Editorial Type:
Article
Article Type:
Research Article

Prognostic Equation for Radar Radial Velocity Derived by Considering Atmospheric Refraction and Earth Curvature

Qin Xu NOAA/National Severe Storms Laboratory, Norman, Oklahoma

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Li Wei Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

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Print Publication:
01 Oct 2013
DOI:
https://doi.org/10.1175/JAS-D-13-011.1
Page(s):
3328–3338
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Abstract

The prognostic equation for the radial velocity field observed with a Doppler radar is derived to include the effects of atmospheric refraction and earth curvature on radar-beam height and slope angle. The derived equation, called the radial velocity equation, contains a high-order small term that can be truncated. The truncated radial velocity equation is shown to be much more accurate than its counterpart radial velocity equation derived without considering the effects of atmospheric refraction and earth curvature. The truncated equation has the same concise form as its counterpart radial velocity equation but remains to be sufficiently accurate as a useful dynamic constraint for radar wind analysis and assimilation (in normal situations) even up to the farthest 300-km radial range of operational Weather Surveillance Radar-1988 Doppler (WSR-88D) scans where its counterpart radial velocity equation becomes erroneous.

Corresponding author address: Dr. Qin Xu, NSSL, 120 David L. Boren Blvd., Norman, OK 73072-7326. E-mail: qin.xu@noaa.gov

Abstract

The prognostic equation for the radial velocity field observed with a Doppler radar is derived to include the effects of atmospheric refraction and earth curvature on radar-beam height and slope angle. The derived equation, called the radial velocity equation, contains a high-order small term that can be truncated. The truncated radial velocity equation is shown to be much more accurate than its counterpart radial velocity equation derived without considering the effects of atmospheric refraction and earth curvature. The truncated equation has the same concise form as its counterpart radial velocity equation but remains to be sufficiently accurate as a useful dynamic constraint for radar wind analysis and assimilation (in normal situations) even up to the farthest 300-km radial range of operational Weather Surveillance Radar-1988 Doppler (WSR-88D) scans where its counterpart radial velocity equation becomes erroneous.

Corresponding author address: Dr. Qin Xu, NSSL, 120 David L. Boren Blvd., Norman, OK 73072-7326. E-mail: qin.xu@noaa.gov
Keywords: Advection; Radars/Radar observations; Differential equations; Error analysis; Data assimilation

1. Introduction

The prognostic equation for radial velocity scanned from a Doppler radar, called the radial velocity equation, has been used in various simplified forms as a dynamic constraint for analyzing and assimilating radial velocity observations in space and time dimensions (Xu et al. 1994, 1995, 2001a,b; Xu and Qiu 1995; Qiu and Xu 1996). As the radial velocity observations used in these previous studies were confined in near-radar ranges (<20 km), the effects of atmospheric refraction and earth curvature on the radar-beam height and slope angle (relative to the earth surface below the measurement point) were neglected in the previously derived simple forms of the radial velocity equation. For operational Weather Surveillance Radar-1988 Doppler (WSR-88D) scans, the maximum radial range for radial velocity measurements has been extended recently to nearly 300 km. As the radial range increases to 300 km, the WSR-88D beam is about 5 km wide and the lowest beam center is above 7 km. In this case, the radial velocity measurements may become not useful for detecting mesocyclones (that can produce tornadoes) and related applications, but they are still useful for mesoscale data assimilation. For the operational North American Mesoscale Forecast System (NAM), the horizontal resolution is 12 km, so operational radar observations, even out to 300-km radial range, still have excessive horizontal resolutions with respect to NAM and therefore should be compressed into fewer superobservations (after quality control) to reduce their resolution redundancy (Xu 2011; Xu and Wei 2011) and then all assimilated into NAM.

To assimilate full volumes of operational radar data, it is necessary to properly consider the effects of atmospheric refraction and earth curvature on the radar-beam height and slope angle. According to Ge et al. (2010), neglecting these effects, especially the earth curvature effect, can cause significant and increasingly large errors in assimilated storm winds as the distance between the storm center and radar location increases to 60 km and beyond. The above effects have been considered in the 3.5-dimensional variational method (3.5DVar) (Gu et al. 2001; Zhao et al. 2006, 2008; Xu et al. 2010) and other radar data assimilation method (Gao et al. 2006, 2008) as well as the widely used Weather Research and Forecasting Model–Data Assimilation Research Testbed (WRF-DART) radar data assimilation package (A. Caya and D. Dowell 2013, personal communication). The radial velocity equation in the 3.5DVar, however, was not derived rigorously to include the effects of atmospheric refraction and earth curvature and it also neglected the perturbation pressure and buoyancy terms. This paper aims to derive the radial velocity equation by considering the above effects and analyze the accuracy of the derived equation. The next section reviews the commonly used equivalent earth model for radar ray path. Section 3 derives the radial velocity equation using the equivalent earth model. Section 4 truncates the derived equation into a concise form and examines the accuracy of the truncated radial velocity equation in comparison with its counterpart equation derived without considering the effects of atmospheric refraction and earth curvature. Conclusions follow in section 5.

