1. Distance Velocity Azimuth Display (DVAD) – A New (Forgotten) Way to Interpret and Analyze Doppler Velocity Wen-Chau Lee Earth Observing Laboratory National Center for Atmospheric Research Outline 1.Introduction 2.Kinematics of linear wind field and Velocity Azimuth Display (VAD) algorithm 3.Distance Velocity Azimuth Display (DVAD) algorithm 4.Summary and future work Lee, W.-C., X. Tang, and B. J.-D. Jou, 2013: Distance Velocity Azimuth Display (DVAD) – New interpretation and analysis of Doppler velocity. Mon. Wea. Rev., 142, 573-589. DOI: 10.1175/MWR-D-13-00196.1.

2. Radar Principles Radio wave energy is transmitted......and scattered back Armijo (1969)

3. Characteristics of V d and rV d /R T AnalyticV d rV d /R T Jou, B. J.-D., W.-C. Lee, S.-P. Liu, and Y.-C. Kao, 2008: Generalized VTD (GVTD) retrieval of atmospheric vortex kinematic structure. Part I: Formulation and error analysis. Mon. Wea. Rev., 136, 995-1012.

4. Previous Work Peace et al. (1969) first used rV d, expanded rV d in polynomial form, and deduced pseudo-dual-Doppler winds. Gal-Chen (1982) recognized that the second derivative of rV d with respect to time is invariant and can be used to determine the optimal advection speed of an atmospheric phenomenon.

5. Identify the Kinematic Properties Represented by These Single Doppler Velocity Patterns Const. Wind Divergence Divergence + Mean Wind DivergenceShearing Deformation One of the main challenges has been to extract KEY kinematic properties of a wind field from a given single Doppler radar observation. Many single Doppler wind retrieval algorithms have been developed based on simplified wind fields. The Velocity Azimuth Display (VAD, Browning and Wexler 1968) has been used to deduce kinematic properties in research and operation.

6. A Taylor expansion is an infinite series of terms that use first, second, and higher order derivatives to determine a periodic function. For simplicity, lets assume that x 0, y 0 is the origin (0, 0) and that we can obtain an adequate estimate of u and v by retaining only the first derivatives. We are assuming that over the small distance the u and v field vary linearly. Then…

7. Let’s take a simple step and write each derivative term as (for example) : Divergence (δ) Relative Vorticity (ζ) Stretching Deformation (D 1 ) Translation Shearing Deformation (D 2 )

8. x y Translation The effect of translation on a fluid element: Change in location, no change in area, orientation, shape

9. x y Divergence (δ > 0) Convergence (δ < 0) The effect of convergence on a fluid element: Change in area, no change in orientation, shape, location

10. x y Positive (cyclonic) vorticity (  > 0). Negative (anticyclonic) vorticity (  < 0) The effect of negative vorticity on a fluid element: Change in orientation, no change in area, shape, location

11. x y E-W Stretching Deformation (D 1 > 0). N-S Stretching Deformation (D 1 < 0). The effect of stretching deformation on a fluid element: Change in shape, no change in area, orientation, location Axis of dilitation Axis of contraction

12. x y SW-NE Shearing Deformation (D 1 > 0). NW-SE Shearing Deformation (D 1 < 0). The effect of shearing deformation on a fluid element: Change in shape, no change in area, orientation, location

13. VELOCITY-AZIMUTH DISPLAY (VAD) PROCESSING OF RADIAL VELOCITY DATA FROM A DOPPLER RADAR A technique for the measurement of kinematic properties of a wind field in widespread echo coverage using a single Doppler radar

14. Equation for radial velocity in terms of wind components: We will assume that the velocity gradient across a ring varies linearly so that:

15. Write previous equation as a Fourier series where and the coefficients are:

16. Contains information about Divergence and total fall velocity of the particles Contains information about the horizontal wind speed and direction Contains information about the flow deformation and its orientation

