRs produced by an automated turbulence reporting system developed by
NCAR and currently operating on a number of commercial aircraft. In addition, analysis of Flight Data
Recorder information provided by the NTSB shows that the NCAR turbulence detection algorithm often
detected hazardous in-cloud turbulence well in advance of the aircraft encounter.
1. INTRODUCTION
Commercial and general aviation aircraft frequently encounter unexpected turbulence that is
hazardous to both aircraft and passengers. For air carriers, turbulence is the leading cause of occupant
injuries, and it occasionally results in severe aircraft damage and fatalities (MCR, 1999). The cost to
airlines due to injuries to flight attendants and passengers, aircraft damage, the need for additional
inspections and maintenance, and associated flight delays is substantial. Moreover, encounters with even
moderate turbulence may reduce passengers’ confidence in airline safety. While clear-air turbulence
forecasts based on numerical weather model data are now routinely generated and possess reasonable skill
for levels above 21,000 ft (Sharman, 2002), a similar system for identifying and disseminating
information about hazardous convectively-induced turbulence remains lacking. This omission is
particularly significant because historical data suggest that over 60% of turbulence-related aircraft
accidents are due to convectively-induced turbulence (Cornman, 1993).
To begin to ameliorate this deficiency, the FAA’s Aviation Weather Research Program has directed
the National Center for Atmospheric Research (NCAR) to develop an improved Doppler radar turbulence
detection capability. The NCAR Turbulence Detection Algorithm (NTDA) makes use of the radar-
measured reflectivity, radial velocity, and spectrum width to produce estimates of eddy dissipation rate
(EDR), an aircraft-independent turbulence metric, along with associated quality control indices, or
confidences. This fuzzy-logic algorithm, which may eventually be installed on the NEXRAD and TDWR
radars that provide coverage of most of the conterminous United States, is expected to be a central
component in a system that will eventually utilize radar, satellite, in situ, and numerical weather model
data to produce a nationwide integrated turbulence detection product.
In this paper, the authors describe an experimental version of the new turbulence detection algorithm
that has been implemented and verified using comparisons between archived NEXRAD data and in situ
data from flight tests, NTSB turbulence encounter cases, and an automated turbulence reporting system
operating on commercial aircraft. The verification process, while not yet comprehensive, suggests that
the turbulence detection algorithm has adequate skill to be of significant operational utility, as will be
shown below.
2. NCAR TURBULENCE DETECTION ALGORITHM
The NCAR turbulence detection algorithm (NTDA) utilizes the first three moments of the Doppler
spectrum—the reflectivity, radial velocity, and spectrum width—to perform data quality control and
produce EDR estimates on the same polar grid as is used for the raw moment data (see Figure 1). Data
quality control is performed by computing a quality control index, or confidence, for each measurement.
For example, the spectrum width confidence computation is based on the signal-to-noise ratio, or SNR
(which for NEXRAD Level II data is inferred from the reflectivity and range from the radar), the value of
the spectrum width, the local variance of the spectrum width field, and image processing techniques
designed to identify known artifacts. The confidence values for each measurement are then propagated
into the EDR computations as weights for local confidence-weighted averaging, and are also used to
determine a confidence value for the resulting EDR. Three distinct methods are used for computing EDR:
a second-moment method, which makes use of the measured spectrum widths; a combined first and
second-moment method, which also makes use of the local variance of the radial velocity; and a structure
function method, which utilizes the radial velocity measurements. The various EDR estimates, along
with their associated confidences, are combined in a fuzzy-logic framework, with a single EDR and
associated confidence produced for each radar measurement location. The structure of the algorithm, as
implemented for NEXRAD Level II data, is diagrammed in Figure 1. For the results presented in this
paper, however, only the output of the second moment module of the algorithm is used.
3. NTDA VERIFICATION
Development, tuning, and verification of the NTDA has been performed over several years using data
from research aircraft (notably the SDSM&T T-28) and Doppler research radars, including the Mile High
radar and CSU CHILL radar. Recently, however, it has become possible to obtain archived NEXRAD
Level II data directly from the National Climate Data Center (NCDC) via a web-based interface, making
it feasible to run the NTDA for any in-cloud turbulence case in which in situ turbulence data are available
for comparison. Sources of high-quality in situ turbulence data include instrumented research aircraft,
flight data recorder information supplied by the National Transportation Safety Board (NTSB), and EDR
values generated by an automated reporting system currently operating on a number of United Airlines
aircraft (Cornman, 1995 and 2004).
3.1 NASA Flight Test Data
In the spring of 2002, a series of eleven flights were performed by the instrumented NASA Langley
B-757 aircraft as part of a successful test of an airborne radar turbulence detection algorithm developed
by NCAR for the NASA Aviation Safety and Security Program’s Turbulence Prediction and Warning
Systems project. The high-rate winds data recorded by the aircraft comprise a dataset that is also ideal for
evaluating the performance of the NTDA, run on archived Level II data from NEXRAD radars along the
flight paths.
The B-757’s 20 Hz vertical winds data were used to estimate eddy dissipation rate (EDR), an
atmospheric turbulence metric, using a single parameter maximum likelihood -5/3 model that assumes a
von Karman energy spectrum form. In particular, a sliding window of width 256 points was used, with
spectral frequency cutoffs set at 0.5 and 5 Hz. This temporal window size corresponds to an along-path
distance of about 3 km at the aircraft’s average cruising speed. In Figure 2, these computed EDR values
are depicted at 30-second intervals along the flight path in for NASA flights 230 and 232, which took
place on April 15 and 30, 2002, respectively. During flight 230, moderate or greater (MoG) turbulence
was experienced over northern and eastern South Carolina and eastern North Carolina, and each of those
encounters was in a region covered by at least three NEXRADs having available archived data. For flight
232, MoG turbulence was encountered over north-central Alabama. Although as many as five
NEXRADs provide coverage of this region, three of these (KGWX, KBMX, and KMXX) had no data
available from the NCDC archives.
Comparisons between the aircraft data and the results from the NTDA running on the archived radar
data were performed by locating the nearest three radar sweeps in space and time to each aircraft location.
These were then utilized in two ways. First, a comprehensive series of overlay plots were generated to
permit comparison between the aircraft EDR and the EDR computed by the NTDA for each sweep. Two
sample plots are depicted in Figure 3 and Figure 4. Although precise collocation and quantitative
matches were not achieved, both plots show that the radar successfully detected hazardous turbulence of
about the right intensity in the region of the aircraft encounter. Moreover, both encounters were in
regions of relatively low reflectivity (< 30 dBZ in Figure 3 and < 15 dBZ in Figure 4) where commercial
aircraft commonly fly. On the other hand, no radar moments data were available near the location of the
smaller MoG turbulence encounter north of the larger encounter in Figure 4; this highlights a limitation of
any turbulence detection algorithm based solely on Doppler weather radar data: it is inherently unable to
measure out-of-cloud turbulence.
