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A GPM Dual-Frequency Retrieval Algorithm: DSD Profile-Optimization Method C. R. ROSE
AND
V. CHANDRASEKAR
Colorado State University, Fort Collins, Colorado (Manuscript received 18 June 2005, in final form 13 February 2006) ABSTRACT A new dual-frequency precipitation radar (DPR) will be included on the Global Precipitation Measurement (GPM) core satellite, which will succeed the highly successful Tropical Rainfall Measuring Mission (TRMM) satellite launched in 1997. New dual-frequency drop size distribution (DSD) and rain-rate estimation algorithms are being developed to take advantage of the enhanced capabilities of the DPR. It has been shown that the backward-iteration algorithm can be embedded within a single-loop (SL) feedback model. However, the SL model is unable to correctly estimate DSD profiles for a significant portion of global median volume diameter Do and normalized DSD intercept parameter Nw combinations in rain because of a multiple-value solution space. For the remaining Do, Nw pairs, another retrieval method is necessary. This paper proposes a supplementary profile-optimization technique to find those DSD profiles in the rain region that the SL model cannot correctly determine. The optimization method is based on a model that both Do and log(Nw) are linear vertical profiles, and that the profiles can be found using an optimization technique from the input reflectivity profiles. Using those assumptions, the optimization method finds the top and bottom Do, log(Nw) values such that a cost function related to the input-measured reflectivity is minimized. A random-restart method is used to generate random top-and-bottom DSD seed values for each optimization cycle. Example cases are shown to demonstrate the performance with and without error in the input reflectivity profiles. Limitations of the method are discussed, including its performance when the input reflectivity profiles are based on nonlinear DSD profiles and values of shape factor different than the algorithm assumed value.
1. Introduction Following the success of the Tropical Rainfall Measuring Mission (TRMM) launched in 1997, the nextgeneration precipitation radar (PR) is expected to be launched aboard the Global Precipitation Measurement (GPM) core satellite around 2009. The TRMM PR operates at a single frequency of 13.8 GHz and uses retrieval algorithms that rely on the surface-reference technique (SRT) to estimate path attenuation and correct the measured Ku-band reflectivity measurements. With the attenuation-corrected reflectivities, a reflectivity-based algorithm is used to retrieve the rain rate (Iguchi et al. 2000). This method works well for moderate-to-heavy rainfall rates where the SRT-derived attenuation value is large compared to its error. The GPM core satellite will use a dual-frequency precipitation radar (DPR) at Ku (13.6 GHz) and Ka (35.6
Corresponding author address: Chris Rose, Colorado State Univeristy, 1373 Campus Delivery, Fort Collins, CO 80523. E-mail:
[email protected]
© 2006 American Meteorological Society
JTECH1921
GHz) bands to measure and map global precipitation with unprecedented accuracy, resolution, and areal coverage. Along with the new DPR come new algorithms to measure and retrieve precipitation parameters, such as the drop size distribution (DSD) parameters in each resolution volume. The underlying microphysics of precipitation structures and DSDs dictate the types of models and retrieval algorithms that can be used to estimate precipitation. Figure 1a is a depiction of the downward-looking GPM core satellite showing two rays, one for each radar frequency, projected through a storm cloud and precipitation region. The small discs represent the resolution volumes of the radar. Generally, there are two main types of dualfrequency algorithms that can be used with a downward-looking radar—1) the forward method, where the DSDs are calculated at each bin starting from the top bin and moving down to the bottom; and 2) the backward method, where the algorithm begins at the bottom bin and moves upward to the top, calculating the DSD parameters along the way. The assumption with the forward method is that there is known or assumed at-
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FIG. 1. (a) A depiction of a downward-looking GPM satellite. The discs represent sampling volumes. The forward method calculates DSD values starting at the top and moving to the bottom. The backward method calculates from the bottom to the top. (b) The definition of bin nomenclature and specific attenuation.
tenuation above the top bin and the integral equations are solved in a single pass through the hydrometeor regions. Forward methods have limited application because of a tendency to diverge in regions of moderateto-heavy attenuation or moderate-to-heavy rainfall (Liao and Meneghini 2004). Backward-calculation algorithms tend to be more stable than the forward types but require an a priori knowledge of the total two-way path-integrated attenuation (PIA) for each ray or an ability to calculate it. Considerable work has been done to evaluate backward-calculating dual-frequency algorithms, such as a hybrid SRT method (Meneghini et al. 1997, 2002). Additionally, another retrieval algorithm being studied for use by GPM is an iterative, dualfrequency algorithm that does not use PIA derived from the SRT but instead estimates it as part of an iterative process (Mardiana et al. 2004). Inherent in the application of any dual-frequency retrieval algorithm are assumptions about the types of hydrometeors in each region: above the melting layer in snow, within the melting layer, and below the melting layer in rain. Because of the potentially large attenuation of the Ka-band radar return in high rain-rate regions, the dual-frequency method has a firm limitation on the maximum rain rate that it can be used to measure. For GPM, the iterative dual-frequency method appears to be best suited for low-to-moderate rain rates below about 12–18 mm h⫺1 (assuming a uniform rain column, 3 km in height), and yields more detailed DSD information, such as the normalized DSD intercept parameter Nw and median volume diameter Do, than does the single-wavelength method used in TRMM. Rose and Chandrasekar (2005) incorporated the dual-frequency iterative algorithm of Mardiana et al.
