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: jkwillia@ucar.edu

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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