8A.6 SINGLE-SCAN RADAR REFRACTIVITY RETRIEVAL: THEORY AND SIMULATIONS

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1 8A.6 SINGLE-SCAN RADAR REFRACTIVITY RETRIEVAL: THEORY AND SIMULATIONS B. L. Cheong 1, and R. D. Palmer 1,2 1 Atmospheri Radar Researh Center, The University of Oklahoma, Norman, U.S.A. 2 Shool of Meteorology, The University of Oklahoma, Norman, U.S.A. 1. INTRODUCTION Surfae refrativity measurements have been reeiving inreasing attention in reent years due to its lose relation with surfae moisture. Previous work has demonstrated that onvetive preipitation initiation is highly dependent on the surfae moisture (e.g., Dabberdt and Shlatter, 1996; Koh et al., 1997; Bodine et al., 29). A radar method to retrieve near surfae refrativity using ground lutter ehoes was first developed by the work of Fabry et al. (1997). Field experiments were onduted and have shown suess of the tehnique (Fabry and Pettet, 22; Wekwerth and Parsons, 23). A similar tehnique has been developed and implemented here at the University of Oklahoma (Cheong et al., 28). The tehnique uses phase measurement from two sans to derive the hange of refrativity between the sans. Absolute refrativity is obtained by hoosing a ondition where referene refrativity an be obtained from surfae measurements from the Oklahoma Mesonet. In this work, a tehnique to derive surfae refrativity using phase measurements from one san is presented. This tehnique is referred to as the single-san radar refrativity retrieval (SR3). Currently, there is no radar suitable for the implementation of SR3 but its feasibility is explored here from theory and simulations. 2. THEORY OF SINGLE-SCAN RADAR REFRACTIVITY RETRIEVAL (SR3) Theoretially, the reeived phase from stationary targets is an integral funtion of the refrative index, this quantity an be expressed as follows (Bean and Dutton, 1968) φ(r) = 4π λ n(γ)dγ (1) where λ represents wavelength of the radar and r is the range. Beause the value of n is lose to 1 for appliations near the earth s surfae, a onvenient term refrativity is frequently used. It an be represented as N = 1 6 (n 1) (2) Corresponding author address: Boon Leng Cheong, University of Oklahoma, Atmospheri Radar Researh Center, 12 David L. Boren Blvd., Rm 464, Norman, OK ; boonleng@ou.edu For most weather radars, the operation wavelength is on the order of m and sine n 1, the phase wraps many times onsidering target range that may span up to 5 km. Therefore, deriving refrativity diretly from a single san using the absolute phase is problemati. However, from Eq. (1), one an also realize that refrative index is a loal derivative of the phase as follows n(r) = λ d φ(r) (3) 4π dr The key problem is to unwrap the gate-to-gate phase hange, rather than to retrieve the absolute phase. This problem is still a hallenging problem but less problemati than retrieving the absolute phase of eah gate. For a given disrete range sampling of a radar, Eq. (3) is applied as n(r) λ 4π φ(r) (4) r = λ [φ(r + r) φ(r)] (5) 4π r It should be emphasized here that r represents the entroid spaing between the two adjaent range gates, whih is a funtion of the ground target distribution in those two ells. This parameter must be estimated aurately in order to suessfully retrieve the refrative index. Clearly, there are multiple solutions that an exist in unwrapping phase measurements. Fortunately, for the appliation near the earth surfae, the refrative index an only be valid for N [2, 4] and the separation of the multiple solutions are suffiient for us to always hoose the unique solution. A numerial example is illustrated as follows. For simpliity, parameters of an X-band radar are onsidered here. Let λ = 3 m, r = 9 m and n(r) = 1.3, using Eq. (1), the theoretial phase hange would be rad. from one gate to the next. Now, onsider the possibility of an inorret unwrapped phase, e.g., add ±2π offsets to the theoretial number. These would be the next possible solutions that are losest to the truth. The estimates of ˆφ = ± 2π = rad, rad would result in refrative index estimates of 1.133, 1.466, whih are invalid for the typial atmospherial onditions near the earth surfae. In pratie, when this senario is enountered, the unique phase an be obtained by adding the appropriate integer multiplies of 2π so that the solution of n(r) falls within the valid range.