2. Review of equivalent earth model

The radial velocity observed by radar is related to the vector velocity v + wTk by
e2.1
where υr ≡ vTc; υrT ≡ wTkTc; superscript T denotes the transpose; v ≡ (u, υ, w)T is the vector velocity of air; wT (<0) is the terminal velocity of hydrometeors; k is the upward unit vector; c ≡ (cosθ sinφ, cosθ cosφ, sinθ)T is the unit vector composed of the three directional cosines along the (curved) radar beam at the measurement point in the local Cartesian coordinate system, denoted by (x, y, z), centered at the radar; φ is the azimuthal angle (positive for clockwise rotation from the y coordinate pointing to the north); and θ is the radar-beam slope angle with respect to the curved earth surface immediately below the measurement point. Because of the atmospheric refraction and earth curvature, the radar beam is curved (usually concave upward relative to a flattened earth surface) as shown in Fig. 1, and
e2.2
where θe is the beam elevation angle at the radar, and ψe can be expressed as a function of (θe, r) in the equivalent earth model reviewed below [also see sections 2.2.3.1 and 9.3.1 of Doviak and Zrnic (2006), and note that the above θ is equivalent to in their (9.9)], where r is the length of the curved beam (ray path) from the radar to the measurement point.
Fig. 1.
Fig. 1.

Radar-beam height z [computed by (2.7) with ae = 4a/3 in the equivalent earth model] plotted by dark curves as functions of the length of ray path r, for seven elevation angles (i.e., θe = 0.5°, 2.4°, 4.3°, 6.2°, 8.7°, 12°, and 19.5° selected from the 14 elevation angles of θe = 0.5°, 1.45°, 2.4°, 3.35°, 4.3°, 5.25°, 6.2°, 7.5°, 8.7°, 10°, 12°, 14°, 16.7°, and 19.5° in volume coverage patterns 11 and 211 used by the operational WSR-88Ds to scan thunderstorms). The shaded band around each dark curve shows the beam thickness (within θe ± 0.5° for each θe). The thin dotted straight line associated with each dark curve is the beam height computed by z = r sinθe in the flat earth model.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

In the equivalent earth model, the refractivity index n is assumed to be horizontally uniform with a constant vertical gradient, typically around dn/dz ≈ −1/(4a) in the lower troposphere, so all low-elevation radar beams have about the same constant curvature and can be mapped into straight beams (with their radius of curvature “inflated” to infinity) by “inflating” the earth radius a to the effective radius ae = a/(1 + adn/dz) ≈ 4a/3 [see (2.27) of Doviak and Zrnic (2006) or section 3 of Heymsfield et al. (1983)]. Approximately, to the first-order accuracy measured by O(ψe) [<O(r/ae) ≪ 1], each mapped straight beam preserves the length r of the originally curved beam and its height above the spherical equivalent earth of radius ae preserves the height z of the originally curved beam above the curved earth surface. In the spherical coordinate system of the equivalent earth model (see Fig. 2), R = ae + z is the radial coordinate of the measurement point, s = aeψe is the great-circle arc distance between the radar and the point immediately underneath the measurement point on the spherical surface of the equivalent earth, and ψe is the angle subtended by the two radii at the radar and measurement points. The related geometry is shown in Fig. 2. Applying the cosine law to the OB side and the sine law to the vertexes O and A of the triangle OAB in Fig. 2 gives
e2.3
e2.4
Note that R sinψe is the length of BE side of the right triangle OEB and r cosθe is the length of BE side of the AEB, and this leads to (2.4) directly. From (2.3) and (2.4), we obtain R2 cos2ψe = R2(1 − sin2ψe) = R2 − r2 cos2θe = (ae + r sinθe)2,
e2.5
e2.6
Here, (2.6) recovers (A1) of Heymsfield et al. (1983), although the symbols used here are not all the same as theirs. Substituting (2.6) into (2.2) recovers (9.9) of Doviak and Zrnic (2006).
Fig. 2.
Fig. 2.

Schematic drawing of radar beam (ray path) in the equivalent earth model with the transformed earth surface plotted by the hatched arc. The equivalent earth center is at point O. The radar is at point A. The radar beam is along line AB of length r. Line BC shows the beam height z at point B above the earth surface. Arc AC shows the horizontal distance s from radar to point C along the great circle.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

Note that s measures the arc distance of the measurement point from the radar, which is horizontal distance in the local Cartesian coordinate system centered at the radar; that is, s = (x2 + y2)1/2, so using (2.3) and (2.4) z and s can be related to r and, ψe by
e2.7
e2.8
which recover (2.28b) and (2.28c) of Doviak and Zrnic (2006). In (2.7), z is expressed as a function of (r, θe) and its derivative to r (for fixed θe) gives
e2.9
e2.10
where (2.3) is used. Substituting (2.2) into (2.9) gives R sin(θe + ψe) = r + ae sinθe; that is, BD = BA + AD as shown in Fig. 2. Substituting the expression of tanθ derived from (2.9)/(2.10)—that is,
e2.11
into the right-hand side of tanψe = tan(θ − θe) = (tanθ − tanθe)/(1 + tanθ tanθe) can lead to the same expression of tanψe in (2.6).
In Fig. 1, the straight beams are computed by z = zf ≡ r sinθe for a flat earth model, which is degenerated from the equivalent earth model in the limit of θ → θe owing to either ae/a → ∞ or r/a → 0. Subtracting (ae + zf)2 = ae2 + r2 sin2θe + 2rae sinθe from (ae + ze)2 = r2 + ae2 + 2rae sinθe gives
e2.12
where ze is the curved-beam height described by (2.7) in the equivalent earth model, and cosθe = 1 − (πθe/180°)2/2 + … = 1 − 0.06 + … ≈ 1 is used for θe ≤ 19.5°. According to (2.12), the difference between the curved- and straight-beam heights for each given θe in Fig. 1 increases quadratically with r almost exactly in the same way for all θe (≤19.5°). Because the r coordinate is compressed relative to z coordinate and the r range is reduced successively with the increase of θe in Fig. 1, the above property is not visually intuitive as viewed from Fig. 1, but this property is important for understanding the numerical results presented (in Fig. 6) for the error analysis in section 4b.