A second level of processing was performed to extract the median reflectivity, SNR, and NTDA EDR
values from a disc of radius 2 km around each aircraft location on each “nearby” radar sweep (time within
3 minutes and vertical displacement less than 3 km). The results are displayed in a series of timeseries
plots for each radar depicting the radar-detected and in situ EDRs and reflectivity and SNR timeseries. In
addition, a plot was designed to visualize the EDR values from all nearby sweeps from all appropriate
radars and compare them with the aircraft EDRs. An example of such a “stacked track” plot for NASA
flight 230, 20:07:10-20:12:30 is shown in Figure 5. Note that the four radars that provide coverage of this
turbulence encounter generate similar EDR estimates and that these match well with the co-located in situ
values, providing compelling evidence of the NTDA algorithm’s skill.
The set of timeseries, overlay, and stacked-track plots generated by the analysis described above were
used to score the ability of the NTDA to detect MoG turbulence encountered by the aircraft from 55 flight
segment “events” drawn from the eleven flights of the NASA flight test. A similar scoring exercise
performed using the output of the airborne radar turbulence detection algorithm identified 34 correct
detections, 8 misses, 4 nuisance alerts, and 9 correct nulls (Cornman, 2003), producing a probability of
detection (PoD) of 81% with a nuisance alert rate of 11%. For the NTDA analysis, 15 events had no
available archived NEXRAD data intersecting them. Of the remaining 40, preliminary scoring identified
32 correct detections, 2 misses, 6 nuisance alerts, and no correct nulls, yielding a PoD of 94% and a NAR
of 16%. This analysis suggests that the NTDA may have skill comparable to that of the airborne radar
algorithm for detecting hazardous turbulence, but that more work needs to be done to reduce the number
of nuisance alerts. However, it should also be noted that research flights aimed specifically at
encountering turbulence may not provide a dataset representative of the conditions encountered by
commercial aircraft in an operational environment, and so care must be taken in interpreting these results.
3.2 NTSB Turbulence Encounter Cases
In addition to flight test data, high-rate accelerometer data from flight data recorders (FDRs) provide
information about turbulence that can be used to verify NTDA performance, and the NTSB has provided
FDR data for several turbulence-related accident cases to NCAR for that purpose. These case studies are
especially compelling because they represent accidental encounters that might have been prevented by an
operational turbulence detection capability. Since these accidents are still under investigation by the
NTSB, these data may not yet be released publicly, and hence a full description of the results cannot be
provided here. However, two severe turbulence encounter cases are described based on the times and
locations of the encounters.
The first case occurred on November 17, 2002, at 23:00 UTC as a regional jet was descending near
Rockville, VA (approximately 37:44 N latitude, 77:43 W longitude, and 18,000 ft) en route to
Washington National Airport. Fortunately, all passengers were in their seats in preparation for landing
and none were injured, though the aircraft required extensive inspection. As Figure 6 shows, the NTDA
successfully detected a coherent region of persistent, very strong turbulence—eddy dissipation rates well
above 1.0 m2/3/s—in that region, despite very low reflectivity values between 10 and 15 dBZ there.
Moreover, this diagnosis was available as early as twelve minutes before the encounter, suggesting the
potential efficacy of an NTDA-based tactical turbulence warning system for this case.
A second turbulence encounter case occurred on August 6, 2003 at 20:57 UTC as an Airbus A340 was at
cruise altitude over Walnut Ridge, AR (approximately 36:33 N latitude, 90:42 W longitude, and 31,000
ft) en route to Houston, TX. Figure 7 depicts the NTDA EDR values produced from the four 2.4° radar
sweeps from KPAH immediately preceding the encounter time. In this case, the NTDA detected
hazardous turbulence at the location of the encounter fourteen minutes in advance, and the extent and
magnitude of the turbulence increased over subsequent scans. The radar reflectivity grew from about 10
to 30 dBZ at the location of the encounter during this time, but its magnitude would likely not have
appeared dangerous to the pilots. It appears that an NTDA-based turbulence warning system would have
been capable of providing adequate warning for this case, and thus possibly prevented the 43 minor
injuries, two serious injuries, and minor damage to the aircraft that resulted from the unexpected
turbulence encounter. However, the quickly-evolving nature of convective turbulence illustrated by this
case will require that the latency between the radar measurement and the communication of the
turbulence hazard to the pilot be very small for the warnings to be effective.
3.2 In situ Turbulence Reports
While FDR data cases like those described above provide valuable information on hazardous
turbulence encounters, the number of events is insufficient to draw statistically meaningful conclusions.
On the other hand, the Cornman in situ turbulence algorithm (Cornman, 1995 and 2004) is currently
installed on about 200 United Airlines B-737 and B-757 aircraft, and efforts are underway to deploy it on
additional aircraft types and airlines in the near future. The in situ algorithm provides median and peak
EDR values reported at intervals of one minute or less, thereby supplying a large dataset of objective
turbulence measurements in locations and conditions where aircraft commonly fly. When an automated
method to quality control these data is completed, several hundred flight hours per day of in situ
turbulence information will be available for use in comparing to NTDA-derived values and producing a
comprehensive statistical analysis.
An illustration is provided using a segment from a flight from Chicago to Salt Lake City that began
just after midnight UTC on November 18, 2003. Figure 8 and Figure 9 depict the peak in situ EDR
measurements obtained from the automated reporting algorithm, represented by colors in circles along the
flight path as the aircraft flew from east to west across Iowa and western Nebraska. In Figure 8, the
aircraft track is overlaid on the radar-measured reflectivity at 31,000 ft—the average cruising altitude for
the flight—obtained by merging data from the KLNX, KUEX, KOAX, KDMX, KDVN and KILX
NEXRADs onto a 2 km x 2 km x 2,000 ft grid. Three distinct instances of elevated in situ EDRs may be
observed: 00:11 - 00:12 over east-central Iowa, 00:27 - 00:33 over western Iowa, and 00:46 - 00:48 UTC
over northeastern Nebraska. The valid time of the radar analysis is 00:30, meaning that it used radar
sweeps collected between 00:24 and 00:30. At that time, which coincides with the second turbulence
encounter, the radar-measured reflectivity was less than 10 dBZ along the flight path. This implies that
the cloud would not have been visible on an airborne radar display, although the pilots appear to have
been deviating around a more intense echo further south. The merged confidence-weighted mean NTDA
EDRs shown in Figure 9 show a good match with the beginning of this turbulence encounter, although
the last and most intense part occurred out of cloud and hence no direct EDR measurement was possible.
This case suggests the importance of developing diagnostics for EDR in the vicinity of convection to
augment the direct in-cloud turbulence detection capability.
4. CONCLUSION
A new Doppler radar turbulence detection algorithm, the NTDA, utilizes the radar reflectivity, radial
velocity, and spectrum width data to perform quality control and produce EDR estimates. Initial
verification studies using archived Level II data and in situ data from flight tests, NTSB turbulence
encounter cases, and automatically-reported EDR data from commercial aircraft suggest that the NTDA
has skill in detecting hazardous turbulence and has the potential to be a valuable new input to decision
support systems that help pilots, air traffic controllers, and dispatchers assess weather-related aviation
hazards. In particular, this capability could improve safety, passenger confidence, and air traffic flow
during convective events.