17. Data taken along a ring in a VAD scan at range 5922 m and altitude 2582 m at elevation angle 17.7° in a winter storm Data points are radial velocities Radial velocity Azimuth angle Line is best fit line Derived Fourier Spectral Components

18. Operational VAD calculation of winds from NEXRAD in Hurricane Jeanne (2004) From Jacksonville, FL radar VAD winds available from College of DuPage, IL http://weather.cod.edu/analysis/analysis.radar.html

19. Details in Performing VAD and EVAD Analysis Matejka and Srivastava (1991): An improved version of the Extended Velocity-Azimuth Display analysis of single-Doppler radar data. J. Atmos. Oceanic Technol., 8, 453-466. Accuracy of the flow model (linear vs nonlinear) Dealiasing Uneven data distribution – Orthogonal basic functions (?) Choice of weighting functions – even sampling biases in range Data gaps – no more than 30-45 degrees

20. Distance Velocity Azimuth Display (DVAD) - A New Paradigm Spherical Coordinate Cartesian Coordinate rV d behaves like velocity potential

21. Taylor Series Expansion of u, v, and W to Second Order Derivatives at the Radar Substitute above equations into rV d =ux+vy+wz and illustrate in two dimensions for linear wind field

22. Linear Wind rV d is a quadratic equation with coefficients the same as those obtained in VAD: Stretching Deformation Sheering Deformation The discriminant: D=(u y +v x ) 2 -4u x v y D<0 : Circle or Ellipse D=0 : Parabola D>0 : Hyperbola

23. Properties of Quadratic Equation Divergence is invariant Deformation is coordinate dependent

24. Constant Wind u(x, y)=C 1, v(x, y)=C 2, and V=C 1 i + C 2 j rV d forms a set of parallel lines

25. Linear Wind Deformation ~10 -4 s -1 Divergence ~10 -4 s -1

26. Divergence + Deformation

27. Divergence + Mean Wind Deformation + Mean Wind

28. Divergence + Deformation + Mean Wind

29. 2 nd Order Non-linear Wind This is a cubic equation. VAD coefficients for non- linear wind fields does not have physical meaning.

30. Divergence + Deformation + Mean Wind + Non-linear u xx ~10 -7 m -1 s -1 u xx ~10 -6 m -1 s -1

31. Computation Two-dimensional polynomial curve fit Use F-test to determine the highest order term (i.e., not limited to linear assumption) Successive differentiation Noise can be a problem Smoothing may be required

33. Two Solutions in DVAD

34. Pseudo Wind

35. A Real Case

36. VAD Analysis of Nonlinear Wind Fields Caya and Zawadzki (1992): VAD analysis of nonlinear wind fields. J. Atmos. Oceanic Technol., 9, 575- 587. Physical Flow Model Mathematic Flow Model Cubic Flow Model

37. VAD Coefficients Have No Physical Meaning for Non-linear Wind Fields Using range dependence relationship of these coefficients, mean wind, divergence and deformation can be deduced at the origin.

38. DVAD on Non-linear Winds Tang, X., W.-C. Lee, and Y. Wang, 2015: Nonlinear wind analysis of single Doppler radar in DVAD framework. J. Appl. Meteor. Climatology, 54, 1538-1555, DOI: 10.1175/JAMC-D-14-0194.1.10.1175/JAMC-D-14-0194.1 Use F-test to determine the highest order term required to fit because of the non-orthogonal base functions. One step process.

39. Summary The characteristics of a “forgotten” rV d and its applications to single Doppler analysis are presented. The wind patterns are easier to identify and interpret graphically. rV d pattern is conserved when mean wind exits. rV d can be analyzed in Cartesian coordinate and physical quantities are easier to deduce compared with the VAD algorithm, especially in non-linear wind field. The full capability of rV d needs to be further explored (e.g., vertical wind profiles, unfolding, data assimilation, etc.)

40. Thank You