(2004) into a single-loop (SL) feedback-control structure, described as the SL model, and showed that about half of global rainfall could be incorrectly estimated (based on Do, Nw retrievals) by the SL model. Their work focused on the rain region and assumed that the PIA down from the top of the storm to the top of the rain could be estimated and used to compensate for the measured radar reflectivity profile values within the rain region. This paper describes a supplementary method, called the DSD profile-optimization method, for DSD retrieval. It is based on a linear model of vertical profiles for Do and log(Nw) and the ability to estimate the Do, Nw values in both the top and bottom bins. (Note that log ⫽ log10, and ln ⫽ loge.) It offers advantages over the SL model in that it does not suffer from the multivalued solution-space problem relating to large Do, Nw, Ai combinations. It also is not susceptible to the bivalued Do ambiguity for rain described in detail by Liao et al. (2003), Mardiana et al. (2004), and Meneghini et al. (1997, 2002). Following this introduction, section 2 offers a brief review of background concepts and necessary mathematical relationships. Section 3 describes the theory behind the applicability and solution space for the linear DSD model solution. In section 4, several test cases are analyzed and discussed showing the performance and limitations of the proposed method. Two simulations are performed using input reflectivity data with no intrinsic measurement error (based on defined DSD profiles), and two are performed using simulated reflectivity profiles with added measurement error. The results of error and error-free data are discussed and analyzed. Data from a 1000-profile simulation are included, showing the mean and standard error of the method when measurement error is randomly added to multiple profiles. Performance of the optimization method with reflectivity profiles based on nonlinear DSD inputs and different shape factor values is demonstrated. Section 5 briefly summarizes the optimization method.
2. Background By way of brief summary, the following is a review of important background concepts on which this paper builds that are described in detail by Rose and Chandrasekar (2005). The drop size distribution N(D) (mm⫺1 m⫺3) is based on the normalized gamma of the form (Bringi et al. 2004; Testud et al. 2001) N共D兲 ⫽ NW f 共兲
冉 冊 D Do
e⫺⌳D,
共1兲
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section and is a function of the drop diameter D; i is the free-space wavelength (m), and Kw is defined as
where ⌳⫽
3.67 ⫹ , Do
共2兲 KW ⫽
共3.67 ⫹ 兲⫹4 , f 共兲 ⫽ 3.674 ⌫共 ⫹ 4兲
6
共3兲
where ⌫ is the gamma function, and Do is the median volume diameter (mm). The value of is fixed ( ⫽ 1) in these algorithms. Note that ⌳ and Do are related according to (2). Figure 1b is a schematic showing the nomenclature for the variables used in this work. The measured radar reflectivity factor Zmi (mm6 m⫺3) at range rj can be expressed in general form as Zmi 共rj 兲 ⫽ Zei 共rj 兲Ai 共rj 兲,
共4兲
where the subscript i (i ⫽ 1, 2) represents the particular frequency (13.6 and 35.6 GHz, respectively), and j is the number of the range bin, 1 ⱕ j ⱕ N, with N equal to the number of bins. Here, Zei(rj) is the effective radar reflectivity and Ai(rj) is the two-way attenuation factor. Although (4) and subsequent equations are general and could apply to the entire path—from snow down through the melting and rain regions—in this paper, they are restricted to the rain region such that r1 is the top of the rain and rN is the bottom. Note that pathintegrated attenuation for each wavelength is PIAi ⫽ 10log(Ai) and is expressed in decibels. The specific attenuation ki(rj) (dB km⫺1) is defined for the region between bins. In the following equations, a tilde (⬃) over a variable name indicates it is an algorithmderived value. ˜ (r) at bin r ⫽ r is calculated Estimated reflectivity Z mi j using (4), ˜ 共r兲A ˜ 共r兲, ˜ 共r兲 ⫽ Z Z mi ei i
共5兲
where ˜ 共r兲 f 共兲D ˜ ⫺I 共D ˜ 兲, ˜ 共r兲 ⫽ N Z ei W o bi o
共6兲
j
˜ 共r兲 ⫽ exp关⫺0.2 ln共10兲h A i
兺 k˜ 共r 兲兴, i
n
共7兲
n⫽1
˜ expressed as where Ibi is a function of D o ˜ 兲⫽C Ibi 共D o Zi
冕
bi 共D兲De⫺⌳D dD,
共8兲
D
CZi ⫽
i4 5 | Kw |
m2 ⫺ 1 m2 ⫹ 2
共9兲
˜ (r ) in (6)–(8). The radar range resolution ˜ ⫽D and D o o N h is equal to 0.25 km; bi is the radar backscatter cross
共10兲
,
where m is the complex index of refraction of water at 20°C at these wavelengths. The specific attenuation k˜i in (7), at a particular range r ⫽ rj , is defined as ˜ f 共兲D ˜ ⫺I 共D ˜ 兲, k˜i 共r兲 ⫽ N W o ti o
共11兲
where ˜ 兲⫽C Iti 共D o ki
冕
ti 共D兲De⫺⌳D dD,
共12兲
D
Cki ⫽ 4.343 ⫻ 10⫺3,
共13兲