It is anticipated that the NTDA will eventually be implemented on all NEXRAD and TDWR radars
so that the EDRs it produces will be readily available to all potential users for operational or scientific
purposes. In addition, NCAR has requested funding from the FAA’s Aviation Weather Research
Program to develop a real-time turbulence detection product based on the NTDA EDRs and confidences
that will support the unique needs of the aviation community. A web-based product is foreseen that will
provide a nationwide, gridded turbulence diagnosis display for specified flight levels, thereby
supplementing the upper-level turbulence forecasts currently supplied by the Graphical Turbulence
Guidance product on the National Weather Service Aviation Weather Center’s Aviation Digital Data
Service (ADDS). Eventually, the NTDA output will be combined with satellite, in situ, and numerical
weather prediction model data to identify and forecast regions of hazardous turbulence.
5. ACKNOWLEDGEMENTS
The authors wish to thank the National Transportation Safety Board for supplying Flight Data
Recorder information for the case studies described herein, and the NASA Aviation Safety and Security
Program and Langley Research Center for providing aircraft data from the spring, 2002 TPAWS flight
tests.
This research is in response to requirements and funding by the Federal Aviation Administration
(FAA). The views expressed are those of the authors and do not necessarily represent the official policy
or position of the FAA.
References
Cornman, L. B. and B. Carmichael, 1993: Varied research efforts are under way to find means of
avoiding air turbulence. ICAO Journal, 48, 10-15.
Cornman, L. B., C. S. Morse, and G. Cunning, 1995: Real-time estimation of atmospheric turbulence
severity from in-situ aircraft measurements, Journal of Aircraft, 32, 171-177.
Cornman, L. B., J. Williams, G. Meymaris, and B. Chorbajian, 2003; Verification of an Airborne Radar
Turbulence Detection Algorithm. 6th International Symposium on Tropospheric Profiling: Needs and
Technologies, 9-12.
Cornman, L. B., G. Meymaris and M. Limber, 2004: An update on the FAA Aviation Weather Research
Program’s in situ turbulence measurement and reporting system. 11th AMS Conference on Aviation,
Range, and Aerospace Meteorology.
MCR Federal, Inc., “Turbulence Benefits Analysis. Historical Safety Impact of Turbulence.” BR-
7100/010-1, 9 June 1999.
Sharman, R., C. Tebaldi, J. Wolff and G. Wiener, 2002: Results from the NCAR Integrated Turbulence
Forecasting Algorithm (ITFA) for predicting upper level clear-air turbulence. 10th AMS Conference
on Aviation, Range, and Aerospace Meteorology, 351-354.
Figure captions
Figure 1: Diagram of the NTDA, as implemented for the WSR-88D (NEXRAD) radar. The Level II
reflectivity, radial velocity and spectrum width data are used to compute EDR and an associated
confidence for each radar measurement point via a fuzzy-logic framework.
Figure 2: (Top) Flight path for NASA flight 230 on April 15, 2002, depicting EDR values scaled from 0
(blue) to 0.7 m2/3/s (red) at 30-second intervals. NEXRAD radar positions and 220-km range rings are
superimposed, with red indicating that the radar intersected the flight path and the archived Level II data
were available. The aircraft took off from Hampton, VA, and traveled counter-clockwise. (Bottom) A
similar plot depicting a portion of the flight path for NASA flight 232 on April 30, 2002; the flight
direction was again counter-clockwise.
Figure 3: Overlay of in situ EDR values depicted along the aircraft track for NASA flight 230, 19:22:00-
19:29:15, superimposed over the NTDA EDR values from the KLTX 2.4° elevation sweep beginning at
19:25:26. Both EDR values are on the same scale as Figure 2, ranging from 0 (blue) to 0.7 m2/3/s (red).
The labels on the range rings and the axes represent the distance from KLTX, in km. The aircraft is
within about 1 km of the sweep throughout this flight segment, and the radar reflectivity ranges from
about 5-30 dBZ within the turbulent region.
Figure 4: Identical to Figure 3, except for NASA flight 232, 18:54:51-19:01:23, and KFFC 2.4° elevation
sweep beginning at 18:57:51. The aircraft is again within about 1 km of the sweep, and the radar
reflectivity ranges between about 5-15 dBZ in the region where the aircraft track intersects the radar-
detected turbulence “hot spot”.
Figure 5: “Stacked track” plot for NASA flight 230, 20:07:10-20:12:30 depicting the colorscaled
timeseries of aircraft EDRs (“AC”, bottom stripe) and the 2-km disc median NTDA EDRs from the three
nearest sweeps of radars KAKQ, KCAE, KCLX, KFCX, KLTX, KMHX, and KRAX. Gray indicates
that the radar was out of range, whereas white depicts times for which a radar sweep was within range but
contained no usable data. The EDR color scale ranges from 0 to 1 m2/3/s.
Figure 6: NTDA EDR from KAKQ 2.4° sweeps at 22:49, 22:55; 23:06, 23:12 UTC on November 17,
2002, ranging from 12 minutes before to 11 minutes after the severe turbulence encounter described in the
text, which occurred at the location marked by the “X”. Note the unusually large EDR scale, from 0 to
1.85 m2/3/s.
Figure 7: NTDA EDR from KPAH 2.4° sweeps at 20:37, 20:43, 20:49, and 20:54 UTC on August 6,
2003, ranging from 20 minutes to 3 minutes before the severe turbulence encounter described in the text.
The EDR color scale ranges from 0 to 0.7 m2/3/s.
Figure 8: Automated in situ reports of peak EDR over 1-minute segments from a flight from Chicago to
Salt Lake City on November 18, 2003, represented as colored circles scaled from 0 (blue) to 0.7 m2/3/s
(red). The flight track is overlaid on the radar reflectivity at 31,000 ft obtained from merging data from
the KLNX, KUEX, KOAX, KDMX, KDVN, and KILX NEXRADs recorded between 00:24 and 00:30
UTC and gridding them onto a 2 km x 2 km x 2000 ft grid. The reflectivity scale, shown below the plot,
ranges from -10 to 30 dBZ.
Figure 9: Identical to Figure 8 but with the aircraft track overlaid on the NTDA EDRs at 31,000 ft
obtained by performing confidence-weighted averaging of the values recorded by the KLNX, KUEX,
KOAX, KDMX, KDVN, and KILX NEXRADs between 00:24 and 00:30 UTC. Both the in situ and
NTDA-derived EDRs are represented on a color scale from 0 to 0.7 m2/3/s.
Figures
Figure 1: Diagram of the NTDA, as implemented for the WSR-88D (NEXRAD) radar. The Level II
reflectivity, radial velocity and spectrum width data are used to compute EDR and an associated
confidence for each radar measurement point via a fuzzy-logic framework.
