Comparison of Meteorological Radar Signal Detectability with Noncoherent and Spectral-Based Processing

Transcription

1 APRIL 2016 M E A D 723 Comarion of Meteorological Radar Signal Detectability with Noncoherent and Sectral-Baed Proceing JAMES B. MEAD ProSening Inc., Amhert, Maachuett (Manucrit received 21 October 2014, in final form 30 October 201) ABSTRACT Detection of meteorological radar ignal i often carried out uing ower averaging with noie ubtraction either in the time domain or the ectral domain. Thi aer conider the relative ignal roceing gain of thee two method, howing a clear advantage for ectral-domain roceing when normalized ectral width i le than ;0.1. A imle exreion for the otimal dicrete Fourier tranform (DFT) length to maximize ignal roceing gain i reented that deend only on the normalized ectral width and the time-domain weighting function. The relative ignal roceing gain between noncoherent ower averaging and ectral roceing i found to deend on a variety of arameter, including the radar wavelength, ectral width, available obervation time, and the fale alarm rate. Exreion reented for the robability of detection for noncoherent and ectral-baed roceing alo deend on thee ame arameter. Reult of thi analyi how that DFT-baed roceing can rovide a ubtantial advantage in ignal roceing gain and robability of detection, eecially when the normalized ectral width i mall and when a large number of amle are available. Noncoherent ower etimation can rovide uerior robability of detection when the normalized ectral width i greater than ;0.1, eecially when the deired fale alarm rate exceed 10%. 1. Introduction Detection of meteorological radar ignal in the reence of noie ha been tudied extenively (cf. Keeler and Paarelli 1990; Doviak and Zrnić 1993). In the context of noncoherent ower averaging, a i ued by the WSR-88D for reflectivity etimation (Doviak and Zrnić 1993), it i well known that the ignal detection threhold reduce a the quare root of the number of amle averaged (Marhall and Hitchfeld 193; Clothiaux et al. 199). Sectral-baed roceing, through the ue of the dicrete Fourier tranform (DFT) to comute the ower ectrum, rovide an alternative method for etimating reflectivity and other ectral moment (Gage and Balley 1978; Farley 198; Kollia et al. 200). Signal detection in ectral roceing ha been hown to imrove linearly with DFT length, ince the DFT imlement coherent integration (Farley 198; Lyon 2004). However, thi linear imrovement in detectability i Denote Oen Acce content. Correonding author addre: Jame B. Mead, ProSening Inc., 107 Sunderland Road, Amhert, MA mead@roening.com trictly true only for zero bandwidth ignal in the reence of noie. In addition, averaging multile DFT-derived ower ectra to etimate the final ower ectrum reult in a reduction in the ignal detection threhold by the quare root of the number of ower ectra averaged (Farley 198). More recently, dualolarization weather radar uing imultaneou tranmiion and recetion (STAR) roceing benefit from aymtotic reduction in noie ower when cro correlating the ignal from orthogonal olarization channel (Keränen and Chandraekar 2014; Ivić et al. 2012, 2014). Given the limited number of radar ule available for ignal etimation, it i ueful to conider the otimal combination of coherent integration through DFT roceing and ectral ower averaging that maximize the robability of detection for a given fale detection (or fale alarm) rate. In thi aer, we how that a imle exreion for the otimal DFT length required to maximize the ignal-to-threhold ratio i a good redictor of the DFT length yielding maximum robability of detection. The otimal DFT length i deendent on only two arameter, a contant that deend on the time-domain windowing function ued in comuting the DFT and the normalized Doler ectral width n. The Doler ectral width rior to normalization y i determined by turbulence DOI: /JTECH-D Ó 2016 American Meteorological Society

2 724 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 withintheradarulevolumeawellaotherfactor,uch a beam broadening and hear broadening (Kollia et al. 2011; Gage 1990). Cloud and reciitation ectral width a meaured uing Doler radar vary from ;0.1 to everal meter er econd [ee Gage (1990); Doviak and Zrnić (1993) for examle of data anning thi range], which reult in n in the range of ;10 2 to ;0. deending on radar wavelength and ule reetition frequency. Thi aer i trictly concerned with otimizing detection robability and doe not conider other iue, uch a meaurement bia or variance in the context of ectral moment analyi for etimating reflectivity, velocity, and velocity ectral width. It i noted in aing that noncoherent method of moment etimation have well-known error (Doviak and Zrnić 1993), although they are unbiaed for reflectivity and velocity. Reflectivity etimated from the zeroth moment of the noie-ubtracted ower ectrum i biaed at low ignal-to-noie ratio if the region of ummation i limited to contiguou ectral oint of oitive value near the ectral eak (Mead 2010). Thee iue are beyond the coe of thi aer, which i rimarily concerned with the detection of weak ignal in the reence of noie. The aer i organized a follow: Section 2 decribe the well-known roblem of ignal detection with threholding uing noncoherent ower averaging. Section 3 addree ectral-baed roceing, deriving exreion for ignal roceing gain and otimal DFT length. Fale alarm threhold factor are derived a a function of fale alarm rate for both method in ection 4.In ection,the formula for ignal roceing gain are comared to imulated value howing excellent agreement. Exreion for robability of detection are derived in ection 6. The relative ignal roceing gain between noncoherent roceing and ectral roceing at the otimal DFT length i hown to be aroximately equal to the relative ignal-to-noie ratio needed to achieve the ame robability of detection. Meaured X-band cloud radar data roceed uing both method are reented in ection 7, which demontrate that the analytical model baed on Gauian ignal tatitic are redictive for real data. A lit of ymbol i rovided for convenience in aendix A. 2. Signal detection with noncoherent ower averaging To comare noncoherent ower averaging to ectralbaed roceing, it i ueful to define a detection threhold et according to an allowable robability of fale alarm. The detection threhold T determine if a articular ower average i declared a detection; that i, P 2 P n $ T, (1) where P Efjj 2 g i the ower average of M amle of the ignal-lu-noie vector and P n i the mean noie ower for the averaging eriod. For imle ower averaging with mean noie ower ubtraction [cf. (6.28) in Doviak and Zrnić 1993], the detection threhold T n a a function of fale alarm rate may be exreed a [cf. (11) in Clothiaux et al. 199] P 2 P n $ T n f P an n ffiffiffiffiffi, (2) M where M i the total number of amle and f an i the fale alarm factor for noncoherent ower averaging that i a function of M and the fale alarm rate. The fale alarm factor raie or lower the threhold baed on the accetable level of fale detection. A a reference oint, a baeline detection threhold Tn 0 P n i etablihed for ingle amle detection (M 1), which correond to a robability of fale alarm of Given thi reference threhold, (2) exhibit a reduction in the threhold by a factor of f an M 20., which i tyically le than one for ractical value of M 1 and reaonable fale alarm rate ( f an, 6 for robability of fale alarm ). An exreion for the fale alarm factor a a function of fale alarm rate i derived in ection 4. The reduction in threhold achieved by ower averaging can be thought of a ignal roceing gain or noncoherent gain G n in minimum detectable ignal a comared to ingle amle detection, and i given a G n T 0 n /T f ffiffiffiffiffiffiffi 21 n an M. (3) The ignal-to-threhold ratio for noncoherent roceing STR n i given a STR n P P ffiffiffiffiffi M SNR G T n P n f n, (4) an where SNR i the ignal-to-noie ratio. The formula reented above for noncoherent roceing and thoe derived in the following ection for ectral roceing, aume that ome method i etablihed for accurately etimating noie ower. Noie ower can fluctuate due to a variety of caue, for examle, change in atmoheric brightne temerature a a function of antenna direction or change in receiver gain and noie figure. Thu, it i critical to imlement method that continuouly etimate noie ower, uch that each averaging eriod ha a noie etimate ecific to the data gathered during that averaging eriod. 3. Signal detection with ectral-baed roceing The ower ectrum of a given random ignal can be etimated from a finite amle of length M a follow.