3. Radial velocity equation derived with equivalent earth model

Projecting the inviscid vector momentum equation on c gives the following form of υr equation:
e3.1
where dt = ∂t + vT∇ is the Lagrangian time differential operator, ∇ = (∂x, ∂y, ∂z)T is the spatial gradient operator, f is the Coriolis parameter, b is the perturbation buoyancy (defined by the ratio between the perturbation and basic-state potential temperatures multiplied by the acceleration of gravity), p is perturbation pressure, and ρ is the basic-state density. The last term in (3.1) is due to the projection of the advection term v · ∇v on c, which gives
e3.2
When dn/dz → −1/a, ae/a → ∞ and θ → θe according to (2.2)–(2.4), so the equivalent earth model degenerates to the flat earth model with r = |x| and c = x/r where x ≡ (x, y, z)T. The equivalent earth model also degenerates to a flat earth model when r/(ae + z) = r/R → 0. For these degenerated cases, we have ∂jci = ∂j(xi/r) = ∂jxi/r + xi∂j(1/r) = δji/r − xixj/r3 or, equivalently, ∇cT = ∇(xT/r) = /r − xxT/r, where ∂j denotes the jth component of ∇ and ci (or xi) is the ith component of c (or x), δji is the Kronecker delta, and is the identity matrix. The last term in (3.1) is then given explicitly by
e3.3
Substituting (3.3) into (3.1) gives the degenerated flat earth υr equation. The components of ∇cT = /r − xxT/r are listed in the first column of Table 1 and expressed in (φ, r, θ) coordinates (with θ = θe) in the second column of Table 1 for latter comparisons.
Table 1.

Components of ∇cT for c = x/r (first column) derived in (x, y, z) coordinates and (second column) expressed in (φ, r, θ) coordinates with θ = θe, where (r2 − z2)/r3 = (x2 + y2)/r3 = cos2θ/r is used.

Table 1.

For nondegenerated cases, c ≠ x/r, but c is still a function of (φ, θ) and θ is a function of (r, θe) as shown in (2.11), while z and s = |(x, y)| are functions of (r, θe) as shown in (2.7) and (2.8). This implies that the last term in (3.1) can be analyzed explicitly by transforming the coordinate system from (x, y, z) to (φ, r, θe), as shown below in two steps.

The first step transforms (x, y, z) to (φ, s, z) or, equivalently, (x, y) to (φ, s) for fixed z. The related Jacobian matrix is
e3.4
and ∂(x, y)/∂(φ, s) = s. The three column vectors of (∇cT)T can be written into
e3.5a
e3.5b
e3.5c
Substituting c defined in (2.1) with (3.4) into (3.5) gives the component terms of ∇cT in (φ, s, z) coordinates, as listed in the second column of Table 2.
Table 2.

Components of ∇cT (first column) formulated in (x, y, z) coordinates, and expressed in (second column) (φ, s, z) coordinates and (third column) (φ, r, θe) coordinates, where e1 ≡ 1 − r cosθ/s and e2 ≡ (1 − cosψe) sinθ + r cos2θ/ae.

Table 2.
The second step transforms (φ, s, z) to (φ, r, θe) or, equivalently, (s, z) to (r, θe) for fixed φ. The related Jacobian matrix can be derived by differentiating (2.4) and (2.3), which gives
eq1
or, equivalently,
eq2
where R = ae + z, s = aeψe and (2.9) and (2.10) are used. The Jacobian matrix is
e3.6
e3.7
where (2.9) and (2.10) are used in the first three steps and (2.5) is used in the last step.
Applying cos2θ∂/∂r|φ,θe and cos2θ∂/∂θe|φ,r to the two sides of (2.11) gives
e3.8
e3.9
where (2.10) and (2.5) are used. The results in (3.8) and (3.9) can be also obtained by applying and to the two sides of (2.6), which gives
e3.10
e3.11
where (2.9) and (2.5) are used. Clearly, (3.10) is consistent with (3.8) because , while (3.11) is consistent with (3.9) because the two sides of (3.9) can be rewritten into
eq3
where (2.9) and (2.10) are used in the last step.
Using (3.6)–(3.9) and (2.10), we obtain
e3.12
Substituting (3.12) into the first six component terms in the second column of Table 2 yields their explicit expressions in (φ, r, θe) coordinates, as listed in the third column of Table 2. Substituting (3.6)–(3.9) with (2.4) and (2.10) into the last vector component term in the second column of Table 2 gives
e3.13
With ∂c/∂θ|φ = (−sinφ sinθ, −cosφ sinθ, cosθ)T, (3.13) gives the explicit expression in the last row of the third column of Table 2.
Substituting the component terms in the third column of Table 2 into the last term in (3.1) gives
e3.14
where υr = vTc but c ≠ x/r unlike in (3.3), e = (e2wυt − e1υht)υhr, e1 and e2 are given in the caption of Table 2, υht = u cosφ − υ sinφ is the tangential component of vh ≡ (u, υ) perpendicular to the beam in the horizontal plane, υhr = u sinφ + υ cosφ is the radial component of vh along the beam azimuth in the horizontal plane, and wυt = w cosθ − υhr sinθ is the vertical tangential component of v perpendicular to the beam in the vertical plane. Substituting (3.14) into (3.1) gives the υr equation derived with the equivalent earth model in the following form:
e3.15
When the effects of atmospheric refraction and earth curvature are neglected, (3.15) degenerates to the flat earth υr equation in which r, c, υr = vTc, and e reduce to |x|, x/r, υr = vTx/r, and 0, respectively.