and ti is the radar extinction cross section and is a ˜ (r ) in (11)–(12). ˜ ⫽D function of drop size D and D o o j The integration limits on D in (8) and (12) are 0.2– 3.2Do (mm), which we view as being sufficient given a Do greater than or equal to 0.5 mm. Liao and Meneghini (2004) stated that under relatively high rain rates the iterative approach does not converge. Rose and Chandrasekar (2005) later showed that the dual-wavelength SL model (which incorporates the iterative model) can converge to incorrect DSD values when the retrieved Nw, Do pairs are in the incorrect convergence region. This incorrect estimation of the true DSDs occurs when the solution space to the integral equations becomes multivalued and the method has insufficient constraints to reach the correct solution. One means of forcing correct convergence (correct DSD retrieval) of the SL model is to reduce PIA, such as eliminating bins (data points) from the profile bottom, which then allows the correct Nw, Do values to be estimated in the remaining higher bins with a corresponding reduction in vertical DSD profile. A simple test has been developed by Rose and Chandrasekar (2005) to determine the approximate region of convergence of a DSD retrieval from the SL method, and the subsequent likelihood of the DSDs being in the incorrect convergence region. Note that this is an approximate test and gives a good indication of region. The relationship test is described by
冋
Do ⫽ a ⫹ , 2
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b ˜ 0.5 N w
册
2
,
共14兲
where a ⫽ 0.9989, b ⫽ 18.31. The test is performed by ˜ (r ⫽ r ) and deterapplying (14) using a retrieved N w 1 ˜ mining the maximum allowable Do. If the retrieved D o
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is larger than the allowable Do shown by (14), then the retrieved Nw, Do pair is in the incorrect convergence ˜ is smaller, then the solution region. If the retrieved D o is in the correct convergence region.
3. Optimization method a. Background The SL method finds the DSD parameters Do, Nw, as well as Ai and ki in each bin without imposing profile constraints on Do and Nw. The profile-optimization method, which has followed as an extension of the SL work, adds two constraints in the rain region in order to retrieve the DSD profiles in that region. It assumes that log(Nw) and Do can be approximated by linear vertical profiles and, given a set of input Zmi(rj) values, that an optimal solution can be found for the top- and bottombin values of Do, Nw such that a cost function between ˜ (r ) is minimized. Like the the Zmi(rj) and estimated Z mi j SL analyses, the profile-optimization method described in this paper assumes that the PIA for each ray can be obtained from the storm top down to the top of the rain region, and that these PIA values can be used to adjust or compensate for the measured reflectivity values in the rain region before retrieval. There has been considerable work done to accurately measure the attenuation above the rain region resulting from the bright band, cloud droplets, and water vapor. This work is ongoing with results showing promise. Recently, Meneghini et al. (2005) described preliminary results for a method using three separate radar frequencies to determine the water vapor content and attenuation. In searching for constraints to impose on the rain region DSD profiles for use with the optimization method, we desired simplicity, such as a minimum number of degrees of freedom (i.e., fewer regression coefficients require less processing and less sophisticated optimization methods), while maintaining a reasonably close relationship to DSD empirical observation. We note that linear constraints on Do and log(Nw) are an approximation to observed DSD profiles. Chandrasekar et al. (2003b) studied the DSD profiles from two precipitation regions, coincident with both TRMM PR overpass and ground radar (GR) measurements, and showed that both the GR and TRMM PR estimated the Do profiles to be fairly linear in the rain region; and the GR showed that the log(Nw) profile could easily be fit by a line. Chandrasekar et al. (2003a), using data from the Texas and Florida Underflights (TEFLUN)-B experiment and the TRMM Large Scale Biosphere–Atmosphere Experiment in Amazonia (LBA), showed that the profiles of both Do and
FIG. 2. Flowchart illustrating the four-variable random-restart optimization method. (a) Input reflectivity values with unknown Do, Nw profiles. (b) Random seed values are generated. (c), (d) The interaction within the optimization routine to find the top and bottom Do, Nw values to minimize a cost function relating to the input Zmi and internally calculated Zmi values.