Second moment Scaled second-moment
method
Structure function
velocity structure functions fit to theoretical curves
Final product
Turbulence (EDR)
EDR estimation methods
Combined First and second moment
combined method
EDR confidence
DZ reflectivity
VE radial velocity
SW spectrum width
SNR signal-to-noise ratio
VE confidence
SW confidence
WSR-88D Level-II Data
Computed quantities
Figure 2: (Top) Flight path for NASA flight 230 on April 15, 2002, depicting EDR values scaled from 0
(blue) to 0.7 m2/3/s (red) at 30-second intervals. NEXRAD radar positions and 220-km range rings are
superimposed, with red indicating that the radar intersected the flight path and the archived Level II data
were available. The aircraft took off from Hampton, VA, and traveled counter-clockwise. (Bottom) A
similar plot depicting a portion of the flight path for NASA flight 232 on April 30, 2002; the flight
direction was again counter-clockwise.
Figure 3: Overlay of in situ EDR values depicted along the aircraft track for NASA flight 230, 19:22:00-
19:29:15, superimposed over the NTDA EDR values from the KLTX 2.4° elevation sweep beginning at
19:25:26. Both EDR values are on the same scale as Figure 2, ranging from 0 (blue) to 0.7 m2/3/s (red).
The labels on the range rings and the axes represent the distance from KLTX, in km. The aircraft is
within about 1 km of the sweep throughout this flight segment, and the radar reflectivity ranges from
about 5-30 dBZ within the turbulent region.
Figure 4: Identical to Figure 3, except for NASA flight 232, 18:54:51-19:01:23, and KFFC 2.4° elevation
sweep beginning at 18:57:51. The aircraft is again within about 1 km of the sweep, and the radar
reflectivity ranges between about 5-15 dBZ in the region where the aircraft track intersects the radar-
detected turbulence “hot spot”.
Figure 5: “Stacked track” plot for NASA flight 230, 20:07:10-20:12:30 depicting the colorscaled
timeseries of aircraft EDRs (“AC”, bottom stripe) and the 2-km disc median NTDA EDRs from the three
nearest sweeps of radars KAKQ, KCAE, KCLX, KFCX, KLTX, KMHX, and KRAX. Gray indicates
that the radar was out of range, whereas white depicts times for which a radar sweep was within range but
contained no usable data. The EDR color scale ranges from 0 to 1 m2/3/s.
Figure 6: NTDA EDR from KAKQ 2.4° sweeps at 22:49, 22:55; 23:06, 23:12 UTC on November 17,
2002, ranging from 12 minutes before to 11 minutes after the severe turbulence encounter described in the
text, which occurred at the location marked by the “X”. Note the unusually large EDR scale, from 0 to
1.85 m2/3/s.
Figure 7: NTDA EDR from KPAH 2.4° sweeps at 20:37, 20:43, 20:49, and 20:54 UTC on August 6,
2003, ranging from 20 minutes to 3 minutes before the severe turbulence encounter described in the text.
The EDR color scale ranges from 0 to 0.7 m2/3/s.
Figure 8: Automated in situ reports of peak EDR over 1-minute segments from a flight from Chicago to
Salt Lake City on November 18, 2003, represented as colored circles scaled from 0 (blue) to 0.7 m2/3/s
(red). The flight track is overlaid on the radar reflectivity at 31,000 ft obtained from merging data from
the KLNX, KUEX, KOAX, KDMX, KDVN, and KILX NEXRADs recorded between 00:24 and 00:30
UTC and gridding them onto a 2 km x 2 km x 2000 ft grid. The reflectivity scale, shown below the plot,
ranges from -10 to 30 dBZ.
Figure 9: Identical to Figure 8 but with the aircraft track overlaid on the NTDA EDRs at 31,000 ft
obtained by performing confidence-weighted averaging of the values recorded by the KLNX, KUEX,
KOAX, KDMX, KDVN, and KILX NEXRADs between 00:24 and 00:30 UTC. Both the in situ and
NTDA-derived EDRs are represented on a color scale from 0 to 0.7 m2/3/s.
1
SEPARATION OF SHEAR AND TURBULENCE CONTRIBUTIONS TO SPECTRUM WIDTH MEASURED WITH WEATHER RADAR
MING FANG
COOPERATIVE INSTITUTE FOR MESOSCALE METOROLOGICAL RESEARCH
UNIVERSITY OF OKLAHOMA, NORMAN, OKLAHOMA ABSTRACT
Shear and turbulence are two main meteorological contributors to spectrum width
measured by radar. A 6-point scheme of linear surface fitting is designed and applied to a
snowstorm case to separate the shear and turbulence contributions from each other. For the
studied case, shear contributes more to the total measured spectrum width than turbulence.
Based on assumptions of horizontally homogeneous turbulence, and a linear change of the
mean radial velocity across the radar beam, the measured shear contribution is calculated
from Doppler measurements. Simulations were performed based on the wind profile obtained
through VAD techniques. Shear calculations based upon wind profiles obtained from the
VAD analysis, and shear obtained from the 6-point fitting method are consistent. For the
studied case of a snowstorm, the contributions from radial velocity shears in radial and
azimuthal directions are negligible compared to the contribution from radial velocity shear in
the elevation direction. Both wind speed shear and directional shear significantly contribute
to the broadening of the Doppler spectrum and can bias estimates of turbulence.
2
1. Introduction
Radar measures spectrum width, σv, that includes all possible contributions to the
variation of the radial component of wind within the radar’s resolution volume. Shear and
turbulence are two main contributors to the spectrum width. Thus it is important to examine
the fields of shear and turbulence, and derive from them properties that are of interest to
meteorologists to improve our understanding of weather, and to aviators to alert them of
possible hazards to safe flight. To extract these two fields from the observed spectrum width,
Istok and Doviak (1986) proposed an approach, called the 9-point scheme herein, to separate
these two components. They successfully used this approach in their study of tornadic storms.
However, the 9-point scheme is a spatial filter that encompasses a wide vertical dimension at
ranges far from the radar site, and thus may not resolve shears that exist within shallow
layers. Based upon the principles proposed by Istok and Doviak (1986), a higher vertical
resolution 6-point scheme is applied to stratiform weather in which vertical shear is quite
large.
Doviak and Zrnić (1993) relate the shear component of spectrum width to uniform
radial velocity shears in the azimuthal, radial, and elevation directions. A set of equations is
derived that relates the spectrum width due to shear in the three orthogonal directions directly
to the vertical shears of the otherwise horizontally uniform wind speed and direction typically
found in stratiform weather. These equations are used to explain the observed pattern of the
shear component of the spectrum width field, and are also used to estimate the shear
contributions to the spectrum width and thence to derive the intensity of turbulence.
Shear contributions to spectrum widths, observed in stratiform weather, can also be
estimated if the wind profile is available. Vertical profiles of wind can be estimated from
3
radar observations using VAD (Velocity Azimuth Display) techniques. Melnikov and Doviak
(2000) examined the pattern of spectrum width field due to shear assuming a vertical profile
of wind. Here, by using a wind profile derived from a VAD analysis of radar observations,
the spectrum width field due to shear in a snowstorm is calculated and compared with one
whereby the spectrum width due to shear is separated, using the 6-point scheme, from the
observed spectrum width σv.
