a. Formalism

The aim of this section is not to recall the reasons that have motivated the development of EODD analysis, as can be found in Chong and Campos (1996), but to remind the reader of its main characteristics. EODD is a variational adjustment of the horizontal Cartesian wind components u and υ (considering a specified vertical velocity w⁰) to observed Doppler velocities in individual horizontal planes, which minimizes the functional

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according to

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and includes the airmass continuity equation to derive the vertical velocity w,

uxυywzκw(3)

Subscript i defines the radar (or beam for dual-beam airborne radars) number that ranges from 1 to at least 2. Here, υ_T is the terminal fallspeed of precipitation particles that can be estimated from an empirical relationship with the observed radar reflectivity; α_i, β_i, γ_i are the direction cosines, in the x, y, and z directions, respectively, of the beam-pointing angles; and V_i is the radial velocity from radar i. Here, J₂ ≡ (∂²/∂x²)² + 2(∂²/∂x∂y)² + (∂²/∂y²)² is a differential operator, κ = −∂ lnρ/∂z accounts for the air density (ρ) decrease with height, μ₁ and μ₂ are normalized weighting parameters that control the relative importance of terms B and C with respect to A. As in the conventional dual-Doppler analysis, solution of (2) is obtained through an iterative process with (3), allowing a step-by-step correction of the vertical air velocity contribution to the measured radial velocity. [At the initial step, w⁰ = 0 in (1); then it is the solution of (3).]

In (1), A is the data adjustment term that links the three wind components to the measured radial velocities. This is the classic least squares minimization formalism of the overdetermined dual-Doppler (ODD) technique (Ray and Sangren 1983), which leads to an equivalent form of the conventional dual-equation system. Term B is the least squares expression of the mass continuity equation, which limits the extent of horizontal wind variations through the divergence term during the iterations. In essence, it allows us to moderate the contribution of errors in determining w from (3) and thus to improve the convergence of the iterative process. Term C is a constraint on second derivatives of u and υ and helps to provide regular horizontal wind field by filtering out small-scale variations. A main advantage of this term is to give an objective solution of the ill-conditioned dual-Doppler analysis that can be caused by the geometrical configuration of the Doppler measurements in both the cone above and below the aircraft that involves high elevation angles. Note that such ill-conditioned analysis also occurs in regions close to the radar baseline when two ground-based radars are involved. In this paper, discussion concerning airborne radar analysis can be readily extended to the case of ground-based radars.

b. Limitations of EODD

Although the variational EODD technique was shown to provide highly reliable flow structure in regions where the traditional approach failed, there remain some limitations that should be clarified in this paper and that justify additional improvements. The first limitation concerns the interpolation of the polar coordinate radar data onto a Cartesian grid, prior to the application of EODD, in order to obtain, at each grid point, one Doppler-radial vector for each radar (or beam). Interpolated grid values are defined as distance-weighted averages of all data falling inside an ellipsoid with an influence radius in the horizontal twice as large as in the vertical. A fixed interpolation function applies at short and long ranges differently. At short range this function forces averaging vectors pointing in quite different directions and creates a sampling problem, as these data are different due to the changes in elevation and/or azimuth angles and, hence, should not be averaged. In the context of airborne radar sampling, which consists of helical scans about the aircraft axis, the averaging process involves such a variety of viewing angles for grid points close to this axis, more specifically above and below it, because of the larger horizontal influence radius. As a result, the interpolated radial velocities and the associated beam-pointing angles are less accurate in these regions than elsewhere, exacerbating the errors in estimating the horizontal components. A solution would consist of incorporating the interpolation step into the least squares data adjustment procedure, allowing the interpolation to take account of a number of measurements (i.e., all the available observed velocity vectors and their actual orientation) instead of a unique mean observation from each radar or beam.

The second limitation results from the actual limits of the proposed approach. Let us examine the functional F [Eq. (1)] to be minimized. Constraint C on the second-order derivatives of the wind field, which acts as a low-pass filter, basically applies to the results of the least squares data adjustment given by term A in order to correct or reduce unreasonable horizontal winds that can be obtained near the flight track (see Fig. 7 in Chong and Campos 1996). However, in some circumstances, this correction may be incomplete. Consider that unstable solutions occur within an interval of 4Δx in the cross-track direction, where Δx is the grid spacing. If their differences with solutions from well-defined regions are maximized at the center of the unstable zone, then they may be interpreted as varying with an equivalent wavelength greater than or equal to 4Δx. Because the cutoff wavelength of the low-pass filter (defined as the wavelength for which the filter response is 0.5 since the response function is not a pure step function) is usually 4Δx, the filtered winds may contain residuals that can reach 50% of the amplitude of the wind bias. As a consequence, a localized wind extremum is obtained and causes an artificial convergence/divergence doublet that is penalizing for the determination of w through (3).