log(Nw) could be approximated by linear functions. Bringi et al. (2004) showed DSD profiles from a convective cell beginning at the early growth stage and developing into a more mature phase, resulting in an intense microburst. In the mature phase, the data show that both Do and log(Nw) can be reasonably approximated as being linear in profile. Using dual-wavelength empirical data, Mardiana et al. (2004), using ⫽ 0 to perform DSD retrievals, presented results showing that Do is substantially linear in the rain region. The profile plots for Nw are in linear scale and are not as obvious as to their linear-log applicability. Based on these observations, we obviously do not conclude that Do and log(Nw) profiles are always linear in the rain region, but in our work of requiring simple constraints to these two profiles, the linear assumption appears to be reasonable. A flowchart of the DSD profile-optimization method is shown in Fig. 2. The Zmi(rj) values (dBZ ) are depicted in Fig. 2a and are inputs to the optimizer shown in Figs. 2c and 2d. Seed values for the optimizer, depicted in Fig. 2b, are generated using the randomrestart method described by Hu et al. (1997). Using both the Zmi(rj) and seed values, the optimizer finds a solution for the top and bottom Do, Nw values, indicated by circles and squares in Fig. 2c. At the beginning of each optimizer cycle (see Figs. 2c and 2d) the top and bottom Do, Nw seed values are used to create linear Do, log(Nw) profiles. Using the DSD ˜ (r ) profiles, the method then calculates estimated Z mi j ˜ (r ) to using (5)–(13). The optimizer compares the Z mi j the input Zmi(rj) profiles during each iteration as it searches the top and bottom DSD variable spaces to
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minimize a cost function. The cost function is the minimum of C1 ⫹ C2 stated as
timum Do, Nw values, the retrieved rain profile at each range r is calculated using
min共C1 ⫹ C2兲,
R共r兲 ⫽ 0.6 ⫻ 10⫺3Nw共r兲f 共兲Do⫺
共15兲
where C1 ⫽
C2 ⫽
冑 冑
1 N
1 N
⫻ m1,dBZ共rj 兲
⫺ Zm1,dBZ共rj兲兴2,
共16兲
j⫽1
共18兲
where
共D兲 ⫽ 4.854De⫺0.195D
N
兺 关Z˜
共D兲D⫹3e⫺⌳共r兲D dD,
D
N
兺 关Z˜
冕
m2,dBZ共rj 兲
⫺ Zm2,dBZ共rj兲兴 , 2
共17兲
j⫽1
and N is the number of bins. Both Zmi,dBZ(rj) and ˜ Z mi,dBZ(rj) in (16) and (17) are in dBZ.
b. Methodology As a general procedure, the SL feedback method should be executed first to retrieve the DSD and rainrate profiles and a test using (14) should be performed to determine if the top bins are in the incorrect convergence region. Depending on the test result, the profileoptimization method can be executed to retrieve the “best fit” Do, Nw profiles for those cases where the SL method is insufficient. The optimizer used in this work allows for the nonlinear optimization of multiple variables. The search space of Do is constrained between 0.5 and 2.5 mm, and Nw is constrained between 103 and 105 [3 ⱕ log(Nw) ⱕ 5]. The random variables used in the random-restart method as seed values are uniformly distributed, 0.75 ⱕ Do ⱕ 2.25 and 103 ⱕ Nw ⱕ 6 ⫻ 104, with the seed having a slightly narrower range than the expected final values. Additionally, both the slopes of the random seed and simulated Do and log(Nw) profiles are opposite to each other such that if the top Do value is greater than the bottom Do value, then the top Nw value will be smaller than the bottom Nw value. Experimental data and analyses indicate that with some frequency the slopes of Do and log(Nw) are opposite one another (Bringi et al. 2004; Chandrasekar et al. 2003b; Mardiana et al. 2004). Liao and Meneghini (2004), in simulating test profiles for a dual-frequency iterative retrieval algorithm, also used oppositely sloped DSD profiles. Of course, we cannot state that oppositely sloped DSDs are always the case, but these are typical example profiles of Do and Nw. It should be noted that the optimization method does not require that the DSDs be oppositely sloped. In normal operation, 10–50 random-restart iterations are performed. After the iterations are completed, the cost function minimum is found along with the corresponding top and bottom Do, Nw values. Using the op-
共19兲
is the terminal velocity (Gunn and Kinzer 1949).
4. Results The simulated datasets, analyzed and discussed in the following examples, are each for a 3-km-height vertical rain column, based on DSD values of Do 1.60–1.75 and Nw log(8000)–log(4000), from top to bottom, respectively. In each of these cases, the simulated, measured input reflectivity profiles Zmi(rj) are derived from this DSD profile. This particular Do, Nw combination is chosen to illustrate the difficulty the SL method has in retrieving the correct DSD profile values and demonstrate the retrieval performance of the optimization method. Many other DSD profile combinations could have been used from the hundreds of simulated pairs, but this example suffices for illustrative purposes. No constant vertical DSD profile datasets are analyzed here because they are viewed as subsets of the more general linear profiles.