3 APRIL 2016 M E A D 72 The M amle are divided into M equal egment of length N. Each egment i roceed by a DFT of length N (Doviak and Zrnić 1993) and then ower averaged to form the ower ectrum etimate P k : P k 1 1 ån21 2 M åm21 w(i) 0 N (i)e 2[( j2ki)/n] MN M, () where j ffiffiffiffiffiffiffi 21 ; i the th N-length egment of ; and w i an N-element vector of time-domain weight, that i, the time-domain window function. The time-domain weight are alied to reduce ectral artifact reulting from the finite amle length (Lyon 2004). The iue i to find the value of N to otimize ignal detectability. An N-oint DFT i a coherent integration algorithm that for zero bandwidth ignal reult in a gain in SNR that cale aroximately linearly with N (Lyon 2004). The ignal roceing gain i evident when comaring the SNR in the frequency domain to that of the original time-domain ignal (Lyon 2004). The deviation from N i due to calloing lo (Prahbu 2014), which can be eentially eliminated by ue of ecialized windowing function (Lyon 2011) or by overamling in the frequency domain by zero adding the time-domain inut (Prahbu 2014). Zero adding reult in a mall correction to the fale alarm threhold, which i addreed in ection 4c. For a nonuniform window, the gain in SNR for a zero bandwidth ignal i given by the following exreion: ån21 2 w(i) G w, (6) N21 å jw(i)j 2 where the numerator i the ignal ower gain and the denominator i the noie ower gain. Thi exreion i derived in aendix D. For the following, it i convenient to define a coherent integration lo factor L w, L w G w N, (7) which give the lo in roceing gain relative to uniform weighting. For examle, L w 2/3 for N. 2 when uing a Hanning window. Lo factor L W i given for variou time-domain window function in Table 1. For ectral roceing, STR i defined a the ratio of eak ectral ower to detection threhold following noie ubtraction, a hown in Fig. 1. The detection threhold for ectral roceing T i exreed in term of the eak ignal in the ower ectrum P0 and the noie ower ectral denity Pn : TABLE 1. Coherent integration lo factor L W, ectral broadening cale factor K W, otimal DFT length cale factor K ot, and reading lo at otimal DFT length L ot for commonly ued window. Window L W K W K ot L ot Uniform Triangular Hamming Hanning Blackman Harri P 0 2 P n $ T f P a n ffiffiffiffiffi, (8) M where f a i the fale alarm factor for ectral-baed roceing. Power ectrum baed ignal roceing gain i defined a the ratio of otroceing STR to ingle-ule SNR and combine both coherent and noncoherent integration gain. Coherent integration gain increae linearly with DFT length N for mall N but aymtotically aroache a aturation level when the ignal ectral eak lobe become ditributed over multile DFT bin for finite bandwidth ignal. To account for thi aturation effect, a reading lo factor L i defined a the ratio of ower in the ignal eak to the total ectral ower, uch that coherent integration gain i equal to NL. For thi analyi, a Gauian ower ectrum of the form P (y) 1 ffiffiffiffiffiffi ex 24 ln2 [y 2 y! 0 ]2 2 y 2 y (9) i aumed, where y i velocity, y 0 i mean velocity, and the ectrum width y i defined a the half-ower full width of the ectral eak 1 (m 21 ). For dicrete Fourier tranform uing common windowing function that do not ignificantly ditort the Gauian hae of the underlying ower ectrum, reading lo i given by the following exreion, derived in aendix C: L erf ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ln(2)dv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 2 w 1, (10) 2 y where erf i the error function, w i the full width at half maximum ectral width broadening ariing from the dicrete Fourier tranform for a articular windowing function, and DV D i the ectral reolution of the DFT, that i, 1 The relationhi between y and the mathematical tandard deviation i y /(2 ffiffiffiffiffiffiffiffiffiffiffiffi 2 ln2).