4
2. Separation of spectrum width due to shear and turbulence
a. The 6-point Separation Scheme
The total spectrum width σv comprises all possible contributions (Doviak and Zrnić,
1993, Section 5.3) to changes of radial velocity within the resolution volume. To separate
these two principal contributions to the total spectrum width a 6-point scheme is introduced
and described herein.
With the 6-point scheme, 6 radial velocities are obtained at three consecutive azimuths
and two elevations at the same range (Fig. 1) and a linear radial velocity field model is least
squares fitted to the data. The matrix, (Eq. (2) given by Istok and Doviak, 1986), is applied to
determine shears using data at six contiguous points. The advantage of the 6-point least
squares fitting (LSF) scheme is that it has better vertical resolution than the 9-point scheme
used by Istok and Doviak. The origin of the matrix is at the asterisk in Fig. 1. In the 6-point
scheme, however, the radar does not directly measure total spectral width at the origin. So,
the total spectral width, that would have been measured had the beam been pointed at the
origin, has to be estimated before the shear and turbulence contributions can be separated.
An exact expression might be obtained under some assumptions, but to simplify the
problem, the total spectrum width for a beam pointed at the origin of 6-point scheme is
estimated using the following simple interpolation formula,
( ) ( ) ( )2
,,
2, 001010
0
θφσθφσθθφσσ vvvv
+=⎟⎠⎞
⎜⎝⎛ +≡∗
where σv(*) is the estimated observed spectrum width at the origin (*), and ( )11,θφσ v the
5
Fig. 1. The 6-point scheme. The asterisk locates the origin to which vr(0) (the
fitted radial velocity at the origin), Kφ (the fitted shear in azimuth direction)
and Kθ (the fitted shear in elevation direction) are referenced.
φ-1 φ0 φ+1
θ1 + + + (θ1+θ0)/2 *
θ0 + + +
6
spectrum width measured by the upper beam at ( )11,θφ , ( )01,θφσ v the spectrum width
measured by the lower beam at ( )01,θφ .
b. Separation of σs and σt in a Snowstorm
The 6-point separation scheme described in the previous section is applied to a case of
a snowstorm. Fig. 2a shows a spectrum width field of a snowstorm that was recorded by the
KLSX WSR88-D Doppler radar in Saint Louis, Missouri. The data that are presented here are
focused on a time when the area of radar detectable snow almost reaches its maximum size,
and is roughly uniformly distributed around the radar; thus a large area of data are available
for analysis. The total spectrum width presented in Fig. 2a was separated into its two
principal constituents, σs, σt,.
Fig. 2b shows the shear contribution, calculated using the 6-point scheme, to the
observed spectrum width. The corresponding contribution due to turbulence is given in Fig.
2c. The generation of those two displays needs velocity data at two consecutive elevations.
The higher the elevation is, the less the available data there are. Therefore the available data
in Figs. 2b and 2c are less than that in Fig. 2a. Because the field of spectrum width due to
shear is computed using the 6-point fitting scheme, the influence from scales less than one
grid length in the elevation direction (in section 4 we will see that the shear in elevation
direction is the main contributor to measured spectrum width) is filtered out. The pattern of
large σs (Fig. 2b) appears like two commas embracing each other. These large values (e.g., σs
> 4 m s-1) in the σs field are located at heights between 430 and 1630m.
7
Fig. 2a The spectrum width field for a snowstorm, 19:36:59, 16 Jan.
1994, Saint Louis, Missouri.
8
Fig. 2b The spectrum width from shear, σs, field obtained using 6-point
scheme.
9
Fig. 2c The spectrum width from turbulence, σt , field obtained using a
6-point scheme.
10
Table 1 lists, samples of σv , σs, and σt obtained using the 6-point scheme. For this
example, we have chosen a direction and range to coincide with regions having large
observed spectrum width values free from overlaid echoes. The 6-point scheme, with data
from elevation angles 0.5o and 1.4o, is used in deriving the data fields shown in Figs. 2b and
2c. Again, the total spectrum width for the 6-point scheme at the center of fitting surface is
the arithmetical average of values obtained at these two elevation angles.
The ‘Xs’ in Table 1 simply indicate σs>σv. There are four practical reasons for such a
result.
1) Istok and Doviak (1986) ascribed those cases (i.e., σs >σv) to the statistical
uncertainties in the estimate of shear and σv.
2) Real σs values could be larger in the calculated σs field, especially if σv is primarily
due to shear, and thus the calculated σs could be larger than the σv.
3) The gradient of the reflectivity field within the radar beam could bias the
measurement of radial velocity which in turn biases the estimation of σv (e.g.,
suppose the radial velocity and reflectivity increase with height within the upper
beam—then the radial velocity measured by upper beam will be overestimated and
the shear, calculated using 6-point scheme, will be overestimated, which in turn leads
to a negative σt2.
4) As mentioned previously, the estimated observed spectrum width associated with
6-point scheme at 0.9o is obtained by averaging those at 0.5o and 1.4o elevation
angles. Thus the total spectrum width could be underestimated due to beam blockage
at 0.5o, which may lead to the values of σv associated with 6-point scheme to be
smaller than it would have been without beam blockage.
11
Table 1. Values of σv, σs, and σt at different ranges obtained with 6-point
scheme sampled at the azimuth of 90.2 degree in a snowstorm.
Range
(km)
σv
(m s-1)
σs
(m s-1)
σt
(m s-1)
40.125 5.250 7.081 x
40.375 5.500 6.726 x
40.625 4.500 6.965 x
40.875 5.000 6.885 x
41.125 5.000 7.435 x
41.375 4.750 7.025 x
41.625 5.250 7.377 x
41.875 4.750 7.183 x
42.125 4.750 6.505 x
Table 2 tabulates the percentage of negative σt2 out of the total number of non-zero
samples of σv. The results show a large percentage of data, separated with the 6-point
scheme, have negative values for σ t2 (i.e., imaginary σt). The percent of imaginary σt does
not change very much with elevation below 3.8o. Above that elevation angle, the reduced
ratios may be ascribed to the coarser resolution due to the increased elevation increment (i.e.,
the shear is underestimated). Because of errors in receiver noise measurements during the
calibration of the radar, negative values of the square of observed spectrum width can occur,
especially in regions where real spectrum widths and signal to noise ratios are small
(Melnikov and Doviak, 2002); these negative values of σv are assigned, by the WSR-88D
12
signal processor, a zero value, and are not included in our separation scheme. Thus the
percent of imaginary σt would even be larger than shown in Table 2. The large percent of
imaginary values in the σt field, at least in this case, is likely due to the fact that σv is
primarily contributed by σs, and statistical uncertainty in the measurements will cause
overestimates of σs with concomitant underestimates of σt ,and consequently some negative
estimates for σ t2 .
Table 2. Percentage of the number of negative σt out of
the number of non-zero σv
6-point scheme
Elevation (θ1+θ0)/2 Percentage
0.9 31.069
1.9 26.486
3.0 29.475
3.8 26.088
5.1 15.712
7.9 10.554
12.2 10.054
13
In Fig. 2c, there exist two areas of large σt values, one to the northeast and another in
the southwest. These areas of large values σv values are not necessarily related to turbulence.