4. Truncated radial velocity equation and error analyses

a. Truncated radial velocity equation and smallness of truncation error

For operational WSR-88D scans, 0.5°(π/180°) = π/360 ≤ θe < 20°(π/180°) = π/9 and r < rmax = 300 km, so z ≪ r ≪ ae (≈4a/3 ≈ 8500 km) < R, ae/R is very close to 1, and ψe ≈ sinψe = r cosθe/R < r/R < r/ae < 0.035r/rmax according to (2.4). Therefore, e1 and e2 are small positive parameters bounded by
e4.1a
e4.1b
where (2.4) and (2.10) are used. Neglecting e1 and e2, the component terms in the third column of Table 2 reduce formally to those in the second column of Table 1 but c ≠ x/r, and (3.15) is truncated into
e4.2
By substituting (4.1a) and (4.1b) into e = (e2wυt − e1υht)υhr defined in (3.14), we can estimate the truncation error relative to |v|2 as follows:
e4.3
Thus, the truncated part in the last term of (3.14) is much smaller than the major part—that is, |v|2 in the last term.

The upper bound of |e|/|v|2 estimated in (4.3) can be verified numerically by computing e/|v|2 at each beam height of the radar scan in a vertical column within a convective storm. Vertical profiles of (u, υ, w, wT) are extracted from the main updraft area within the simulated Oklahoma squall line (valid at 2200 UTC 2 June 2004) produced by the control run in Xu et al. (2010). The extracted vertical profiles are plotted in Fig. 3. By assuming that this vertical column is located to the east (φ = 90°) of the radar at variable distance s (so that r is also variable), e/|v|2 is computed and plotted as a function of r in Fig. 4 for each θe selected in Fig. 1. As shown in Fig. 4, |e|/|v|2 is bounded at least by (r/rmax) × 1.8% regardless of the variations of e/|v|2 with (r, θe). This main feature remains the same for different settings of φ and different vertical profiles of v (not shown), and the numerically estimated upper bound verifies and tightens the analytically estimated upper bound of |e|/|v|2 in (4.3). Since |e|/|v|2 is indeed very small, the truncated υr equation in (4.2) has virtually the same accuracy as the full υr equation in (3.15).

Fig. 3.
Fig. 3.

Vertical profiles of (u, υ, w, wT) and 2 log10Z (plotted in units of 5 dBZ) extracted from the main updraft area within the simulated Oklahoma squall line (valid at 2200 UTC 2 Jun 2004) produced by the control run in Xu et al. (2010), where Z is the reflectivity factor (mm6 m−3) [see section 4.4.5 of Doviak and Zrnic (2006)].

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

Fig. 4.
Fig. 4.

Relative truncation error e/|v|2 (%), defined in (4.3), as a function of r for each θe selected in Fig. 1.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

b. Error of degenerated flat earth radial velocity equation

The truncated υr equation in (4.2) has the same concise form as the degenerated υr equation but the latter contains significant errors owing to the flat earth approximation. When (4.2) is used as a dynamic constraint for radar wind analysis or assimilation, υr contained in the first and last terms should match the processed radial velocity observations, denoted by , that passed proper data quality control, including a bias correction step after dealiasing [see section 3.1 of Xu et al. (2010)] to produce , where (<0) is the terminal velocity estimated from the observed reflectivity [see (5) of Xu et al. (2010)]. The flat earth model can have a significant error in the radar-beam height assigned to and therefore cause a displaced-sampling error in especially if the true υr changes significantly from the assigned height to the true height of . This can cause a large error in the υr equation as evaluated below.

First, we compute as a function of (r, θe) for given φ from the vertical profiles of (u, υ, w, wT) in Fig. 3 by using the equivalent earth model (with a/ae = ¾) and denote the result by . We then compute also as a function of (r, θe) but using the flat earth model (with a/ae = 0) and denote the result by . Considering the effect of finite beamwidth, is computed as a function of (θe, r) by
e4.4a
where ∫(·) dθ′ denotes the integral of (·) computed by the summation of (·) over −Δθ/2 ≤ θ′ ≤ Δθ/2 with a sufficiently high resolution, Δθ = 1° is the beamwidth in elevation angle, and G(θ′) = exp[−(4 ln4)(θ′/Δθ)2] is the two-way power-gain distribution within the radar beam [(5.40), (5.48), and (5.53) of Doviak and Zrnic (2006)], and Z is the reflectivity factor [see section 4.4.5 of Doviak and Zrnic (2006)]. Here, for the integral term in (4.4a), Z, v = (u, υ, w)T, and wT are functions of z given in Fig. 3, with z computed as a function of (θe + θ′, r) by setting θe to θe + θ′ in (2.7), and c is a function of (θ, φ) defined in (2.1), with θ computed as a function of (θe + θ′, r) by setting θe to θe + θ′ in tan−1[(2.11)]. For the last term in (4.4a), z and θ are computed just as functions of (θe, r) from (2.7) and tan−1[(2.11)], respectively. Similarly, is computed as a function of (θe, r) by
e4.4b
where Z, v = (u, υ, w)T, and wT are still given by the vertical profiles in Fig. 3, but z is computed as a function of (θe + θ′, r) by z = r sin(θe + θ′) and x/r is computed as a function of (θe + θ′, φ) by x/r = [cos(θe + θ′) sinφ, cos(θe + θ′) cosφ, sin(θe + θ′)]T for the integral term in (4.4b). For the last term in (4.4b), z and θ are computed just as functions of (θe, r) by z = r sinθe and x/r = (cosθe sinφ, cosθe cosφ, sinθe)T, respectively.
The computed () with is plotted by a thick (thin) curve in Fig. 5 as a function of r for each θe (selected in Fig. 1) with φ = 90°. Since the height assignment error for is much smaller than that for (as will be seen later), the error in relative to |v| can be denoted and estimated by
e4.5
where v(θe, r) = v[z(θe, r)] is taken from the vertical profiles in Fig. 3 at the beam height—that is, z computed from (2.7) for given (θe, r). The estimated REpof(θe, r) is plotted in Fig. 6. As shown, Epof(θe, r) becomes larger than 10% as r exceeds 75 km on θe = 2.4°, and becomes larger than 25% as r exceeds 180 km on the lowest tilt (θe = 0.5°). According to (4.4) and (4.5), Epof is essentially the difference between vTc at z = ze(θe, r) and vTx/r at z = zf(θe, r) normalized by |v| at z = ze(θe, r), where ze(θe, r) and zf(θe, r) are defined in (2.12) and plotted as paired functions of r for each θe in Fig. 1. Since c and x are smooth functions of (θe, r), the rapid variations of Epof with r are caused mainly by the rapid differential variations of v between ze(θe, r) and zf(θe, r), while the upper bound of these variations are largely controlled by ze − zf ≈ r2/(2ae), independent of θe as shown in (2.12). This explains the gross pattern of the envelope (not shown but can be perceived) of all the REpof(r) curves plotted for the different θe in Fig. 6. This feature and the qualitative aspect of the above results remain the same for different settings of φ and different selections of vertical profile of v (not shown). Note that υr is linear in dtυr, so the relative error caused in the first term of (4.2) owing to the use of the flat earth model can be estimated roughly by REpof(θe, r). On the other hand, since υr is quadratic in , the relative error caused in the last term of (4.2) by the flat earth model should be estimated by the square of REpof(θe, r). Because [REpof(θe, r)]2 < REpof(θe, r) according to the results in Fig. 6 and becomes a small term in (4.2) as r increases to 80 km and beyond, the error of the flat earth υr equation is mainly in the first term of (4.2), and this error can be estimated by REpof(θe, r).
Fig. 5.
Fig. 5.