4 726 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 FIG. 1. (left) Power ectrum howing noie ower ectral denity (dahed line). The unambiguou velocity i denoted a y a. (right) Power ectrum following noie ubtraction howing ignalto-threhold ratio (STR). Noie ike exceeding the threhold level contitute fale alarm. DV D 2y a N F l 2N, (11) where l i the radar wavelength (m), F i the ule reetition frequency (Hz), and y a F l (12) 4 i the unambiguou velocity (m 21 ). Sectral broadening due to the DFT i linearly related to ectral reolution when alying a uitable windowing function, that i, W K w DV D, (13) where K W i a window-deendent cale factor, ranging in value from 1.02 to 1.72 for common window function, or zero for a uniform window. Subtituting (11) and (13) into (10) yield a normalized form for reading lo: 0 1 ffiffiffiffiffiffiffiffiffiffiffi B ln(2) C L erf@ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi KW 2 1 A, (14) N2 2 n where the normalized ectral width n i given by n y 2y a. (1) To find K W for a articular time-domain window, we note that Pareval theorem can be ued to comute L directly when n 0 and y 0 0. By definition, the reading lo for zero ectral width and zero velocity L 0 i given by L 0 P k0 N21 å k0 P k, (16) where P k0 i the ower in the zero-velocity bin of the ower ectrum. When n 0 and y 0 0, (i) A 0, Pareval theorem i FIG. 2. Sreading lo L, comuted from (14), a a function of N for variou normalized ectral width, auming a Hanning timedomain window. A 2 0 N ån21 jw(i)j 2 N21 å P k. (17) k0 Combining (16) and (17) demontrate that L 0 L W: L 0 P k0 A 2 ån21 0 N A 2 N N å ån21 2 w(i) w(i) N 2 L jw(i)j 2 N21 A 2 0 å jw(i)j 2 N21 W. N å jw(i)j 2 (18) Equating (14) to (18) for n 0 yield the deired olution: ffiffiffiffiffiffiffi ln2 K W erf 21 (L W ). (19) The term L i lotted a a function of N in Fig. 2. For mall N, L L W then begin to dro in magnitude for N. 0. 1/ n. To oit an exreion for ectral ignal roceing gain, the effect of coherent integration ffiffiffiffiffi gain (NL ), noncoherent integration gain ( M due to averaging of M individual ower ectra), ectral fale alarm threhold, f a (derived in ection 4b), and a correction factor for finite averaging, C m (decribed below), are combined. Power ectrum baed ignal roceing gain G i equal to the roduct of thee four factor: ffiffiffiffiffi G NL Mf 21 a C (20) m and the ectral ignal-to-threhold ratio i

5 APRIL 2016 M E A D 727 FIG. 3. Power ectrum with y 0. 27y a, M 4, and N 26 howing multile eak in the main lobe region. The correction factor C m i hown (db) relative to the exected value of the ectrum (red). STR SNR G. (21) Note that G G n in the limit of M 1. The correction factor C m i required when the number of ectral average M i mall. Thi iue i hown grahically in Fig. 3, where everal eak aear in the main lobe region. Each bin of the ower ectrum i tatitically indeendent of it neighbor, with chi-quare ditribution of order 2M. Furthermore, any one of the eak in the region near the ectral eak ha the otential of being the eak value in the ectrum. Thu, the exected value of the eak exceed the mean value of the central eak by the factor C m. Through imulation, an aroximate exreion for C m wa found: n C m lnm m$ 1, (22) M 0.6 where m i an aroximation of the number of DFT bin in the eak ignal region of a Gauian ectrum: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 m 2 n N2 2 n N $ ffiffi 2 and m 1 n N, ffiffiffi 2, (23) where m can take on noninteger value. The term C m i lotted in Fig. 4 a a function of DFT length for two value of n auming M 2. Thi figure how that tatitical fluctuation due to finite averaging in the ower ectrum rovide enhanced detection, equivalent to a roceing gain of everal decibel under condition of large N and mall M. Detail of the model for C m are reented in aendix B. FIG. 4. Simulated and modeled C m a a function of DFT length for M 2 with Hanning weight alied to the time-domain ignal: modeled value (olid line), imulated value for n 0. 3 (triangle), and imulated value for n (diamond). The otimal DFT length can be found by finding N that maximize the ignal roceing gain (20). However, (20) i a nontrivial function of N and cannot be readily differentiated. An aroximate formula for the otimal DFT length i given by N ot ffi K ot K F l ot, (24) n 2 y where K ot i a contant that deend on the articular window function alied, for examle, K ot for the Hanning window. Figure lot the aroximation of otimal DFT length N ot (24) and the actual DFT length that maximize ignal roceing gain (20) determined numerically, a a function of ectral width and two fale alarm robabilitie. The aroximate and exact value are een to be in cloe agreement, demontrating that (24) i a good redictor of the otimal DFT length for maximizing the ignal roceing gain. Otimal ignal roceing gain occur when N ffi N ot : ffiffiffiffiffiffiffiffiffiffiffiffiffi G ot ffi N ot L ot ffiffiffiffiffi M f 21 ao Lot f 21 ao K ot M n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ot f 21 ao F K ot lt ob, (2) 2 y where L ot i the reading lo comuted at N N ot, T ob i the total obervation time, and f ao i the fale

6 728 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 FIG.. From (24), N ot (olid trace) when alying a Hanning window, and DFT length N that maximize ignal roceing gain (20) v normalized ectral width for two fale alarm rate: fa (dahed red trace) and fa 0. 2 (dotted red trace). alarm factor comuted at N ot. Note that C m 1 when N N ot. For all of the weighting function tudied, it wa found that L ot ffi 0. 8L W. The formula for G ot how that otimal DFT-baed roceing gain cale a the quare root of the radar wavelength and obervation time, and cale linearly with ule reetition frequency. Furthermore, otimal DFT-baed roceing gain relative to noncoherent gain (3) i given by G R G ot G n Lot f an f ao f 21 an G ot q ffiffiffiffiffiffiffiffiffiffiffiffiffi F T ob ffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ot n L ot f ao f an ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ot lf. (26) 2 y Thi exreion how that ignal roceing gain of ectral-baed roceing a comared to noncoherent roceing cale a the quare root of wavelength and ule reetition frequency, and i inverely roortional to the quare root of the ectral width. Furthermore, G R i found to increae monotonically with T ob, ince the ratio f an /f ao increae a the total number of amle increae. In Fig. 6 G R i lotted a a function of normalized ectral width for fale alarm robabilitie between and 0.2 with the total number of amle M a a arameter. For a given normalized ectral width, G R increae a the fale alarm rate i decreaed. In addition, the ratio f an /f ao in (26) increae with M, aymtotically aroaching a contant value that deend on the fale alarm rate and the otimal DFT length. Thu, larger amle length M imrove the relative gain of FIG. 6. Relative gain at otimal DFT length G R (26) a a function of normalized ectral width, with total number of amle M a a arameter (M ), for four fale alarm robabilitie. DFT-baed roceing to noncoherent roceing due to a more favorable ectral detection threhold. A an aid to interreting Fig. 6, normalized ectral width i lotted in Fig. 7 a a function of ule reetition frequency for two ectral width, y 0. 1 m 21 and 1.0 m 21 with radar frequency a a arameter. Cloud above the turbulent boundary layer can exhibit ectral width on the order of y 0. 1 m 21, for which G R i greater than 1 for all of the radar frequencie lotted auming ule reetition frequencie above a few kilohertz and a fale alarm robability below 0.1. Cloud and clear-air turbulence within the atmoheric boundary layer often exhibit ectral width on the order of y 1m 21. In thi cae, horter wavelength radar how little enitivity imrovement from ectral roceing. Auming, a wa found emirically, that L ot ffi 0. 8L W, the following formula can be ued to olve for K ot : K ot ffi ffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ln2 [erf 21 (0. 8L W )] 2 1. (27) 2 [erf 21 (L W )] 2 However, thi formula cannot be ued for a uniform window, ince the exreion for reading lo (14) i only accurate for taered weighting function. Therefore, K ot wa determined emirically for the uniform window. By equating DFT-baed ignal roceing gain (20) to the noncoherent integration gain (3), the critical DFT length i found, above which the DFT-baed roceing gain fall below the ure noncoherent integration gain:

7 APRIL 2016 M E A D 729 FIG. 7. Normalized ectral width n a a function of ule reetition frequency F for variou oerating frequencie for ectral width (left) y 0. 1 m 21 and (right) y 1m 21. N crit f 2 a f 2 an C2 m L2. (28) Thi equation i mot eaily olved numerically, ince f an, f a, C m, and L are nontrivial function of N. Table 1 ummarize the variou contant in the formula given above for commonly ued window function. While the reading lo (14) ha ignificant error for the uniform window, the equation for otimal DFT length (24), otimal ignal roceing gain (2), and relative gain (26) are accurate for the uniform window uing the contant in Table Fale alarm factor a. Derivation of fale alarm factor for noncoherent roceing The um of M noie ower amle of a bivariate Gauian voltage ditribution rereenting the noieignal enveloe ha a chi-quare ditribution with 2M degree of freedom, x 2 (2M) (Ulaby et al. 1982, ). The cumulative ditribution function (CDF) for a chi-quare ditribution with 2M degree of freedom i denoted by D [Abramowitz and Stegun 1964, their (26.4.1); cf. Wolfram 2012]: x D 2M (x 2 ), x 2 2M c P n, (29) which give the robability x that a articular average of M ower amle i le than a threhold ower c for a roce having mean noie ower P n. For examle, if the threhold c P n, then x D 2M (2M) 0. for large M. The invere function i written a c P N D21 2M ( ) x 2M, (30) which give the threhold ower a a function of x. The robability of fale alarm fa (Skolnik 1990) i equal to the robability that a noie amle in the FIG. 8. Cumulative ditribution function CDF of the average of 32 noie amle for mean noie ower P n 1 (olid curve) and CDF after mean noie (dahed curve) for fale alarm robability fa.0, howing the renoie ubtraction threhold c fa and the threhold after noie ubtraction for noncoherent roceing T n. abence of a ignal exceed c, thu fa 1 2 x. Prior to mean noie ubtraction, the threhold for a given robability of fale alarm i therefore c fa P N D21 2M (1 2 ) fa. (31) 2M After noie ubtraction, thi threhold dro by the mean noie ower: T n c fa 2 P n c fa $ P n, (32) where T n and c fa are hown grahically in Fig. 8 along with the cumulative ditribution function for the average of 32 noie amle before and after the mean noie ubtraction. For thi analyi, negative ower after the noie ubtraction are dicarded. The maximum robability of fale alarm for noncoherent roceing occur when c fa P n (i.e., T n 0). Thi maximum i lotted in Fig. 9 a a function of the amle length. Equating (32) to (2) give the fale alarm factor f an for noncoherent roceing: f an ffiffiffiffiffi ( D 21 2M M [1 2 ] ) fa 2 1. (33) 2M b. Derivation of fale alarm factor for ectral roceing For ectral roceing, the robability of fale alarm for each bin in the ower ectrum i denoted by 0 fa. The robability that all oint in the ower ectrum fall below the threhold i given by

8 730 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 FIG. 10. Fale alarm factor: (left) f an and (right) f a a a function of total amle length M for ectral lot, howing N 64 (black trace) and N 12 (red trace). FIG. 9. Maximum robability of fale alarm v amle length M for noncoherent roceing. null (1 2 0 fa )N. (34) Thu, the fale detection rate for the entire ower ectrum fa i fa 1 2 null 1 2 (1 2 0 fa )N. (3) Solving for the er-bin fale alarm rate, 0 fa 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi N 1 2 fa. (36) The threhold rior to noie ubtraction c fa i found by ubtituting (36) in (31): P c n D21 N ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 fa 2M fa, (37) 2M where Pn i the noie ower ectral denity. The ectral roceing detection threhold i T c fa 2 P n. (38) The maximum fale alarm rate for ectral-baed roceing i very cloe to 1.0, ince there i a high robability that at leat one oint in the ower ectrum will exceed the etimate of the mean noie ower ectral denity. Equating (38) to (8) give the fale alarm factor f a for ectral roceing: 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 ffiffiffiffiffi D 21 N >< 1 2 2M fa >= f a M 2 1. (39) >: 2M >; The fale alarm factor f an and f a are lotted in Fig. 10 for variou fale alarm rate a a function of the total number of amle. When comaring the roceing gain of ectral and noncoherent roceing, the fale alarm rate fa and fa hould be equal. Note that the fale alarm factor comuted at N ot, f ao, i equal to f a evaluated at N N ot. c. Imact of zero adding on ectral fale alarm threhold To avoid reduction in ignal ower due to calloing lo, the time-domain ignal and the windowing function may be zero added rior to comuting the ower ectrum; that i, in () the DFT i taken over N 0. N with N 0 2 N zero aended to the vector of ignal-lunoie voltage and the time-domain weight vector w. For examle, if N 0 2N, then the calloing lo reduce from 1.42 to 0.3 db when uing a Hanning time-domain window function. With zero adding, the fale alarm factor i not accurately modeled by relacing N with N 0 in (39), ince adjacent ectral value of noie become increaingly correlated a N 0 increae. Through imulation, it wa found that f a rie by no more than 6% relative to the nonzero added value, with little change a N 0 increae beyond 3N. If thi mall change in the threhold level i ignored, then the fale alarm rate will increae by a few ercent.. Simulation of ignal roceing gain An Interactive Data Language (IDL) rogram wa written to imulate ignal of variou ectral width in the reence of noie and to imulate ectral-baed and noncoherent ignal roceing gain. The imulated ectra were generated uing the method decribed in Sirman and Bumgarner (197). Noncoherent gain (3) and DFT-baed ignal roceing gain (20) are lotted in Fig. 11 for total amle length M 26 and M Thee figure how excellent agreement between (3) and (20) and the imulated gain value. The ratio of eak ectral roceing gain to noncoherent gain i een to be greater for the larger total amle length a redicted by (26).