a. Linear vertical profile for Do, Nw, without measurement error The SL retrieval method is demonstrated to establish a baseline retrieval to which the remaining simulations can be compared. Figure 3 shows the output profiles from both the SL retrieval and optimization methods. The input Zmi profiles (dBZ ), with no measurement error in Zmi(rj), are shown in Fig. 3d by the solid and dashed curves. Because this Do, Nw combination is in the incorrect convergence region both in the top and bottom bins, the SL method incorrectly estimates the DSD values in the lower bins. Figure 3a both shows the SL-retrieved Do profile (dotted curve, labeled SL Do) and that it is correct at the top bin at 1.60 mm and incorrect at the bottom bin at 2.14 mm (true value 1.75). Figure 3a also shows the SL-retrieved Nw profile (log scale, labeled SL logNw), and a correct value at the top and an inaccurate Nw value in the bottom bin of 2.922 [log(835)] (true value 4000). The SL-retrieved rain-rate profile is shown in Fig. 3b as the dotted curve. The
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FIG. 3. Profiles from the SL and profile-optimization methods for a vertical rain column 3 km in height based on Do: 1.60–1.75; log(Nw): log(8000)–log(4000). (a) The true (solid line) and estimated (dots) Do profiles. The right portion of (a) shows the true log(Nw) (dashed) and estimated (asterisks) profiles. The single-loop outputs are indicated. (b) The rain-rate profile obtained via the optimizer (dots), and the true profile (solid) along with the single-loop output. (c) The effective radar reflectivity factors (solid and dashed lines) along with the estimated profiles (asterisks and dots). (d) The input Zmi values (solid and dashed) and optimizer outputs (dots and asterisks) at both frequencies. Twenty random-restart cycles were used, with the second cycle being selected. Single-loop convergence tolerance was 0.01% and required 3966 iterations.
value at the top bin is correct at 18.4 mm h⫺1, but the retrieved bottom value of 7.3 mm h⫺1 should be 13.9 mm h⫺1, yielding a 47% underestimation. Retrievals using the optimization method are shown for comparison. Figure 3a shows profiles of the true Do ˜ values (solid line on the left) and optimizer-found D o (indicated by dots). The heavy dashed line on the right is true log(Nw) and asterisks are optimizer-found ˜ ). Note that the optimizer-retrieved values for log(N w both Do and Nw overlay exactly. The final DSD values from the optimizer are shown at the top and bottom of the profiles. Figure 3b shows the true (solid line) and retrieved rain-rate profiles (dots). Figure 3c compares the Zei(rj) profiles (heavy solid and dashed lines, dBZ ) based on the true Do, Nw and retrieved values shown as dots and asterisks. Figure 3d shows comparisons of the
input Zmi(rj) (heavy solid and dashed lines) and retrieved Zmi(rj) values (dots and asterisks) based on optimized Do, Nw values. The minimum value of the cost function was found at the second iteration of the 20 random-restart operations performed. Optimization tolerance was set to 10⫺6.
b. Linear vertical profile for Do, Nw, with measurement error In this section, effects of measurement error on the SL and profile-optimization retrieval methods are described. The same DSD-based Zmi profiles are used from section 4a, but with Gaussian error, 0.5-dBZ standard deviation, and zero mean added to the Zmi profiles in each bin to simulate measurement error in the system. Figure 4d shows the input reflectivity profiles
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FIG. 4. Profiles from the single-loop and profile-optimization methods for a vertical rain column 3 km in height based on Do: 1.60–1.75; log(Nw): log(8000)–log(4000) with added measurement error. Error added to the 13.6-GHz profile was 0.49 dBZ, and error added to the 35.6-GHz profile was 0.64 dBZ, both one standard deviation. (a) The true Do, log(Nw) profiles with the estimated values from the optimizer and single-loop methods. (b) The true and estimated rain-rate profiles, including single-loop rain estimation. (c) True and estimated effective reflectivity. (d) The input and estimated measured reflectivity profiles. Fifty random-restart cycles were performed with the residual minimum found at cycle number 35. Single-loop convergence tolerance was 0.01% and required 42 iterations.