To explain these bands refer to Fig.3 and keep in mind that σt is obtained by taking the square
root of the difference σ σv s2 2− where σs is obtained using the observed velocity field at the
two lowest elevation angles (i.e., 0.5o and 1.4o). Fig.3 shows an assumed vertical profile of vr,
ss
Fig.3 The artificial radial velocity profile used to explain how the 6-point
scheme can underestimates the shear contribution.
and radar beams at the lowest two elevation angels. If vr is homogeneous in the azimuthal
direction Kφ will be zero. Assuming reflectivity is uniform with height, the measured radial
velocity will be the same at the two elevation angles, and thus the Kθ values calculated
through surface fitting will be zero. So, the calculated σs at this location will be zero although
Vr profile
Beam axis
1.4o Beam axis 0.5o
14
it is not. The observed σv would not be zero at either elevation. The σv at elevation of 0.9o,
which is an arithmetic average of σv at 0.5o and 1.4o, will not be zero either. After σt and σs are
separated, all σv are ascribed to σt due to the underestimated σs. The areas of large σt values,
seen in Fig. 2c, is caused by a situation similar to the described by Fig. 3 as we will see in
later discussions.
3. Relating shear contributions to profiles of horizontally uniform wind fields
Doviak and Zrnić (1993) showed that, if the wind is linear within the radar’s
resolution volume, the spectrum width due to shear may be separated into the following three
contributions:
σs2=σsr
2+σsφ2+σsθ
2 (1)
where σs is the total spectrum width due to shear; σsr, σsφ, σsθ are the spectrum widths due to
shear in the radial, azimuth and elevation directions respectively (in a spherical coordinate
system), where
σsr2=(roKrσr)
2 (2)
σsφ2=(roKφσφ)
2 (3)
σsθ2=(roKθσθ)2 (4)
where Kr, Kφ and Kθ are shears in radial, azimuth and elevation directions respectively, σθ2
and σφ2 are defined as the second central moments of the two-way antenna power pattern in
the indicated direction, σr2 the second central moment of the range weighting function, and
ro is the range from radar to the center of resolution volume. For a circular symmetric beam
σθ2=σφ
2=θ12/(16ln2)
15
where θ1 is radar beam width at the –3 dB level. For a rectangular transmitted pulse and a
receiver with a matched Gaussian shaped response,
σr2=(0.35cτ/2)2
where c is light speed in the air and τ is the radar pulse width.
Assuming reflectivity is uniform within the resolution volume and that the wind field
is horizontally uniform, the mean radial velocity observed by the radar can be written as
vr=Vhcos(φ-φw)cosθ+Vpsinθ (5)
or
vr=Vcosφ cosθ+Usinφ cosθ+Vpsinθ (6)
where vr is radial velocity observed with the radar; Vh the horizontal wind speed; Vp the fall
speed of precipitation particles (snow for the case analyzed herein), φw is the azimuth of the
wind (i.e., the direction from the radar that the wind is blowing); φ is the azimuth of the radar
beam; θ is the elevation angle of the radar beam, and U and V are eastward and northward
components of the horizontal wind respectively (these are assumed to be functions of height
z). Here, vr is positive away from radar and Vp is negative downwards. Fig. 4 illustrates the
relationship among these variables.
By neglecting the effect of fall velocity of the snowflakes, Eq. (5) can be rewritten as
)cos()cos( θφφ whr Vv −= (7)
Based on the definition, Kθ may be written in the form
θθ ∂
∂=r
vK r .
where r is the slant range from radar to the resolution volume. From Eq. (7), Kθ can be
written as
)cos()sin()cos()cos( θθ
φφφθφφθθ ∂
∂−+−
∂∂
=r
Vr
VK w
whwh
16
)sin()cos( θφφ wh
r
V−− .
Fig. 4 The relationship among variables.
Because the wind profiles are expressed in terms of height z above ground, it is convenient to
express Kθ in terms of the vertical gradients of the wind field using the following equation
Vr N Vp
φw hV hV Vp
θ
φ
17
)(cos)sin()(cos)cos( 22 θφφφθφφθ whw
wh V
zz
VK −
∂∂+−
∂∂=
)sin()cos( θφφ wh
r
V −− (8)
Similarly, we can obtain the expressions for Kr and Kφ. They are
)sin()cos()cos( θθφφ wh
r z
VK −
∂∂=
)sin()cos()sin( θθφφφw
wh z
V −∂
∂+ (9)
( ) ( )
( )( )r
V
r
VK whwh φφ
θθφφ
φ−−=−−= sin
cos
cossin (10).
Through Eqs. (1)-(4), and (8)-(10), the shear contribution to spectrum width is directly related
to the vertical profile of the horizontal wind vector.
4. Simulation of spectrum width due to shear
Using an assumed vertical profile of wind, Melnikov and Doviak (2002) examined the
pattern of spectrum width field due to shear. They concluded that two bands of enhanced
spectrum width, looking like two commas embracing each other, can result if wind veers with
height. They also concluded that the pattern of spectrum width due to isotropic turbulence is
roughly a circle. Here an algorithm is designed to simulate the spectrum width field due to
shear by using an actual wind profile generated from a VAD analysis.
The computation of σs requires, as shown in Eqs (2)-(4), radial velocity shears in
elevation, azimuth and radial directions. These shears are computed using a LSF method as
described in section 2. But now we shall use the vertical profile of a horizontally uniform
wind field, obtained through a VAD analysis, to compute throughout the observation domain
a so-called simulated spectrum width field σs due to shear. Because a horizontally uniform
18
wind field is assumed for the VAD analysis, some non-uniform features that could have been
exhibited in the separated field will not appear in the simulated field. The vertical resolution
of LSF method depends on the beam width and the range of fitted surface, whereas the
vertical resolution of the wind profile generated through VAD is at a fixed increment of 300
meters.
The VAD analysis is applied to a snowstorm in which the reflectivity is relatively
uniform, and the derived wind profile should not be severely biased by reflectivity gradients.
Simulations cannot only verify the algorithm used to separate σs and σt, but also allow us to
respectively investigate the effects of vertical shears of wind speed and direction.
a. The profile of the wind speed and direction
To simulate the spectrum width field σs due to the horizontally averaged shear a wind
profile is required. The wind obtained from regular daily soundings 12 hours apart is too
coarse to represent reliable winds about the radar site. Furthermore, the sounding does not
necessarily represent an accurate horizontal average of the wind field because the sounding is
along a single curve that the balloon follows. Thus the simulation performed with daily
sounding data may not give accurate results. The best choice is a direct retrieval of the wind
profile by using the radial velocity field obtained by radar itself. Although the profile
obtained through the VAD technique may not apply everywhere within the range covered by
radar, it should be more representative than that obtained from the daily balloon soundings.
Since the radial velocity field and spectrum width field are observed at same time, the errors
in simulation introduced by non-synchronic observations of velocity and spectrum width field
is reduced to a minimum by making use of VAD profiles.