Radial velocity observation , simulated by using (4.4a) with the equivalent earth model, plotted by each dark curve as a function of r for each θe selected in Fig. 1. The thin dotted curve associated with each dark curve is the radial velocity observation simulated by using (4.4b) with the flat earth model.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

Fig. 6.
Fig. 6.

Relative error estimated in (4.5) for [REpof(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

c. Error of truncated equivalent earth radial velocity equation

As reviewed in section 2, the equivalent earth model assumes that the vertical gradient of refractivity index is constant in the lower troposphere, and this gradient is commonly set to the typical value of dn/dz = −1/(4a). In the real atmosphere, however, dn/dz is not constant and its value can fluctuate over a wide range around the above typical value in the lower troposphere (z ≤ 2 km). Occasionally, adn/dz can decrease drastically below −1 within a temperature inversion and/or strong moisture gradient layer in the lower troposphere to cause the radar beam on the lowest tilt (θe, = 0.5°) to bend back toward Earth's surface and therefore produce ground clutter and near-zero values in radial velocity observations. Since ground clutter and associated near-zero velocity must be removed through data quality control, this type of rarely occurring abnormal situation is not considered here. In normal situations, the equivalent earth model can be used to estimate the beam height (Gao et al. 2006, 2008), but setting adn/dz = −¼ can cause an error in the estimated beam height if this commonly used value of adn/dz = −¼ is different from the true value of adn/dz in the lower troposphere. This can cause a small error in the υr equation as evaluated below.

To evaluate the aforementioned error, vertically averaged values of adn/dz over the depth of 0 ≤ z ≤ 2 km are computed from the vertical profiles of temperature and specific humidity collected over the period from 0000 UTC 22 November 2012 to 1200 UTC 14 March 2013 at the operational sounding station LMN at Lamont, Oklahoma. As shown in Fig. 7, the values computed from the early morning soundings at 0600 LT (plotted by the gray dots) are generally more negative than those computed from the latter afternoon soundings at 1800 LT (plotted by the black dots), and this is simply because the vertical stratifications of temperature and specific humidity in the lower troposphere are generally stronger in the early morning than in the latter afternoon. From the time series in Fig. 7, we can also see that the vertically averaged value of adn/dz fluctuates between −0.357 and −0.133, the time mean is −0.233 and the standard deviation is 0.036. These results indicate that the error for the commonly used value of adn/dz = −0.25 is time dependent and fluctuates between 0.107 and −0.107, while the bias is −0.017 and the RMS error is 0.040. The RMS error for a/ae = 1 + adn/dz is thus also 0.04, which is much smaller than the dramatic change of value in a/ae (from ¾ to 0) caused by using the flat earth model. Therefore, the error in caused by fixing adn/dz = −0.25 should be much smaller than the error in caused by the flat earth model. To verify this, we set adn/dz = −0.25 ± 0.04 to compute from (4.4a) with , denote the two computed values by and , respectively, and then use
e4.6
to estimate the RMS error in (caused by using adn/dz = −0.25) relative to |v|. As shown in Fig. 8, REpoe(θe, r) increases in general as θe becomes small and r becomes large, but it does not exceed 5% even on the lowest tilt (θe = 0.5°). The qualitative aspect of the above results remains the same for different settings of φ and different selections of vertical profile of v (not shown). Again, for the same reason as explained for REpof(θe, r) in section 4b, the relative RMS errors caused by setting adn/dz = −0.25 can be estimated by REpoe(θe, r) and [REpoe(θe, r)]2 for the first and last terms in (4.2), respectively, while the relative RMS error caused by setting adn/dz = −0.25 for the υr equation in (4.2) is mainly in the first term and thus also less than 5%.
Fig. 7.
Fig. 7.