(solid line for 13.6, and dashed line for 35.6 GHz) used for both retrievals. The standard deviation of the random error for the 13.6-GHz reflectivity profile is 0.49 dBZ, and the standard deviation for the 35.6-GHz profile is 0.64 dBZ, where these error numbers are calculated as the standard deviation of the Zmi(rj) noise minus the Zmi(rj) true profile without noise at each bin in the profile. Note that because the error is random the amount of error in each bin will vary from simulation to simulation, as will the total error in each profile. Figure 4a shows the SL-retrieved Do profile (solid curve labeled SL Do). At the top bin, the SL-retrieved value of 1.78 mm differs from the true value without the error of 1.60 mm. Even more error is present in the bottom-bin retrieval with a value of 4.93 mm. The error in the lower bins is caused both by the incorrect convergence region
and the error in the reflectivity profiles. Figure 4a also shows the SL-retrieved log(Nw) profile (dotted curve labeled SL logNw) from log(3797) at the top to log(1.02) at the bottom. Figure 4b both shows the SL rain-rate profile (dotted curve) and that it varies significantly from the true profile shown by the solid curve. The top rain-rate value should be 18.4 mm h⫺1, but 14.3 mm h⫺1 is estimated. The bottom value should be 13.9 mm h⫺1, but 0.31 mm h⫺1 is retrieved—a 98% underestimation. For comparison, the optimization method retrievals are shown in Fig. 4. Figure 4a shows the true Do and log(Nw) profiles (Do is the solid line and Nw is the dashed line) with superimposed Do, log(Nw) profiles (dots and asterisks) from the optimizer. Note that the top Do points agree but the bottom true value is 1.75
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and the retrieved value is 1.85 mm. The optimizer slightly overestimates the log(Nw) value at the top at 3.923 but underestimates it at the bottom. The bottom true value is 3.602 ⫽ log(4000), but the retrieved value is 3.429. Figure 4b shows the true and estimated rainrate profiles. The top value should be 18.4 but is estimated as 19.2 mm h⫺1. At the bottom, the expected value is 13.9, but is retrieved as 11.9 mm h⫺1—a 14% underestimation. Figure 4c shows the true (solid line, dashed line) and estimated Zei values (dots and asterisks) at both 13.6 and 35.6 GHz. The retrieved values for 13.6 GHz are very close to those expected but there is some underestimation in the 35.6-GHz values in the lower part of the profile. Of the 50 random-restart cycles used in this example, the optimization method found the best approximation in cycle number 35 given the input reflectivity data and constraints. Note that the estimated reflectivity profiles (see Fig. 4d) closely overlay the input reflectivity profiles. The optimization method is not able to retrieve exactly the true DSD profiles because of errors in the input Zmi(rj) data. Note that even in this scenario of inaccurate input Z mi (r j ) values, the optimization method was able to find reasonably estimated Zmi(rj) profiles and DSD profiles that are much more accurate than those retrieved using the SL method with the same input data. Because the added measurement error to the reflectivity profiles in this example was random, the retrieved rain-rate values based on this method will vary from simulation to simulation in accordance with the amount and location of random error in each of the profiles. Figure 5 shows a histogram of the bottom-binestimated rain-rate values using the profileoptimization method from 1000 simulated profiles [based on Nw log(8000)–log(4000), Do 1.75–1.60, vertical rain column 3 km high], each with added 0.5-dBZ standard deviation zero mean random error, as described above. Additionally, for each simulated profile, the optimizer was configured for 15 random-restart cycles. The results show that the estimated mean of the histogram is 13.86 mm h⫺1, close to the correct value of 13.9 mm h⫺1, demonstrating the important result that the optimization process is not biased. The standard error of the distribution is 1.47.
c. Nonlinear vertical profile for Do, Nw optimization, without measurement error In this section, we examine how the profileoptimization method retrieves the DSD and rain-rate profiles when the input DSD profiles are nonlinear, that is, they do not meet our linear profile assumption. For this case, we use the same DSD pairs of section 4a,
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FIG. 5. Plot showing a histogram of the bottom-bin rain-rate estimation for 1000 simulated profile pairs, each with added random 0.5-dBZ standard deviation measurement error and using 15 random-restart iterations per simulation. The reflectivity profiles are for a 3-km-high vertical rain column based on Nw: log(8000)– log(4000); Do: 1.75–1.60. The mean of the histogram is 13.86 mm h⫺1 close to the true mean of 13.9. The standard deviation is 1.470.
but make the profiles second order (parabolic) instead of linear, with the perturbations occurring in the middle bins of each DSD profile. We mention that many other possible DSD profile shapes could be used based on myriad functions including sine, cosine, or others. For example, if a cosine function is used, in the middle, the DSD values tend to zero. For a sine-type function, at the top and bottom bins, the DSD values are zero. It is doubtful that we will see such cases in nature. Therefore, we used part of a sine (a sine function with a an offset), which approximates a parabola or second-order function. No noise (random measurement error) was added to the datasets to be able to clearly describe performance of the profile-optimization method. Figure 6 shows input and optimizer-found Do and log(Nw) profiles. True Do is shown by the solid curve on ˜ values are indithe left, and optimizer-estimated D o cated by dots. The DSD values are shown at the top and bottom of the profiles. The heavy, dashed curve on the right shows true nonlinear log(Nw) ranging from 3.903 to 3.602 at the bottom and the asterisks are optimizer˜ ) ranging from 3.834 to 3.606 at the found linear log(N w bottom. Note that there is some error between the optimizer-retrieved values for both Do and Nw and their true profiles because of the algorithm’s linear assumptions. Figure 7 shows true (solid line) and retrieved rainrate profiles (dots). The true rate in the bottom bin, as before, is 13.9 mm h⫺1, but the rate calculated by the optimizer is 15.2 mm h⫺1—a 9.4% overestimation. Figure 8 compares the input Zmi(rj) (heavy solid and
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FIG. 6. Graphic showing true nonlinear and linear estimated Do, log(Nw) profiles. The optimizer input reflectivity profiles were based on these nonlinear Do and log(Nw) profiles. True Do is the solid curve; true log(Nw) is dashed. Estimated profiles are dots and asterisks. Because of the nonlinearity, there is some error between the retrieved and true values.