19
Assuming the wind is uniform about the radar site, the mean radial velocity observed
by the radar can be written as Eq. (5) or (6). Browning and Wexler (1968) showed that vr can
be expressed in terms of a Fourier series
( ) ( )[ ]∑∞
=++=
10 sincos
2
1
nininr nbnaav φφ (11)
For uniform wind,
a0=m
2 ∑=
m
i 1
Vri =Vpsinθ (12)
a1=m
2 ∑=
m
i 1
Vricosφi=Vcosθ (13)
b1=m
2 ∑=
m
i 1
Vrisinφi=Ucosθ (14)
Where m is the total number of uniformly spaced samples; V and U are, as noted earlier, the
north and east component of horizontal velocity. The WSR-88D records the radial velocity at
a spacing about 1o during each scan, but, it should be noted that samples are not equally
spaced. Furthermore, because of the reasons presented by Ming (2003), some data might not
be available along the circle on which the VAD is performed. An algorithm, based on the
least-square fitting proposed by Rabin and Zrnić (1980), is designed to compute the a0, a1 and
b1 even though the data are not uniformly spaced, and even if some data are missing. The
wind direction and radial velocity are then computed using Eq. (11)-(14).
By taking into account errors due to the non uniformities in the fall speed of
precipitation, height measurement errors, and inhomogeneities of reflectivity, Browning and
Wexler (1968) recommend:
1) The elevation angel of radar beam should less than 27o in snow and 9o in rain, and
20
2) the radius of the circle on which VAD is performed should be less than 20 km from
radar, to keep the error of VAD derived horizontal wind speed due to each of
mentioned error sources less than 1 m s-1.
Based on these constrains, a set of heights at which VAD analysis is performed are chosen
for the studied case. The lowest height chosen is 20 meters above ground which is about the
height above ground of the beam center at the lowest elevation angle, 0.4o, and the nearest
range (≈ 3km) where data are used in the VAD analysis. The height increments are 300
meters, about the resolution of radar beam at a range of 17 km.
The horizontal wind speed and direction profiles obtained through the VAD analysis
of data from a snowstorm are presented in Figs.5a and 5b. At the 20-meter height, the wind
derived through VAD is unreliable due to the likely influence of ground clutter at near ranges
to the radar. Thus the wind speed and direction profiles below 320 m are obtained by
interpolating the observed values at the surface, recorded at the Saint Louis international
airport at 19:29 UTC, and the VAD derived values at 320 m. The top elevation angle of the
volume scan for this case at chosen time is 19.5o, well within the 27o elevation angle
constraint suggested by Browning and Wexler (1968).
The maximum height of the profiles is about 6 kilometers. From these profiles one
sees that there exists a strong shear of horizontal wind speed below 1.2 kilometers. Horizontal
wind speed changes from 3.0 m s-1 at 20 meters to 32.6 m s-1 at 1220 meters above the
ground. Also, within this layer, the horizontal wind direction changes significantly. At the 20-
meter height the wind blows towards 293o (i.e. SE wind), but the wind blows towards to 41.3o
(i.e. SW wind) at 1220 meters. The rate of wind direction change is about 90 degree/km, and
the wind veers with height.
21
Fig.5a. The horizontal wind speed profile derived from a VAD
analysis of Doppler data having the spectrum width field shown in
Fig. 2.
22
Fig.5b. Changes of the wind direction with height. Zero (or 360) degree
is the wind direction (i.e., direction of air motion) towards north.
Positive angles are those measured clockwise from north while negative
angles are those measured anti-clockwise.
23
b. Computation of σs
If the radar beam is sufficiently narrow and wind is approximately linear across the
resolution volume, then shears can be computed from the wind profile at the heights
intersected by the beam. Because the beam has a finite width, and shear and reflectivity are
not uniform across it, the accurate computation of the shear contribution to spectrum width
requires a reflectivity-weighted integration of the velocity field (principally over the
resolution volume) to compute the Doppler spectrum following the procedure given by
Doviak and Zrnić (1993, section 5.2). But the reflectivity field within the beam is not known.
If the reflectivity is roughly horizontally uniform, one could, as with the velocity field, obtain
a vertical profile of reflectivity from the VAD scans and use it in the integration. However we
simply use the VAD derived wind profile that, under the constraints imposed by Browning
and Wexler, should have errors less than about 1 m s-1.
Using Eq.(1), we calculate the spectrum width associated with shear for each
resolution volume. But to calculate the spectrum width due to shear, we first need to calculate
the shear components φθ KK , and Kr, using Eqs. (8)-(10) and the wind profile obtained from
the VAD analysis. The wind speed and direction at beam top (T) and bottom (B) (i.e., at the
3 dB points; Fig. 6) are determined by interpolating the wind speed and direction between
levels 3 and 4, and between levels 1 and 2, where levels 1 to 4 are those heights at which the
VAD derived wind speed and direction are available. Next we compute the wind speed and
direction at ‘C’ by linearly interpolating the wind speed and direction between T and B. The
wind speed shear and the rate of change of wind direction with height are computed using the
differences of speed and direction between beam top (T) and bottom (B) obtained in first
step divided by the vertical distance between T and B. Such a two-step interpolation
smoothes out some detail variations of wind, but it is an approximation that gives σs fields
24
Fig.6 The origin ‘O’ is the radar location; C is the center of the beam axis; T
is the location of the beam top; B the location of the beam bottom. Levels 1 to
4 are the heights where VAD are available. θ1 is the beam width (1o for WSR-
88D), and ZT, ZC and ZB are heights at T, C and B.
Level4 T ZT Level3 beam axis C ZC
r
Level2 B ZB
Level1 θ1 θ O
25
consistent with the one separated from the observed spectrum width σv using the 6-point
scheme as will be shown in the next section. The finite difference forms of Eqs. (8)-(10)
used to calculate the shears φθ KK , and Kr are:
( ) ( ) ( )[ ] θφφθ2coscos Cw
BT
BhTh ZZZ
ZVZVK −
−−=
( ) ( ) ( ) ( )[ ] θφφφφ 2cossin CwBT
BwTwCh Z
ZZ
ZZZV −
−−
+
( ) ( )[ ] θφφ sincos Cw
Ch Zr
ZV −−
( ) ( ) ( )[ ] θθφφ sincoscos CwBT
BhThr Z
ZZ
ZVZVK −
−−=
( ) ( ) ( ) ( )[ ] θθφφφφsincossin Cw
BT
BwTwCh Z
ZZ
ZZZV −
−−
+
( ) ( )[ ]
r
ZZVK CwCh φφ
φ−−= sin
c. Comparison of separated and simulated fields
The simulated spectrum width field due to shear, computed using the shears obtained
from the VAD wind profile, is displayed in Fig. 7 for the interpolated elevation angle of 0.9o.
The corresponding field, separated from the observed spectrum width with the 6-point
scheme, is presented in Fig. 2b. Because the horizontal velocity field is assumed to be
homogeneous, the simulated field symmetrically distributes around origin----the radar site.
By comparing these two figures, one finds that:
26
1) The shapes of the two patterns roughly agree.