Time series of adn/dz (averaged over the depth of 0 ≤ z ≤ 2 km) computed from the vertical profiles of temperature and specific humidity collected every 6 h over the period from 0000 UTC 22 Nov 2012 to 1200 UTC 14 Mar 2013 at the operational sounding station LMN at Lamont, Oklahoma, within the Southern Great Plains (SGP) site established by the Department of Energy's Atmospheric Radiation Measurement Program (ARM). The gray (black) dots plot the values computed from the early morning (late afternoon) soundings at 0600 (1800) LT, while the plus signs plot the values computed from the soundings at the local noon and midnight.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

Fig. 8.
Fig. 8.

Relative error estimated by (4.6) for [REpoe(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-13-011.1

So far, by setting in this and previous subsections, we have not yet considered possible errors in the estimated terminal velocity. In fact, even if , the estimated terminal velocity is still not exactly accurate because of not including the effect of finite beamwidth, although the error is very small (within ±0.18 m s−1 and within ±0.7% of |v|). The accurate estimate should satisfy exactly. Nevertheless, since is usually estimated empirically from the observed reflectivity, the error can be as large as ±20% of the true |wT| (or even ±50% of the true |wT| especially in the graupel or hail region aloft where the air density is low within a severe storm), but its resulting errors in and are within ±0.8 (or ±2.0) m s−1 and within ±12% (or ±30%) of |v| for the cases considered above, according to our computations (omitted here). These errors are independent of the displaced-sampling errors estimated above for and , and they also cause errors independently in the υr equations.

5. Conclusions

In this paper, the prognostic equation for radial velocity is derived using the equivalent earth model to include the effects of atmospheric refraction and earth curvature on radar-beam height and slope angle. The derived equation contains a high-order small term that can be truncated without degrading the accuracy achieved (and also limited) owing to the use of the equivalent earth model. In particular, as estimated analytically [see (4.3)], the upper bound of the truncated term [that is, e/r in (3.25)] is only a few percent of |v|2/rmax, where rmax = 300 km is the maximum radial range of the operational radar scans. This upper bound is verified numerically and tightened [by about 50%, as shown in Fig. 4 versus (4.3)]. The truncated radial velocity equation is shown to be much more accurate than its counterpart radial velocity equation (Xu et al. 2001b) derived without considering the effects of atmospheric refraction and earth curvature. In particular, using the equivalent earth model with the vertical gradient of refractivity index set to the commonly used typical value (dn/dz = −0.25/a) in the lower troposphere can cause only a small relative RMS error (<5%, as shown in Fig. 8) in the truncated radial velocity equation if the true vertical gradient of refractivity index is not too different from the typical value in the lower troposphere (see Fig. 7), but the relative error caused by neglecting the effects of atmospheric refraction and earth curvature in the counterpart radial velocity equation can become larger than 25% as the range distance exceeds 180 km on the lowest tilt (see Fig. 6).

The truncated equation has the same concise form as that in Xu et al. (2001b) and can be used as a dynamic constraint for radar wind analysis in the same way as in Xu et al. (2001b), but the effects of atmospheric refraction and earth curvature are no longer neglected so operational WSR-88D radial velocity observations can be used to full radial ranges (up to 300 km). The radial velocity equation in the 3.5DVar included the above effects but neglected the perturbation pressure and buoyancy terms [see (15) of Xu et al. (2010)], so the radial velocity equation derived in this paper can be used to upgrade the dynamic constraint in the 3.5DVar for radar data assimilation. This is under current research.

Acknowledgments

The authors are thankful to Guoqing Ge of the University of Oklahoma (OU), Richard Doviak of NSSL, and anonymous reviewers for their comments and suggestions that improved the paper and to Kang Nai and Yuan Jiang of OU for their help in producing Figs. 2 and 7. The research was supported by ONR Grant N000141010778 to OU. Funding was also provided by NOAA/OAR under NOAA-OU Cooperative Agreement NA17RJ1227, U.S. Department of Commerce.

REFERENCES

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  • Gao, J., K. Brewster, and M. Xue, 2006: A comparison of the radar ray path equations and approximations for use in radar data assimilation. Adv. Atmos. Sci., 23, 190–198.

    • Search Google Scholar
    • Export Citation

  • Gao, J., K. Brewster, and M. Xue, 2008: Variation of radio refractivity with respect to moisture and temperature and influence on radar ray path. Adv. Atmos. Sci., 25, 1098–1106.

    • Search Google Scholar
    • Export Citation

  • Ge, G., J. Gao, K. Brewster, and M. Xue, 2010: Impacts of beam broadening and earth curvature on storm-scale 3D variational data assimilation of radial velocity with two Doppler radars. J. Atmos. Oceanic Technol., 27, 617–636.

    • Search Google Scholar
    • Export Citation

  • Gu, W., H. Gu, and Q. Xu, 2001: Impact of single-Doppler radar observations on numerical prediction of 7 May 1995 Oklahoma squall line. Preprints, Fifth Symp. on Integrated Observing Systems, Albuquerque, NM, Amer. Meteor. Soc., 139–142. [Available online at https://ams.confex.com/ams/annual2001/techprogram/paper_17251.htm.]

  • Heymsfield, G. M., K. K. Ghosh, and L. C. Chen, 1983: An interactive system for compositing digital radar and satellite data. J. Climate Appl. Meteor., 22, 705–713.

    • Search Google Scholar
    • Export Citation

  • Qiu, C., and Q. Xu, 1996: Least-square retrieval of microburst winds from single-Doppler radar data. Mon. Wea. Rev., 124, 1132–1144.