9 APRIL 2016 M E A D 731 which can be reexreed in term of the ignal-tothrehold ratio (4):! 2Mc d 1 2 D fa /T n 2M. (41) STR n 1 P N /T n Eliminating T n by ue of (32): 2 3 c 2M fa c d 1 2 D fa 2 P n 2M 6 STR n 1 P 7 4 n c fa 2 P n M B c A fa 2 1 P 1 2 D n 2M STR n 1 1. (42) c 6 fa P n FIG. 11. Combined DFT-baed (20) and noncoherent (3) ignal roceing gain v DFT length with ectral width a a arameter uing a Hanning time-domain window. (to) Total amle length NM 26 and (bottom) NM , howing noncoherent integration gain [(3); horizontal dahed red trace], DFT-baed roceing gain [(20); black trace], fa 0. 01, N ot (vertical red line), and N crit (vertical green line). Simulated value hown with ymbol rereent the average of 200 trial for NM 26 and 0 trial for NM Probability of detection analyi a. Probability of detection for noncoherent roceing Firt, conider a imlified cae where all ignal-lunoie amle are indeendent, bivariate Gauian ditributed random variable. The robability of detection, d,(skolnik 1990) i the robability that a given realization of the ignal-lu-noie exceed the fale alarm threhold c fa : 2Mcfa d 1 2 D 2M, (40) S 1 P N Thi formula give the robability of detection for imle noncoherent integration of M uncorrelated amle of the ignal and noie. Equation (42) i trictly valid only when each ignal amle i indeendent. However, ignal amle are often artially correlated, thu the ditribution of the um of M ignal-lu-noie amle i not alway accurately decribed by chi-quare tatitic. The ditribution function of the average ignal-lunoie ower can be obtained from the convolution of two indeendent ditribution, the firt ditribution being ure noie and the econd ditribution coniting of indeendent ignal amle in the reence of noie. Although thi i an artificial contruct, the reultant ditribution function i correct, rovided the ower of the indeendent ignal amle i caled to yield the original ignal-to-noie ratio. Conider the ower average of M artially correlated ignal amle in the reence of noie. The ditribution of the um of two indeendent ditribution i equal to the convolution of the individual ditribution (Paouli 1984). Two ditribution are created, length N n and N n, coniting of ure noie and ignal lu noie: N n M 2 N ind (43) N n N ind, (44) where N n 1 N n M. The effective number of indeendent ignal amle N ind i given by an aroximate formula valid for normalized ectral width u to 0. (6.12 in Doviak and Zrnić 1993): rffiffiffiffiffiffiffiffiffiffi N ind M 2 ln2 n n # 0.. (4)

10 732 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 FIG. 12. Probability of detection for noncoherent ower averaging with noie ubtraction a a function of SNR. Theoretical d given by (42) auming all ignal amle indeendent (red trace); (42) with the modified CDF D 0 2M ubtituted for D 2M (black trace); fa 0.02, and M 192. Simulated value hown with ymbol rereenting the average of amle. The total ignal ower aociated with each of thee ditribution i chi-quare ditributed, with 2N n and 2N n degree of freedom for noie and ignal lu noie, reectively. The mean noie ower in both ditribution equal P n. The ignal-to-noie ratio of the indeendent ignal amle i caled to account for the fact that fewer ignal amle are emloyed than the original M amle: SNR 0 SNR M N n. (46) The mean ower of each of the ignal-lu-noie amle i given by P n P n (1 1 SNR 0 ). (47) In thi way, the um of the two ditribution maintain the original ignal-to-noie ratio. In addition, the um of the two ditribution contain the correct number of indeendent amle of the ignal (N ind ) and the correct number of indeendent amle of the noie (M). The cumulative ditribution function D 0 2M reulting from the convolution of thee two ditribution i ubtituted in (42) to yield the robability of detection. A cloed form olution for D 0 2M wa derived, but factorial term involving M were too large to comute for large M, thu D 0 2M wa evaluated numerically. Analytical and imulated reult for the robability of detection for noncoherent roceing are reented in Fig. 12 for four ectral width, howing excellent FIG. 13. Simulated and theoretical robability of detection /n d (3) for NM amle and SNR 224 db, Hanning window alied. (to) Term fa 0.01; (bottom) fa 0.2. Noncoherent roceing [(42); horizontal red line] and imulated value rereent the average of 1000 trial (ymbol), and N ot (vertical red line mark). agreement. Thi figure how a ignificant imrovement in detection robability for ignal with wide ectral width when the ignal-to-threhold ratio i greater than one. Thi trend revere for low ignal-to-threhold ratio (below 0.8). b. Probability of detection for ectral roceing The robability of detection er DFT bin in the region of the ectral eak lobe, 0 d, i given by 0 d 1 2 D00 2M ( ) 2Mc fa S 1 Pn, (48) which can be reexreed in term of the ectral ignalto-threhold ratio (21):