The above limitation concurrently combines with the problem arising from a separate estimation of the horizontal and vertical wind components. Indeed, the iterative process forms another important limitation of EODD in this region of unstable solution near the flight track, where vertical motion accounts contribute more than 70% of the measured radial velocity at elevation angles greater than 45°. In some conditions, prescribing w⁰ = 0 in the initial step can be excessive, and biased u and υ estimates can be obtained due to associating nonconsistent vertical and horizontal velocities during the iterative process.

In fact, the dual-Doppler equation system and the continuity equation define a mathematically complete system to retrieve the three wind components, and ill-conditioned problems could disappear if they are solved simultaneously (noniterative process). Such a solution has been already investigated by Scialom and Lemaître (1990) in their adjustment technique of observations with an analytical wind field and more recently by Gamache (1995), who found, however, that the technique was computationally intensive and required high memory storage in a three-dimensional context. An alternative approach consists of resolving the above-mentioned equations in individual planes. The EODD approach offers such an opportunity by simply extending the minimization problem to the variable w. Improvements are, however, necessary to preserve the plane-to-plane wind synthesis. These will be described in the next section, which presents the basis for MUSCAT, a method that can be applied to airborne or ground-based radar data.

a. Data fit

As previously explained, the use of grid values, obtained with a fixed interpolation function, in the data fit expression A of (1) may cause problems in regions above and below the aircraft. Because data distribution is not coincident with the grid mesh used to recover the three-dimensional wind field, the interpolation process must be maintained. The alternative proposed in the previous section, consisting of inclusion of this process in the data fit, may be expressed for a grid point (i, j) of a horizontal plane as

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where u, υ, w, and υ_T are defined for the considered grid point; subscript q defines the qth measurement of a total number n_q that can be observed from the pth radar and that falls inside an ellipsoid of influence centered on the grid point; N is the cumulated n_q’s over the considered domain; ω_q is the Cressman weighting function depending on the distance between measurement q within the ellipsoid and grid point (ω_q = 1 at the center of the ellipsoid and ω_q = 0 on its surface); and n_p is the total number of radars that must be equal to at least 2. Weighted data fit was also used by Roux and Sun (1990) to evaluate wind components from a series of conical single-Doppler scans of ground-based radar.

In (6), the terminal particle fallspeed υ_T is evaluated from preinterpolated radar reflectivity Z, instead of individual reflectivity observations, because the radar signal may undergo strong attenuation in regions of heavy precipitations and, to further mitigate this effect, the maximum gridpoint value from more than two radars can be considered. Another possibility may be provided by the stereoradar analysis of dual reflectivity observations proposed to estimate corrected reflectivity fields (Testud and Amayenc 1989; Kabèche and Testud 1995).

Minimizing (6) with respect to u, υ, and w requires, for each radar dataset, the estimation of nine terms for each individual grid point. They are defined as follows:

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such that

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Summations in the matrix arrays are performed over the number n_p (≥2) of available radars that contribute to the observations at the considered grid point. Because υ_T is derived from an independent processing and is a constant during the summation, term P_w can be split into two terms:

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where P_ω = P^′_ω − P^"_ωυ_T. It should be noted that if the υ_T–Z relationship is changed, the above terms need not be reevaluated, as long as these are saved. This is what is done for MUSCAT software.

b. Continuity equation adjustment

The adjustment term B′ for the continuity equation is exactly the same as in the EODD approach and is rewritten as

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In essence, B′ must be discretized by considering grid point values of u, υ, and w. Due to the vertical derivative term, this necessarily involves two successive planes. The discrete form of (9) allowing a plane-to-plane formulation is described in appendix A. An off-centered finite-difference scheme is used to express B′ at the center of a grid box in terms of net mass flux through the box sides. This differentiating scheme accounts for the boundary conditions at the surface when starting the wind synthesis from the lowest level, or at the top in a downward plane-to-plane analysis, so that the integration of the continuity equation is implicitly performed.

c. Filtering process

As in EODD, the role of term C′ in (4) is to smooth out small-scale variations of u, υ, and also w in the horizontal. Considering B′ as a weak constraint, it can be shown (appendix B) that minimizing second derivatives of wind components through J₂(u), J₂(υ), and J₂(w) with a weight μ₂ as in (1) is equivalent to a low-pass filter defined by a simple transfer function,

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where k = (k_x, k_y) is the wave vector in the x–y plane. Such a transfer function is isotropic and has a 3-dB cutoff wavenumber k_c controlled by μ₂ according to k_c = μ^−1/4₂, that is, a cutoff wavelength λ_c = 2πμ^1/4₂.