  • Xu, Q., 2011: Measuring information content from observations for data assimilation: Spectral formulations and their implications to observational data compression. Tellus, 63A, 793–804.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., and C. Qiu, 1995: Adjoint-method retrievals of low-altitude wind fields from single-Doppler reflectivity and radial-wind data. J. Atmos. Oceanic Technol., 12, 1111–1119.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., and L. Wei, 2011: Measuring information content from observations for data assimilation: Utilities of spectral formulations for radar data compression. Tellus, 63A, 1014–1027.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., C. Qiu, and J. Yu, 1994: Adjoint-method retrievals of low-altitude wind fields from single-Doppler wind data. J. Atmos. Oceanic Technol., 11, 579–585.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., C. Qiu, H. Gu, and J. Yu, 1995: Simple adjoint retrievals of microburst winds from single-Doppler radar data. Mon. Wea. Rev., 123, 1822–1833.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and C. J. Qiu, 2001a: Simple adjoint retrievals of wet-microburst winds and gust-front winds from single-Doppler radar data. J. Appl. Meteor., 40, 1485–1499.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and S. Yang, 2001b: Simple adjoint method for three-dimensional wind retrievals from single-Doppler radar. Quart. J. Roy. Meteor. Soc., 127, 1053–1067.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., L. Wei, W. Gu, J. Gong, and Q. Zhao, 2010: A 3.5-dimensional variational method for Doppler radar data assimilation and its application to phased-array radar observations. Adv. Meteor.,2010, 797265, doi:10.1155/2010/797265.

  • Zhao, Q., J. Cook, Q. Xu, and P. Harasti, 2006: Using radar wind observations to improve mesoscale numerical weather prediction. Wea. Forecasting, 21, 502–522.

    • Search Google Scholar
    • Export Citation

  • Zhao, Q., J. Cook, Q. Xu, and P. Harasti, 2008: Improving short-term storm predictions by assimilating both radar radial-wind and reflectivity observations. Wea. Forecasting, 23, 373–391.

    • Search Google Scholar
    • Export Citation
Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Doviak, R. J., and D. S. Zrnic, 2006: Doppler Radar and Weather Observations. 2nd ed. Dover Publications, 562 pp.

  • Gao, J., K. Brewster, and M. Xue, 2006: A comparison of the radar ray path equations and approximations for use in radar data assimilation. Adv. Atmos. Sci., 23, 190–198.

    • Search Google Scholar
    • Export Citation

  • Gao, J., K. Brewster, and M. Xue, 2008: Variation of radio refractivity with respect to moisture and temperature and influence on radar ray path. Adv. Atmos. Sci., 25, 1098–1106.

    • Search Google Scholar
    • Export Citation

  • Ge, G., J. Gao, K. Brewster, and M. Xue, 2010: Impacts of beam broadening and earth curvature on storm-scale 3D variational data assimilation of radial velocity with two Doppler radars. J. Atmos. Oceanic Technol., 27, 617–636.

    • Search Google Scholar
    • Export Citation

  • Gu, W., H. Gu, and Q. Xu, 2001: Impact of single-Doppler radar observations on numerical prediction of 7 May 1995 Oklahoma squall line. Preprints, Fifth Symp. on Integrated Observing Systems, Albuquerque, NM, Amer. Meteor. Soc., 139–142. [Available online at https://ams.confex.com/ams/annual2001/techprogram/paper_17251.htm.]

  • Heymsfield, G. M., K. K. Ghosh, and L. C. Chen, 1983: An interactive system for compositing digital radar and satellite data. J. Climate Appl. Meteor., 22, 705–713.

    • Search Google Scholar
    • Export Citation

  • Qiu, C., and Q. Xu, 1996: Least-square retrieval of microburst winds from single-Doppler radar data. Mon. Wea. Rev., 124, 1132–1144.

  • Xu, Q., 2011: Measuring information content from observations for data assimilation: Spectral formulations and their implications to observational data compression. Tellus, 63A, 793–804.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., and C. Qiu, 1995: Adjoint-method retrievals of low-altitude wind fields from single-Doppler reflectivity and radial-wind data. J. Atmos. Oceanic Technol., 12, 1111–1119.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., and L. Wei, 2011: Measuring information content from observations for data assimilation: Utilities of spectral formulations for radar data compression. Tellus, 63A, 1014–1027.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., C. Qiu, and J. Yu, 1994: Adjoint-method retrievals of low-altitude wind fields from single-Doppler wind data. J. Atmos. Oceanic Technol., 11, 579–585.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., C. Qiu, H. Gu, and J. Yu, 1995: Simple adjoint retrievals of microburst winds from single-Doppler radar data. Mon. Wea. Rev., 123, 1822–1833.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and C. J. Qiu, 2001a: Simple adjoint retrievals of wet-microburst winds and gust-front winds from single-Doppler radar data. J. Appl. Meteor., 40, 1485–1499.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and S. Yang, 2001b: Simple adjoint method for three-dimensional wind retrievals from single-Doppler radar. Quart. J. Roy. Meteor. Soc., 127, 1053–1067.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., L. Wei, W. Gu, J. Gong, and Q. Zhao, 2010: A 3.5-dimensional variational method for Doppler radar data assimilation and its application to phased-array radar observations. Adv. Meteor.,2010, 797265, doi:10.1155/2010/797265.

  • Zhao, Q., J. Cook, Q. Xu, and P. Harasti, 2006: Using radar wind observations to improve mesoscale numerical weather prediction. Wea. Forecasting, 21, 502–522.