11 APRIL 2016 M E A D 733 FIG. 14. Term G R (red trace) and DSNR (black trace and ymbol) a a function of normalized ectral width; DSNR i lotted for five different detection robabilitie, d 0.1 (quare), 0.3 (triangle), 0. (*), 0.7 (1), and 0.9 (diamond) M41 1 c fa >< d 1 2 P >= D00 n, (49) 2M STR 1 1 C m c fa 2 1 >: Pn >; where D 00 i a modified verion of 2M D0 (generated by 2M convolution a in ection 6a) to account for the fact that all ectral value at a articular value of k (rior to ectral averaging) are indeendent for N ind. M, while the ectral value rior to ower averaging are artially correlated for N ind, M: D 00 2M D0 2M N, M ind and (0) D 00 2M 2M N. M. ind (1) The condition N ind M i equivalent to rffiffiffiffiffiffiffiffiffiffi 2 ln2 N , n n which i omewhat le than N ot. Note that the multile eak correction factor C m i removed from STR by diviion in (49), ince 0 d i comuted er DFT bin. Probability of detection for the entire ectrum i given by d 1 2 (1 2 0 d )m, (2) where it i aumed that the m amle near the ectral eak have equal robability of detection, 0 d. For ectral roceing, the robability that either noie or ignal FIG. 1. Otimal DFT length N ot, auming Hanning timedomain weight, a comuted from meaured ectral width in a high SNR region of cloud layer meaured at X band, 6 May 201 (range gate ). The average value of 312 (dahed red line) correond with n ( y 0. 1 m 21 ). Data were roceed with zero adding to avoid calloing lo: N 0 3N. lu noie will exceed the fale alarm threhold and thu be detected i given by /n d 1 2 (1 2 0 d )m (1 2 0 fa )N2m. (3) Thi latter equation i ueful when the robability of detection i cloe to or le than the robability of fale alarm. Simulated and theoretical value of robability of detection are lotted in Fig. 13, howing excellent agreement between (42) and (3) and imulated data. The maximum robability of detection i een to occur when the DFT length i aroximately equal to N ot. Thi reflect the fact that (3) i near to it maximum value when STR i maximized. It wa found that N crit (28) doe not accurately redict the oint at which the robability of detection i equal between the two roceing method. Thi i not urriing, ince STR n STR when N N crit,butthe formula for noncoherent detection robability (42) and ectral-baed detection robability (3) have a ignificantly different deendence on STR. A direct comarion between ignal roceing gain and robability of detection can be made by comaring the required difference in SNR needed to achieve the ame detection robability for the two roceing method. In Fig. 14, the relative ignal roceing gain at the otimal DFT length, G R (26) in decibel, i comared to the difference in the ignal-to-noie ratio, DSNR in decibel, required to achieve the ame robability of detection for the two ignal roceing method when the DFT length i et to N ot. Secifically,

12 734 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 FIG. 16. Range rofile of cloud STR with fa 0.01 uing (to left) noncoherent averaging and mean noie ubtraction for amle length M 24 76; (to right) eak STR with ectral roceing with DFT length N 32, and number of ectral average M 768; (bottom left) N 312, M 78; and (bottom right) N 2048, M 12. Data roceed with N 0 3N. FIG. 17. Range rofile of cloud STR with fa 0.2 uing (to left) noncoherent averaging and mean noie ubtraction for amle length M 24 76; (to right) eak STR with ectral roceing with DFT length N 32 and number of ectral average M 768; (bottom left) N 312, M 78; and (bottom right) N 2048, M 12. Data roceed with N 0 3N. DSNR (db) SNR n (db) 2 SNR (db), where the ubcrit n and refer to noncoherent and ectral roceing, reectively, and the SNR value are elected to equate robability of detection. Thi comarion wa made at five detection robabilitie, howing agreement within 60.4 db for d Thu, G R, which i comuted at N N ot, i een to be a good redictor of the relative ignal-to-noie ratio required for the two roceing method to achieve the ame detection robability for a given robability of fale alarm. an FFT algorithm with Hanning time-domain weight uing three different DFT length of N 32, 312, and 2048 with the robability of fale alarm et to The ignal-to-threhold ratio i uerior for all of the ectral lot a comared to the noncoherently roceed ignal, with the bet enitivity een for a DFT length of 312. Note that the cloud layer between 9.4 and 9.8 km i oorly detected in the noncoherently roceed data and that it ha the highet STR in the data roceed 7. Exerimental reult A 2.4- record (M amle er range gate) of raw in-hae and quadrature (I/Q) data from a cloud layer located between 9.2 and 10.9 km wa gathered in Amhert, Maachuett, at 1401 UTC 6 May 201 uing the DOE ARM rogram X-band canning ARM Program cloud radar (SACR; DOE 2012). The radar ule reetition frequency wa 10 khz, the range reolution wa 4 m, and the antenna beamwidth wa The otimal DFT length wa etimated by comuting the normalized ectral width from a ower ectrum comuted uing a 2048-oint FFT with Hanning weight with M 12 ectra averaged to form the ower ectrum. To comute N ot, n wa etimated from the meaured ectral width uing (C2) to remove the ectral broadening term, w. Term N ot i lotted a a function of range in Fig. 1, howing an average value of 312 in the range between 7.9 and 8.6 km. Figure 16 lot the ignal-to-threhold ratio roceed noncoherently and with ectral roceing emloying FIG. 18. Probability of detection for noncoherent roceing (red trace) for M and for ectrally roceed data (black trace) for DFT length N 32, and number of ectral average M 768 (olid trace); N 312, M 78 (dahed trace); and N 2048, M 12 (dotted dahed trace). (left) Term fa 0.01 with dahed line howing d 0. 0 and d 0. 9; (right) fa 0.2 with dahed line howing d and d The theoretical roceing gain are a follow: G n db; G db for N 32; G db for N 312; and G db for N 2048.