In fact, the adjustment for the continuity equation cannot be considered as a simple constraint since it is an integral part of MUSCAT (in contrast to EODD). Therefore, the transfer function is no longer isotropic, as shown in appendix B. Additional terms to J₂(u), J₂(υ), and J₂(w) are then required to obtain an isotropic response of the filtering process, which are

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Finally, term C′ is given by

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The discrete forms of the various derivatives involved in the expression of J₂ and J^′₂ are developed in appendix C.

d. Resolution procedure

Finally, the sparse matrix equation is solved with the standard conjugate gradient method. Because the obtained solution of u, υ, and w satisfies the continuity equation in a least squares sense, and to limit the errors in the vertical velocity component, a refinement of the w estimates is then performed by integrating this equation according to the approach proposed in Chong and Testud (1983) and applied in EODD.

a. Real data analysis

In Chong and Campos (1996), the EODD analysis was applied to the data collected within a squall-line system occurring on 22 February 1992 during TOGA COARE, by the Doppler radars aboard the two NOAA P-3 aircraft (N42 and N43, respectively). Observations from a specific leg simultaneously performed by the two aircraft within the time period 2110–2121 UTC were used to test EODD, by considering, respectively, data from only the N43 track (dual-Doppler scan) and data from both N42 and N43 tracks (quad-Doppler scan). These dual- and quad-Doppler configurations yielded quite comparable results. Because of this, we will focus on comparisons of EODD and MUSCAT applied to dual-Doppler observations only.

To ensure some continuation with the previous work, the same dataset relative to the above-mentioned N43 leg is used in this study. Detailed description of the observations and data preprocessing can readily be found in Chong and Campos (1996). Application of MUSCAT is performed with the same Cartesian grid and interpolation function characteristics as in the EODD analysis, that is, a domain of 73.5 × 73.5 × 14.5 km³ with a grid resolution of 1.5 × 1.5 × 0.5 km³ and a Cressman weighting function with horizontal and vertical influence radii of 2.5 and 1.2 km, respectively. Also, the weights μ₁ and μ₂ for the continuity equation adjustment and the filtering process of MUSCAT are set to S² (S is the area of an elementary horizontal grid mesh) and 0.164S⁴, respectively. With these specifications, direct comparison with results presented in Chong and Campos (1996) can be done.

Figures 1 and 2 compare the vertical cross sections of the line-transverse (west–east section) relative flow (with the corresponding radar reflectivity) and of the vertical velocity at y = 33 and y = 60 km, respectively, obtained from EODD (Figs. 1a and 2a) and MUSCAT (Figs. 1b and 2b). Locations of the aircraft that flew at 170-m altitude are also indicated at x = 16 km and x = 0 km, respectively. The overall airflow structures from both approaches appear similar (see left panels of Figs. 1 and 2), and this mutual consistency substantiates their validity in solving the three wind components from a minimal configuration of dual-Doppler observations. The components from the two analyses are within a few meters per second of each other. However, the detailed features are significantly different, and this can be attributed to improvements due to the simultaneous solution of MUSCAT as opposed to the iterative process of EODD (this will be confirmed in the next section). The differences are noticeable for the cross section at y = 33 km (Fig. 1), which displays an upper-level updraft (>10 m s⁻¹ above 13-km altitude) in the EODD analysis (Fig. 1a), while MUSCAT provides a main updraft reaching a maximum of 4 m s⁻¹ at 10-km altitude and then decreasing progressively to less than 1 m s⁻¹ at top levels (Fig. 1b). Moreover, an area of downward motions less than −1 m s⁻¹ is observed at x = 18–23 km, beneath the EODD-derived upper-level updraft. There is a similar updraft–downdraft structure in the EODD analysis in Fig. 2, at x < 8 km, which is not revealed in MUSCAT results. In fact, this major discrepancy occurs in regions close to the flight track, where different conclusions about storm mechanisms would result from using the two different approaches. EODD depicts convergent flows composed of the main upward front-to-rear flow (from right to left) opposing a rear-to-front flow (from left to right) at high levels greater than 8 km. This contrasts with the overall upward front-to-rear flow from MUSCAT.