dashed curves) and retrieved Zmi(rj) values (dots and asterisks). The minimum value of the cost function was found at the second iteration of the five random-restart operations performed in this example. Optimization tolerance was set to 10⫺6. Even though the input reflectivity profiles were based on nonlinear DSD profiles, the profile-optimization method closely matched the input reflectivity profiles and provided best-fit lin-
FIG. 7. Graphic showing true (solid curve) and optimizerestimated (dots) rain-rate profiles from reflectivity profiles based on nonlinear Do and log(Nw) of Fig. 6. True and estimated rainrate profiles are shown with some overestimation of rain rate in the bottom.
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FIG. 8. Graphic showing true and optimizer-estimated reflectivity profiles. Input reflectivity profiles based on nonlinear Do and log(Nw) profiles. Dashed lines with dots and asterisks are profiles estimated by the optimization procedure. Input-measured reflectivity profiles are shown as solid and dashed curves. Note that the method found very good approximations to the input reflectivity profiles. The nonlinear Do, log(Nw) profiles of Fig. 6 were used to simulate these true profiles.
ear DSD profiles and a reasonable estimation of rain rate. Again, we emphasize that an equivalent linear fit is made.
d. Linear vertical profile for Do, Nw, with different values Various authors have assumed and used different values of in developing, analyzing, and testing retrieval algorithms. From Bringi et al. (2003), the value of has been shown to be in the range of ⫺1 ⬍ ⱕ 5. Mardiana et al. (2004) presented algorithm development and DSD retrieval results based on simulated and empirical data using ⫽ 0. Liao and Meneghini (2005) assumed ⫽ 2 to test retrieval methods. Using groundbased polarimetric radar data to perform DSD retrieval, Bringi et al. (2004) also showed that ⫽ 3 is a reasonable estimate under some circumstances. In version 5 of the TRMM algorithm, ⫽ 3 was used to perform the calculations and regressions for the k–Z and Z–R power-law relationships (Kozu and Iguchi 2000). The profile-optimization algorithm described in this paper assumes ⫽ 1. We note that if the simulated datasets are based on one value of and the retrievals use another [e.g., (5)–(13)], then the retrieved DSD profiles will be skewed. In this section, we describe how the profile-optimization method retrieves DSD and rain-rate profiles using input reflectivity profiles based
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FIG. 9. Retrieved profiles using input data based on ⫽ 0 with retrieval based on ⫽ 1. (a) The true and estimated rain-rate profiles. True rain rate is solid; retrieved values are dots. The profile-optimization method underestimates the bottom-bin rain rate by 7.6%. (b) The optimization method closely matches both the 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks are optimizer-generated values at 13.6 and 35.6 GHz.
on ⫽ 0, 2, and 3 and the same DSD pairs detailed in section 4a. The case for ⫽ 1 datasets has already been shown in Figs. 3 and 4. The optimizer-found results for ⫽ 0 are shown in Fig. 9. True (solid curve) and estimated rain rate (dotted curve) are shown in Fig. 9a. The optimizer-found bottom-bin rate is 7.3 and the true rate is 7.9 mm h⫺1—a 7.6% underestimation. The 13.6- and 35.6-GHz input and estimated reflectivity profiles for ⫽ 0 are shown in Fig. 9b. Solid lines are true values while dots
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FIG. 10. Retrieved profiles using input data based on ⫽ 2 with retrieval based on ⫽ 1. (a) The true and estimated rain-rate profiles. True rain rate is solid; retrieved values are dots. The profile-optimization method overestimates the bottom-bin rain rate by 3.8%. (b) The optimization method closely matches both the 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks are optimizer-generated values at 13.6 and 35.6 GHz. The reflectivity profiles have been shortened because of noise floor constraints.
and asterisks are estimated. Note that they are in very good agreement because the optimizer outputs match the input reflectivity profiles. The results for ⫽ 2 are shown in Fig. 10. True (solid curve) and estimated (dots) rain rate are shown in Fig. 10a. The optimizer overestimates the bottom-bin rain rate by 3.8%. Figure 10b shows the true reflectivity profiles (solid and dashed curves) with estimated profiles (dots and asterisks). The input and estimated re-
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flectivity profiles overlay exactly. Because of large attenuation (resulting from a relatively high rain rate) in the Ka-band signal, a reduced number of reflectivity points were used to remain above a defined 10-dBZ noise floor. Figure 11 shows the optimizer results for ⫽ 3. True (solid curve) and estimated (dotted curve) rain-rate profiles are shown in Fig. 11a. The optimizer overestimates the rainfall rate by 7%. The true (solid and dashed curves) and estimated (dots and asterisks) reflectivity profiles are shown in Fig. 11b. We note that, again, the estimated and true reflectivity profiles closely match. Note that with this DSD and combination, the rain rate is very high leading to large attenuation of the Ka-band signal. As before, reflectivity data points below the noise floor have been truncated. We note that in each of these cases ( ⫽ 0, 1, 2, 3), the profile-optimization method is able to closely match the input reflectivity profiles (although they are based on different values of ) with profiles based on ⫽ 1 and thereby estimate reasonable rain-rate profiles.