13 APRIL 2016 M E A D 73 FIG. 19. Meaured G R (black trace with aterik) at X band, 6 May 201, for the average of range gate (range km) and imulated G R (red trace) auming N ot 312 for (left) fa and (right) fa 0.2, howing N ot (vertical red line) and N crit [vertical green line in (right)] for fa 0.2. Term N crit i greater than M for fa Data roceed with N 0 3N. with N 312. Thee figure are reeated in Fig. 17 with the robability of fale alarm increaed to 0.2. The ignal-to-threhold ratio STR n for the noncoherently roceed ignal and STR for the ectrally roceed data with DFT length 32 and 2048 are aroximately equal when fa 0. 2, ince all three cae have nearly identical ignal roceing gain ( db). However, the robability of detection i coniderably lower for the noncoherently roceed ignal for the cloud layer between 9.4 and 9.8 km where the STR i le than db. Thi behavior i exlained by Fig. 18, which how that for fa 0. 2 the SNR required to achieve robability of detection of 9% i 221 db for N 32, 222 db for N 2048, and 216 db for noncoherent roceing. In general, the detection robability curve lotted againt SNR are teeer for the ectrally roceed data, eecially at higher fale alarm rate. Thi i becaue the robability ditribution of the eak noie ower after noie ubtraction i narrower for ectrally roceed data than the ditribution of noie ubtracted ower for the noncoherently roceed data. Steeer robability of detection curve are more favorable, ince a maller difference in SNR above the threhold i needed to yield a high detection robability. Relative ignal roceing gain G R i lotted in Fig. 19 for thi ame dataet along with imulated value for fale alarm robabilitie of 0.01 and 0.2 howing excellent agreement. The critical DFT length N crit exceed M for fa but wa correctly etimated to be 2400 for fa The effect of calloing lo can be een in Fig. 20, where G R wa comuted with uniform and Hanning weight uing zero adding (N 0 3N) and no zero adding (N 0 N). The eak ignal ower i underetimated by a much a 2.4 db in the uniform weight cae without zero adding due to calloing lo. The maximum calloing lo oberved for the cae of Hanning weight i 1 db. Note that the effect of calloing lo are only aarent for N, N ot, a the FIG. 20. (left) Meaured G R at X band, 6 May 201, for the average of range gate (range km) roceed uing uniform weight with N 0 3N (olid black trace with aterik), and N 0 N (dahed black trace) howing deleteriou effect of calloing lo. Simulated G R (red trace), N ot 206 with uniform weight (vertical red line), and fa (right) A in (left), but for Hanning time-domain weight and N ot 312. ectral eak lobe i read over multile DFT bin when N. N ot and i thu not ubject to calloing lo. 8. Concluion The analyi reented above how that ectral roceing ha the otential to rovide ignificant enitivity imrovement a comared to noncoherent ower averaging. However, it ha been hown that ectral roceing can reult in lo of enitivity relative to ower averaging when the normalized ectral width i greater than ;0.2. The imle exreion reented for N ot wa hown to accurately redict the DFT length that maximize the robability of detection. The fale alarm rate wa found to lay an imortant role in determining the relative detection erformance of the two roceing method. Exerimental data taken at X-band howed excellent agreement with the analytical exreion for roceing gain N ot and the robability of detection. Data roceed at N N ot uing a uniform time-domain window exhibited an 8-dB roceing gain advantage a comared to noncoherently roceed data. Acknowlegment. The author thank the U.S. Deartment of Energy for roviding acce to the canning ARM cloud radar AMF-2 facility ued to gather the cloud data reented in thi aer. Alo, Dr. Mark Goodberlet and Dr. Andrew Pazmany of ProSening, Inc., are thanked for their helful dicuion and uggetion related to the analye reented herein. C m x APPENDIX A Lit of Symbol Gain cale factor to account for ignal fading Chi-quare ditribution variable

14 736 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 D 2M Cumulative ditribution function for chi-quare ditribution with 2M degree of freedom D 0 2M Cumulative ditribution function reulting from the um of two indeendent chi-quare ditribution D 00 2M Modified verion of D 0 2M to account for the effect of DFT length on amle indeendence DSNR Difference in ignal-to-noie ratio between noncoherent and ectral-baed roceing required to achieve the ame robability of detection DV D Sectral velocity reolution (m 21 ) f an Fale alarm cale factor regulating the threhold level for noncoherent roceing f ao Fale alarm cale factor regulating the threhold level for ectral roceing at otimal DFT length f a Fale alarm cale factor regulating the threhold level for ectral roceing F Pule reetition frequency (Hz) G n Noncoherent (ower averaging) roceing gain G ot Otimal ectral ignal roceing gain G R Ratio of otimal ignal roceing gain to noncoherent roceing gain G Sectral-baed roceing gain G W SNR gain of weighting function K ot Weighting function deendent contant ued to determine otimal DFT length K W Weighting function deendent cale factor relating ectral broadening factor W to ectral reolution DV D l Radar wavelength L Sreading lo L ot Sreading lo at otimal DFT length L W Lo in the SNR gain of weighting function relative to uniform weight M Total number of amle, that i, M F T ob M Number of ectra averaged together to form the final ower ectrum etimate m Effective number of ectral bin in the ignal eak lobe N DFT length N 0 DFT length after zero adding N ind Effective number of indeendent ignal amle N n Number of noie amle when creating earate noie and ignal-lu-noie vector N n Number of ignal-lu-noie amle when creating earate noie and ignal-lu-noie vector N crit Critical DFT length above which the noncoherent roceing gain exceed ectralbaed roceing gain N ot Otimal DFT length to maximize the ignal roceing gain P Exected value of ignal lu noie d Probability of detection 0 d Probability of detection er DFT bin fa Probability of fale alarm 0 fa Probability of fale alarm for each bin of the ower ectrum fa Probability of fale alarm for the entire ower ectrum d Probability of detection for ectral roceing /n d Probability of detecting either ignal or noie for ectral roceing x Probability that a amle from the chi-quare ditribution i le than threhold c P k Power ectrum etimate for the kth ectral bin P n Mean noie ower Pn Power ectral denity of noie P 0 Exected value of eak ignal ower in the ower ectrum P n Exected value of ower amle in the artificial ignal-lu-noie vector P Signal ower ectrum model c General threhold variable c fa Fale alarm threhold for ectral roceing c fa Fale alarm threhold for noncoherent roceing n Normalized ectral width of the meteorological target y Sectral width of the meteorological target (m 21 ) W Sectral broadening due to the time-domain weighting function (m 21 ) Signal ower M-length vector of ignal-lu-noie amle Segment of the ignal-lu-noie amle vector ued in comuting DFT SNR Signal-to-noie ower ratio SNR 0 Signal-to-noie ower ratio in the artificial ignal-lu-noie vector SNR n Signal-to-noie ower ratio for noncoherent roceing SNR Signal-to-noie ower ratio for ectral roceing STR Signal-to-threhold ower ratio STR n Signal-to-threhold ower ratio for noncoherent roceing STR Signal-to-threhold ower ratio for ectral roceing T Detection threhold T n Detection threhold et by the allowable fale alarm rate for noncoherent roceing. T Detection threhold et by the allowable fale alarm rate for ectral-baed roceing Tn 0 Tn 0 P n i the reference threhold for M 1. T ob Total obervation time () y Velocity (m 21 )