2) The locations of large values (i.e., σs>3 m s-1) of σs are consistent with each other.
3) There are more values larger than 7 m s-1 in the simulated field (Fig. 7) than in the
separated field (Fig. 2b) (this is due to the fact that the simulated field has a
higher vertical resolution than 6-point scheme at ranges larger than 17 km).
4) Beyond 80 km, the pattern of simulated field differs considerably from the
observed one (Again this could be ascribed to the coarser vertical resolution of 6-
point scheme).
In conclusion, the spectrum width field due to shear, derived from a VAD analysis of the
wind field, strongly supports the spectrum width field derived from application of the 6-point
filter. This latter method, although having the possibility of errors due to the poorer vertical
resolution at longer ranges, could better represent the horizontal dependence of the shear
contributions to spectrum width
27
Fig. 7 The spectrum width field due to shear obtained using wind fields
derived from a VAD analysis at an elevation angle of 0.9o.
28
d. Relating patterns to the wind profile
The very good consistency between simulated and separated fields implies that
equations of (8)-(10), on which the simulation was based, can be used to explain the patterns
of spectrum width due to shear shown in Fig. 2b. The contribution from radial velocity shear
in elevation direction dominates the contributions from shears in the azimuthal and radial
directions as will be shown in section 5. Furthermore, only the first two terms in the right
hand side of Eq. (8) are significant (section 5). So, the two bands of large values in Figs. 2b
(and Fig. 7) are associated with the large vertical shear of wind speed and direction. Also,
large wind speed itself might play an important role as indicated by the second term in the
right hand side of Eq. (8). The large values of σs (e.g., σs > 4 m s-1) in Fig. 2b are located at
heights between 430 and 1630m where the largest speed and directional shears are located
(Fig. 5a, and b). Above 1500 m, both speed and directional shears are small (although speed
itself is large), which causes a relatively lower contribution from shear. The change of wind
direction with height is also responsible for the displayed spiral signature in Fig. 2b (and Fig.
7) as was pointed out by Melnikov and Doviak (2002).
5. Contributions of shears
a. Contributions of shears in radial and azimuth directions
Equation (1) shows that the spectrum width due to shear is related to radial velocity
shears in the radial, azimuth and elevation directions. However, they are not equally
contributing to σs. Table 3 tabulates some values, obtained using 6-point scheme, of Kθ, Kφ
and Kr at an azimuth angel of 90.2o at ranges where the observed spectrum width values are
large and free from overlaid echoes. Listed values show that Kθ is usually one order
29
magnitude larger than Kφ while Kr is negligible. Therefore the contributions from shears in
radial and azimuth directions can be neglected in comparison to the contribution from shear
in the elevation direction.
Table 3. Values of Kθ, Kφ and Kr at different ranges in snowstorm
(Azimuth = 90.2o)
Range(km) Kθ Kφ Kr
40.125 6.080 0.113 0.000
40.375 6.725 0.074 0.000
40.625 6.960 0.266 0.000
40.875 6.881 0.227 0.000
41.125 6.434 0.151 0.000
41.375 6.017 0.335 0.000
41.625 6.377 0.075 0.000
41.875 6.182 0.113 0.000
42.125 6.503 0.151 0.000
b. Contribution of shear in the elevation direction
Because Kφ, and Kr can be neglected, shear in elevation direction is the only
remaining significant contributor. From Eq. (8), Kθ consists of three terms. Simulation (not
presented) shows that contribution from third term on the right hand side of Eq. (8) is less
than 1x10-3 s-1, which can be verified by substituting Vh=30 ms-1, r=3 km (generally, r is
larger than 3 km), φ=φw and θ = 0.4o into that term. Therefore contribution from term 3 can
also be ignored.
Thus the most significant contributors are the first and second terms of Eq. (8). The
first term depends on the vertical shear of the horizontal wind speed, and second term
30
depends on the vertical shear of wind direction, and the horizontal wind speed. Figs.8a and 8b
display the two fields respectively generated by term 1 and term 2 in which the wind profile
presented in Figs. 5a and 5b is used. Two kidney shaped patterns of large values are shown in
Fig. 8b whereas in Fig.8a there is one, but values of σs there exceed 7 m s-1, and are much
larger than those seen in Fig. 8b. Because term 1 depends on the vertical shear of horizontal
wind speed, large maximum values in narrow bands imply strong vertical shear in a relatively
narrow layer. Another interesting feature in Fig.8a is that the two kidney shaped bands within
80 km show a spiral signature around radar site while the four bands (i.e., one at about105
km, and the other at about 120 km) beyond 80 km do not. The spiral feature is related to the
cos(φ-φw ) factor in term 1 (i.e., the changes of wind direction with height). As can be seen
from Fig. 5b, wind direction does not change significantly at the heights where the four bands
of enhanced σv are seen and thus the spiral feature is absent. On the other hand term 2
produces two wider kidney shaped patterns of relatively large values, and two maximum
cores, but its maximum values are less than that determined by term 1. The centers of these
peaks are located about 45 km and 75 km respectively and lie in the vicinity of the strong
shear layer. They should coincide with the peaks of the product of wind speed and wind shear
that are part of term 2. Since term 1 and term 2 include factor of cos(φ-φw) and sin (φ-φw)
respectively, the pattern associated with term 2 is rotated clockwise 90o relative to the pattern
associated with term 1.
31
Fig.8a The σs field generated by the first term of Eq. (8) using the wind profile
presented in Fig.5a, b, and when this first term is substituted into Eq. (4).
32
Fig.8b The σs field generated by the second term of Eq. (8) using the wind
profile presented in Fig.5a, b, and when this second term is substituted into
Eq. (4).
33
6. Summary
A 6-point scheme of linear surface fitting is designed to separate the shear and
turbulence contributions to the spectrum width data collected in a snowstorm with a WSR-
88D. There are unexpectedly large numbers of negative values in the separated σt2 field.
There exist four practical reasons for these negative values, but, for this studied case, most
these negative values are simply the result of both σs dominating the turbulence contribution
σt , and the limitations of the radar data. Spectrum width due to shear, σs, comes from the
shears in the radial, azimuthal, and elevation directions, but they do not equally contribute.
For the studied case, contributions from shear in the radial and azimuth directions are
negligible compared to the contribution from shear in the elevation direction.
By assuming a horizontally homogenous wind field, neglecting the effect of fall
velocity, and assuming a linear change of radial velocity across the radar beam, the part of
shear contribution to the measured spectrum width is related to the vertical shear of wind
speed and direction obtained from a VAD analysis. Two bands of large values in the
separated shear contribution field (and simulated field) can be ascribed to the large vertical
shears of wind speed and direction. The change of wind direction with height is responsible
for the observed spiral signature. The fields of shear induced spectrum width σs obtained
from the 6-point least squares fitting technique is strongly supported by the simulated σs
fields obtained using wind profiles derived form a VAD analysis. The VAD derived results
also indicate that the pattern associated with the term of vertical shear of wind direction
clockwise rotates 90o relative to the pattern associated with the term of vertical shear of wind
speed.
34
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