These differences on the line-transverse components are also observed on the line-parallel ones, as shown in Fig. 3, which compares the u and υ components obtained from EODD (Fig. 3a) and MUSCAT (Fig. 3b) at y = 33 km. The associated line-transverse (∂u/∂x) and total (∂u/∂x + ∂υ/∂y) divergence patterns are shown in Fig. 4. In most respects, MUSCAT provides a smooth structure in wind components and divergence, while EODD can produce strong gradients, especially in regions above the aircraft. A rapid reversal of the components is obtained from the EODD analysis, which contrasts with the gradual variation found in the MUSCAT-derived components. The high level region of strong vertical velocities from EODD (Fig. 1a) clearly results from the convergence zone above 8-km altitude (Fig. 4a). The existence of such an area of convergence is questionable at storm top, where virtually any numerical or conceptual model would indicate an area of divergence. Numerical simulations of the present observational case (Trier et al. 1996) show a progressive decrease of the upward motions at top levels. This corroborates the MUSCAT results, which show a divergent upper-level flow (Fig. 4b) quite consistent with the detrainment of updraft air toward the rear of the system, as previously documented (see, e.g., Houze and Betts 1981; Chong et al. 1987). At these levels above the aircraft, the contribution of the incorrect vertical velocities from the EODD analysis is maximized, which increases the errors in the determination of the horizontal components during the iterative process.

At low levels, although the u and υ components (Fig. 3) depict similar features, there are small-scale variations in the EODD analysis that strongly affect the convergence zone between the deep rear-to-front flow (positive u) and the shallow downdraft-generated diverging outflow toward the rear of the convective cells (negative u), as revealed by MUSCAT (Fig. 4b). As discussed in section 2b, incorrect winds can explain the occurrence of the doublets of convergence/divergence that can be easily observed on the total divergence field of Fig. 4a and that extend between x = 10 km and x = 22 km, which is still in regions close to and above the aircraft. Note that in these areas in which EODD fails other dual-Doppler approaches will likely fail also.

b. Numerical tests

Comparison of the application of EODD and MUSCAT to real data has clearly shown noticeable differences in the derived wind structure, particularly in regions above the aircraft track. Expected limitations of EODD were put forward to explain these differences, while previously documented airflow structure within squall lines was suggested to evaluate the consistency of the wind retrieval from MUSCAT. However, this comparison remains qualitative since the reference wind field (exact solution) cannot be observed in nature. An alternative consists of using simulated radar observations and performing a quantitative analysis of the improvements of MUSCAT against EODD. Such an approach was also used by Chong and Testud (1996) in their study of the coplane methodology to synthesize airborne Doppler radar data. We adopt the same procedure to obtain numerical tests close to experimental conditions, that is, we consider the dual-beam sampling from the leg analyzed in the previous section and substitute the observed radial velocities with those derived from an analytic form of the wind components. To simulate the radar statistical error, random errors with an rms value of 1 m s⁻¹ are then added to these radial components.

1) Simulated wind field

The two-dimensional wind field used in Chong and Testud (1996) is considered, and it derives from a description of the vertical velocity w in terms of a product of two sine functions defined as

wAk_zzk_Hr_Hazb(13)

where A is the amplitude; k_H and k_z are, respectively, the horizontal and vertical wavenumbers; r_H is the horizontal distance; and a and b are parameters defining a linear variation of the phase shift with height. Note that w = 0 is always verified at the surface and that implicitly k_H and r_H are colinear. In the vertical plane containing the wave vectors, r_H = az + b defines the line for which w = 0, as do the parallel lines at a horizontal distance that is a multiple of half the horizontal wavelength. Considering the mass continuity equation (3), the horizontal wind V_H in the r_H direction can be expressed as

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In this study, the horizontal wave vector was assumed along the x direction (i.e., u = V_H and υ = 0), and horizontal and vertical wavelengths were equal to 40 and 24 km, respectively. Amplitude A was set to 10 m s⁻¹ and κ, the inverse of the scale height of the air density decrease, was assumed to be equal to 0.1. For simplicity, the analysis domain was limited to 48 km in the x direction since no observation was available beyond that limit and to 12 km in the vertical where w vanishes. Parameters a and b were chosen to be equal to −0.7 km⁻¹ and 17 km, respectively. The corresponding y-independent two-dimensional wind field and the associated u and w components are shown in Figs. 5a–c, respectively. With the above specifications, the simulated airflow consists of two opposite vortex circulations sloping to the left with an angle of 35° and centered at x = 13.8 and 33.8 km, and z = 4.6 km. Upward motions are in the central part of the domain and reach 10 m s⁻¹, as do the downdrafts on either side. Horizontal velocities range from 0 to ±18 m s⁻¹.