2 8A.6 2 From Eq. (5), we an also solve for r if n(r) is known, as r = λ [φ(r + r) φ(r)] (6) 4πn(r) Similar to estimating n(r), multiple solutions for r exist but unfortunately there is no distint signature that allows us to hoose the unique solution. If two frequenies are available, however, then two phase measurements an be made to mitigate the limitation. Essentially, the so alled frequeny domain interferometry (FDI) is applied here (e.g., Kudeki and Stitt, 1987; Franke, 199; Palmer et al., 199). For the SR3, we are interested in the range entroid spaings (rather than the absolute range) so the effets of initial phase of the radar an be negleted. With two phase measurements, Eqs. (1) and (3) beomes φ 1 (r) = 4πf 1 φ 2 (r) = 4πf 2 n(γ)dγ (7) n(γ)dγ (8) φ 1 (r) φ 2 (r) = 4π(f 1 f 2 ) n(γ)dγ (9) d dr [φ 1(r) φ 2 (r)] = 4π(f 1 f 2 ) n(r) (1) where is the speed of light. Assuming the refrative index is uniform within the two sampling points in range, we an then derive the spaing between the two points if n(r) is known. r = 4π(f 1 f 2 )n(r) [φ 1(r) φ 2 (r)] r (11) In pratie, r may be estimated during a ondition where n(r) is less omplex. In that ase, objetive analysis from surfae stations an be used to derive n(r). From Eq. (11), one an think of the dual-frequeny ombination effetively provides us a lower operating frequeny (f 1 f 2 ) (longer wavelength). Multiple solutions still exist for Eq. (11) but the separation among the solutions beome larger and thus, allows for finding a unique solution. Of ourse, with the ambiguity of phase wrapping, there is a limit to r estimation, whih is r [, 2(f 1 f 2 ) ] m (12) Sine it is very rare to have entroid spaing lose to zero, The aquired phase an be rewrapped by a π-equivalent into the [π, 3π] interval, whih then maps the entroid spaing into [ r 4(f 1 f 2 ), 3 4(f 1 f 2 ) ] m (13) It should also be mentioned here that range oversampling may be applied to aid this proess. In the next setion, a numerial simulation will be presented to demonstrate the proess deriving refrative index from a single san measurement using two frequenies. 3. RESULTS FROM SIMULATIONS A simple refrativity distribution is used for simulation, illustrated in Figure 1(a). Here, we onsidered an X-band radar system with two frequenies, i.e., f 1 = 9.55 GHz, f 2 = GHz, r = 9 m and N(r) [2, 4] Estimating Centroid Spaings With the seleted frequeny separation, we have the ability to estimate entroid spaings within [41.67, 125] m, aording to Eq. (13). For simpliity, we assume that ground lutter is present in all range gates. Sine only a pair of adjaent measurements are onsidered for eah point of refrativity retrieval, regions without ground lutter simply may not be used in pratie. A 3-m uniformly distributed random perturbation is introdued to the range gate spaing to simulate random ground target distribution along the range. In addition, a.1 uniformly distributed random phase is added to simulate instrumental noise. For now, a quite preise equipment is assumed. Later in this paper, different phase noise will be presented to show the effets of suh ontamination to this tehnique. Using Eq. (3), the absolute returned phase for eah frequeny is shown in Figure 1(b). They are wrapped into the [ π, π] range in Figure 1(), whih represent the atual phase measurements that would be obtained in pratie. Using the simulated phase measurements in Figure 1(), the first step is to alulate the gate-to-gate phase differene for the two measurements from the two frequenies, i.e., the term [φ 1 (r) φ 2 (r)] r in Eq. (11). The phase is then rewrapped into the [π, 3π] interval. Using the rewrapped phase, the entroid spaing are derived and is shown in Figure 2(b). With a preise phase measurement (.1 noise), the orresponding estimation error is small and shown in Figure 2() Single-san Radar Refrativity Retrieval Again, for simpliity, the same radar system is used. One the entroid spaing of ground targets are identified, whih an be done ahead of time, radar refrativity an be retrieved by using the phase measurements from one san using the method presented in Setion 2. In this example, we assumed that the entroid spaings have been obtained with an auray on the orders of 1 m, as depited in Figure 2(), and proeed with refrativity retrieval based on phase measurement from one of the two frequenies. With the estimated entroid spaings, we an first derive the initial guess of the gate-to-gate unwrapped phase as shown in Figure 3(a). Proeed with refrativity derivation using Eq. (5) and Eq. (2), the initial estimated refrativity is shown in Fig-

3 8A.6 3 Refrativity (N) Phase Differene (rad.) Phase (rad.) (a) Simulated Refrativity Field (N) x 16 (b) Absolute Phase () Measured Phase (rad.) Figure 1: Simulation setup with an X-band radar with two frequenies at f 1 = 9.55 GHz, f 2 = GHz. Random range perturbations and measurement noise are added to simulate radar ehoes. (a) shows the refrativity distribution in the experimental setup, (b) shows the theoretial absolute phase with range perturbation and added noise while () shows the wrapped phase measurements. f 1 f 2 Error (m) Estimated and Rewrapped Phase Differene (rad.) Centroid spaing (m) x 1 Estimation Error (m) Est Figure 2: Range-gate entroid spaing an be estimated if Refrativity is known. In pratie, this an be ahieved by using a refrativity field from objetive analysis using surfae measurements, e.g., ASOS or Oklahoma Mesonet, during onditions where the field is less omplex.