5. Summary This paper has described a supplementary, dualfrequency method to retrieve rain region DSD values based on assumed linear vertical profiles for the DSD using a nonlinear profile-optimization technique. The optimization technique requires as inputs the Zmi(rj) values for both wavelengths and top and bottom seed values for a random-restart process. Outputs from the method are the top and bottom values of Do, Nw that minimize a cost function relating to the input Zmi(rj) ˜ (r ) profiles. From the reand internally calculated Z mi j trieved top and bottom DSD values, linear profiles for Do, Nw are calculated from which the rain-rate profile is estimated. To illustrate the performance of the technique, it was compared with the SL method both with and without measurement error using simulated linear DSD profiles based on Do, Nw pairs found in the incorrect convergence region. As expected, the SL model incorrectly retrieved the DSD values in the lower altitudes both with and without measurement error. The optimization technique was able to retrieve the correct DSD profiles throughout the vertical profile when no measurement error was included. When measurement error was present, the optimization technique retrieved a “best estimate” of the true DSD profiles that closely matched the known values. A simulation of 1000 profiles with added random measurement error showed that the optimizer-found mean value of the bottom-bin rain rate indicated no bias in the retrievals.
FIG. 11. Retrieved profiles using input data based on ⫽ 3 with retrieval based on ⫽ 1. (a) The true and estimated rain-rate profiles. True rain rate is solid; retrieved values are dots. The profile-optimization method overestimates the bottom-bin rain rate by 7.0%. (b) The optimization method closely matches both the 13.6- and 35.6-GHz reflectivity profiles. Dots and asterisks are optimizer-generated values at 13.6 and 35.6 GHz. The reflectivity profiles have been shortened because of noise floor constraints.
Based on observation, we have assumed that the vertical profiles of Do and log(Nw) can often be approximated as linear in the rain region. In the optimization process, any nonlinearity of the DSD profiles (noncompliance with the linear model assumptions) will contribute to error in the retrievals. We showed retrieval results from a dataset based on nonlinear DSD profiles and that the optimizer estimated reasonable linear fits to the DSD, rain rate, and measured reflectivity pro-
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files. Many other nonlinear DSD profiles could be used with varying degrees of optimizer compliance. The profile-optimization method detailed in this work is based on ⫽1 [see (1)–(3)]. To demonstrate how the optimizer performs with datasets based on unexpected values of , three retrieval test cases were shown based on ⫽ 0, 2, and 3. For the ⫽ 0 case the retrieved rain-rate error was ⫺7.6%, for ⫽ 1, 0%; for ⫽ 2, 3.8%; and for ⫽ 3, 7%. We note that in all of these cases, the optimizer was successful in matching the input reflectivity profiles at 13.6 and 35.6 GHz and providing reasonable best estimates for the DSD and rain-rate profiles. The profile-optimization method assumes that the PIA to the top of the rain region can be measured and used to adjust the rain region reflectivity values. Research is ongoing to find accurate and reliable means of measuring this attenuation caused by cloud water vapor, water droplets, and melting region. (The attenuation in the melting region is more readily measured and characterized than the attenuation caused by water vapor and cloud droplets because it is directly associated with DSDs.) The TRMM algorithm accounts for the water vapor and droplet attenuation by its use of the total PIA from the SRT, though by using the PIA from the SRT, other errors are introduced into the algorithm retrieval (Iguchi et al. 2000). At present, several retrieval algorithms are being considered for use by GPM. One algorithm being considered is the SL model, which does not use the SRT, and another uses GPM dual-frequencies in conjunction with the SRT (Meneghini et al. 1997, 2002). Iguchi (2005) recently described a dual-frequency algorithm similar to the TRMM algorithm but that uses two frequencies and an optimization method constrained on rain rate calculated for each wavelength. At present, the optimization method described in this paper could be used as a supplement to the SL method where the SL retrieval, using appropriate models for snow/ice and melting region, estimates the PIA at the top of the rain region. These PIA values, one at each frequency, are then used to adjust the rain-region reflectivity values. The shortcoming of this method is the omission of direct measurements of attenuation resulting from cloud water vapor and droplets. It can also be used with the modified DF model using SRT, or perhaps with the modified dual-frequency Hitschfeld–Bordan (DFHB) method. The authors view this work as contributing to the ongoing research into suitable algorithms for GPM. Acknowledgments. This research was jointly supported by the NASA Precipitation Program and Los Alamos National Laboratory.
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