    • Search Google Scholar
    • Export Citation

  • Zhao, Q., J. Cook, Q. Xu, and P. Harasti, 2008: Improving short-term storm predictions by assimilating both radar radial-wind and reflectivity observations. Wea. Forecasting, 23, 373–391.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Radar-beam height z [computed by (2.7) with ae = 4a/3 in the equivalent earth model] plotted by dark curves as functions of the length of ray path r, for seven elevation angles (i.e., θe = 0.5°, 2.4°, 4.3°, 6.2°, 8.7°, 12°, and 19.5° selected from the 14 elevation angles of θe = 0.5°, 1.45°, 2.4°, 3.35°, 4.3°, 5.25°, 6.2°, 7.5°, 8.7°, 10°, 12°, 14°, 16.7°, and 19.5° in volume coverage patterns 11 and 211 used by the operational WSR-88Ds to scan thunderstorms). The shaded band around each dark curve shows the beam thickness (within θe ± 0.5° for each θe). The thin dotted straight line associated with each dark curve is the beam height computed by z = r sinθe in the flat earth model.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Schematic drawing of radar beam (ray path) in the equivalent earth model with the transformed earth surface plotted by the hatched arc. The equivalent earth center is at point O. The radar is at point A. The radar beam is along line AB of length r. Line BC shows the beam height z at point B above the earth surface. Arc AC shows the horizontal distance s from radar to point C along the great circle.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Vertical profiles of (u, υ, w, wT) and 2 log10Z (plotted in units of 5 dBZ) extracted from the main updraft area within the simulated Oklahoma squall line (valid at 2200 UTC 2 Jun 2004) produced by the control run in Xu et al. (2010), where Z is the reflectivity factor (mm6 m−3) [see section 4.4.5 of Doviak and Zrnic (2006)].

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Relative truncation error e/|v|2 (%), defined in (4.3), as a function of r for each θe selected in Fig. 1.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Radial velocity observation , simulated by using (4.4a) with the equivalent earth model, plotted by each dark curve as a function of r for each θe selected in Fig. 1. The thin dotted curve associated with each dark curve is the radial velocity observation simulated by using (4.4b) with the flat earth model.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Relative error estimated in (4.5) for [REpof(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Time series of adn/dz (averaged over the depth of 0 ≤ z ≤ 2 km) computed from the vertical profiles of temperature and specific humidity collected every 6 h over the period from 0000 UTC 22 Nov 2012 to 1200 UTC 14 Mar 2013 at the operational sounding station LMN at Lamont, Oklahoma, within the Southern Great Plains (SGP) site established by the Department of Energy's Atmospheric Radiation Measurement Program (ARM). The gray (black) dots plot the values computed from the early morning (late afternoon) soundings at 0600 (1800) LT, while the plus signs plot the values computed from the soundings at the local noon and midnight.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Relative error estimated by (4.6) for [REpoe(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.

Fig. 1.

Radar-beam height z [computed by (2.7) with ae = 4a/3 in the equivalent earth model] plotted by dark curves as functions of the length of ray path r, for seven elevation angles (i.e., θe = 0.5°, 2.4°, 4.3°, 6.2°, 8.7°, 12°, and 19.5° selected from the 14 elevation angles of θe = 0.5°, 1.45°, 2.4°, 3.35°, 4.3°, 5.25°, 6.2°, 7.5°, 8.7°, 10°, 12°, 14°, 16.7°, and 19.5° in volume coverage patterns 11 and 211 used by the operational WSR-88Ds to scan thunderstorms). The shaded band around each dark curve shows the beam thickness (within θe ± 0.5° for each θe). The thin dotted straight line associated with each dark curve is the beam height computed by z = r sinθe in the flat earth model.
Fig. 2.

Schematic drawing of radar beam (ray path) in the equivalent earth model with the transformed earth surface plotted by the hatched arc. The equivalent earth center is at point O. The radar is at point A. The radar beam is along line AB of length r. Line BC shows the beam height z at point B above the earth surface. Arc AC shows the horizontal distance s from radar to point C along the great circle.
Fig. 3.

Vertical profiles of (u, υ, w, wT) and 2 log10Z (plotted in units of 5 dBZ) extracted from the main updraft area within the simulated Oklahoma squall line (valid at 2200 UTC 2 Jun 2004) produced by the control run in Xu et al. (2010), where Z is the reflectivity factor (mm6 m−3) [see section 4.4.5 of Doviak and Zrnic (2006)].
Fig. 4.

Relative truncation error e/|v|2 (%), defined in (4.3), as a function of r for each θe selected in Fig. 1.
Fig. 5.

Radial velocity observation , simulated by using (4.4a) with the equivalent earth model, plotted by each dark curve as a function of r for each θe selected in Fig. 1. The thin dotted curve associated with each dark curve is the radial velocity observation simulated by using (4.4b) with the flat earth model.
Fig. 6.

Relative error estimated in (4.5) for [REpof(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.
Fig. 7.

Time series of adn/dz (averaged over the depth of 0 ≤ z ≤ 2 km) computed from the vertical profiles of temperature and specific humidity collected every 6 h over the period from 0000 UTC 22 Nov 2012 to 1200 UTC 14 Mar 2013 at the operational sounding station LMN at Lamont, Oklahoma, within the Southern Great Plains (SGP) site established by the Department of Energy's Atmospheric Radiation Measurement Program (ARM). The gray (black) dots plot the values computed from the early morning (late afternoon) soundings at 0600 (1800) LT, while the plus signs plot the values computed from the soundings at the local noon and midnight.
Fig. 8.

Relative error estimated by (4.6) for [REpoe(θe, r)] (%), plotted by each dark curve as a function of r for each θe selected in Fig. 1.
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