4 8A.6 4 ure 3(b). Note that there are some refrativity values that an be onsidered outliers. Using a simple interval hek where N [2, 4], we an reover these points into the valid range and the result is shown in Figure 3(). Refrativity (N) (a) Refrativity with Phase Noise of.1 deg Phase (rad.) Refrativity (N) Refrativity (N).5 x (a) Initial Unwrapped Phase (rad.) (b) Estimate of Refrativity (N) first pass () Estimate of Refrativity (N) seond pass Figure 3: SR3 to retrieve refrativity using one phase measurement. (a) shows the initial unwrapped phase measurements and (b) shows the orresponding initial refrativity estimate. It an be seen that there are several outliers. Sine N [2, 4], they an be orreted and () shows the results by re-mapping the outliers into the valid range. Refrativity (N) Refrativity (N) (b) Refrativity with Phase Noise of.5 deg () Refrativity with Phase Noise of.1 deg Figure 4: With noise added to the phase measurements, the performane of refrativity retrieval deteriorated. 5. FUTURE WORK Here at the ARRC at OU, a travelling wave tube (TWT) based X-band radar is urrently being developed. Simulation parameters presented in this paper have been hosen to mimi this system and this radar will be the first testbed for the SR3 tehnique. Future work also inludes investigation of using multiple frequenies to solve for entroid spaings and refrative index simultaneously. Referenes 4. INSTRUMENTAL PRECISION The preision of refrativity retrieval diretly depends on the preision of phase measurements, this an be realized from Eq. (5). Figure 4 illustrates the effets of phase noise on refrativity retrieval. It an be seen that the results of refrativity suffers tremendously even with phase noise of.5 (uniformly distributed in [.25,.25] ). There are several open questions on the pratiality of the SR3 tehnique. The most important ones inlude the instrumental preision and the stability of entroid spaings of ground lutter. These parameters diretly ontrol the viability of the tehnique and we are urrently exploring the pratial impliations of SR3. Bean, B. R., and E. J. Dutton, 1968: Radio Meteorology. Dover Publiations. Bodine, D., B. L. Cheong, P. L. Heinselman, R. D. Palmer, and D. Mihaud, 29: Convetive initiation and storm evolution foreasting using radar refrativity retrievals. Wea. Foreasting, under review. Cheong, B. L., R. D. Palmer, C. D. Curtis, T.-Y. Yu, D. S. Zrnić, and D. Forsyth, 28: Refrativity retrieval using the Phased Array Radar: First results and potential for multifuntion operation. IEEE Trans. Geosi. Remote Sens., 46(9), Dabberdt, W. F., and T. W. Shlatter, 1996: Researh opportunities from emerging atmospheri observing modeling apabilities. Bull. Amer. Meteor. So., 77,

5 8A.6 5 Fabry, F., C. Frush, I. Zawadzki, and A. Kilambi, 1997: On the extration of near-surfae index of refration using radar phase measurements from ground targets. J. Atmos. Oeani Tehnol., 14(4), Fabry, F., and C. Pettet, 22: A primer to the interpretation of refrativity imagery during IHOP 22. IHOP 22 Refrativity Manual. Franke, S. J., 199: Pulse ompression and frequeny domain interferometry with a frequeny-hoped MST radar. Radio Si., 25, Koh, S. E., A. Aksakal, and J. T. MQueen, 1997: The influene of mesosale humidity and evapotranspiration fields on a model foreast of a old-frontal squall line. Mon. Weather Rev., 125, Kudeki, E., and G. R. Stitt, 1987: Frequeny domain interferometry: a high-resolution radar tehnique for studies of atmospheri turbulene. Geophys. Res. Lett., 14(3), Palmer, R. D., R. F. Woodman, S. Fukao, M. F. Larsen, M. Yamamoto, T. Tsuda, and S. Kato, 199: Frequeny domain interferometry observations of tropo/stratospheri sattering layers using the MU radar: Desription and first results. Geophys. Res. Lett., 17, Wekwerth, T. M., and D. B. Parsons, 23: An overview of the International H 2 O Projet (IHOP 22). Bull. Amer. Meteor. So., 85(2),