15 APRIL 2016 M E A D 737 y 0 Mean velocity (m 21 ) y a Unambiguou Doler velocity w Time-domain weight vector APPENDIX B Decrition of Model for C m The ignal in any of the m ectral bin in the ectral eak lobe region ha a ignificant robability of being the highet value. Each bin of the ower ectrum i tatitically indeendent of it neighbor, with a chiquare ditribution of order 2M. Regardle of the underlying robability denity function, the exected value of the maximum of m amle of a random variable x i of the form E(x max ) ffi E(x) 1 b lnm, (B1) where E(x) i the mean value of x (Watkin 2012). Thi form aume that the mean value of x i the ame for all m amle. While thi i not true for the cae of a Gauian ectrum hae, where the central bin ha the highet mean value, (B1) wa found to correctly model the exected value of the highet eak in the region near the ectral eak relative to the mean value of the ectral eak. Data for curve fitting wa generated by imulating a Gauian ectrum uing the method decribed in Sirman and Bumgarner (197). Thearameterb in (B1) wa found to deend on the number of individual ower ectra averaged M ued to form the ower ectrum and on the normalized ectral width n. The condition that C m 1for n N, ffiffiffi 2 i enforced when mot of the ectral ower fall in the DFT bin aociated with the ectral eak. Note that C m 1whenN # N ot for all of the windowing function lited in Table 1. APPENDIX C Derivation of Sreading Lo For a dicrete ower ectrum comuted uing (), the reading lo i equal to the ratio of the energy contained in the central ectral bin to the total energy in the ectrum. Referring to Fig. C1 and auming a Gauian ectrum hae, the reading lo for the dicrete frequency ectrum with ectral reolution DV D can be aroximated by the error function FIG. C1. Doler ower ectrum (red trace) with ectral width howing relative energy of dicrete frequency amle (blue rectangle) for ectral amle marked with aterik. where L erf erf(x) ffiffiffiffiffiffiffi! ln2 DVD, (C1) ð x 0 e 2t2 dt, and i the tandard ffiffiffiffiffiffiffi deviation of the ectrum. The cale factor ln2 in (C1) wa determined by etting e 2x2 0. when the ectral reolution equal the ectral width, that i, DV D. The exreion for reading lo (10) ha a ectral width that combine ectral broadening due to the windowing function W with the underlying ectral width of the meteorological ignal y. The reultant ectrum rereent the convolution of the meteorological ignal ower ectrum with the ower ectrum time-domain weighting function; thu, in (C1) i relaced by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W 1 2 y. (C2) The exreion for reading lo (10) wa comared to imulated reading lo and wa found to be accurate to within one ercent when uing the nonuniform time-domain window function lited in Table 1. Ue of a uniform window reult in a non-gauian ower ectrum, which introduce a maximum reading lo overetimation of aroximately 0.7 db near the region of eak roceing gain.

16 738 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 33 APPENDIX D Derivation of G W Conider a fixed frequency dicrete time ignal, (i) Ae jv o i, where v o 2k 0 /N for ome ecific frequency index k 0. Uing thi ignal in () with M 1, the reultant ectrogram i given by A P k ån21 N w(i)e jv o i e 2[( j2ki)=n] 2 The ectral eak occur at k k 0 : A P k 0 ån21 N 2 w(i). Next, conider a dicrete-time comlex-valued noie ignal, n(i) B i e ju i, where the amlitude B i i a Rayleigh ditributed random variable with mean value B 0 and u i i a uniformly ditributed random variable. The ectrogram of the noie i P n k 1 ån21 N 2 w(i)b i e ju i e 2[( j2ki)=n]. Since u i i uniformly ditributed and all amle are uncorrelated, the exected value of P n k i equal to the ower um: hp n k i B2 0 N ån21 jw(i)j 2. Setting the inut SNR to 1.0 (A B 0 ), the ignal roceing gain i given by the outut SNR: ån21 2 G w P w(i) k hp n k i. N21 å jw(i)j 2 REFERENCES Abramowitz, M., and I. A. Stegun, Ed., 1964: Handbook of Mathematical Function with Formula, Grah, and Mathematical Table. Alied Mathematic Serie, Vol., National Bureau of Standard, Clothiaux, E. E., M. A. Miller, B. A. Albrecht, T. P. Ackerman, J.Verlinde,D.M.Babb,R.M.Peter,andW.J.Syrett, 199: An evaluation of a 94-GHz radar for remote ening of cloud roertie. J. Atmo. Oceanic Technol., 12, , doi:10.117/ (199)012,0201: AEOAGR.2.0.CO;2. DOE, 2012: Scanning ARM cloud radar (X/Ka/W-SACR) handbook. U.S. Det. of Energy Tech. Re. DOE/SC-ARM/ TR-113, 30. [Available online at htt:// ublication/tech_reort/handbook/wacr_handbook.df? id6.] Doviak, R. J., and D. S. Zrnić, 1993: Doler Radar and Weather Obervation. 2nd ed. Academic Pre, 62. Farley, D. T., 198: On-line data roceing technique for MST radar. Radio Sci., 20, , doi: / RS020i Gage, K. S., 1990: Radar obervation of the free atmohere: Structure and dynamic. Radar in Meteorology: Battan Memorial and 40th Anniverary Radar Meteorology Conference, D. Atla, Ed., Amer. Meteor. Soc., 34 6, doi: / _37., and B. B. Balley, 1978: Doler radar robing of the clear atmohere. Bull. Amer. Meteor. Soc., 9, , doi:10.117/ (1978)09,1074:drpotc.2.0.co;2. Ivić, I. R., D. S. Zrnić, and T. Yu, 2012: Threhold calculation for coherent detection in dual-olarization weather radar. IEEE Tran. Aero. Electron. Syt., 48, , doi: / TAES , R. Keränen, and D. S. Zrnić, 2014: Aement of cenoring uing coherency-baed detector on dual-olarized weather radar. J. Atmo. Oceanic Technol., 31, , doi:10.117/ JTECH-D Keeler, R. J., and R. E. Paarelli, 1990: Signal roceing for atmoheric radar. Radar in Meteorology: Battan Memorial and 40th Anniverary Radar Meteorology Conference, D. Atla, Ed., Amer. Meteor. Soc., Keränen, R., and V. Chandraekar, 2014: Detection and etimation of radar reflectivity from weak echo of reciitation in dualolarized weather radar. J. Atmo. Oceanic Technol., 31, , doi:10.117/jtech-d Kollia, P., B. A. Albrecht, E. E. Clothiaux, M. A. Miller, K. L. Johnon, and K. P. Moran, 200: The Atmoheric Radiation Meaurement rogram cloud rofiling radar: An evaluation of ignal roceing and amling trategie. J. Atmo. Oceanic Technol., 22, , doi:10.117/ JTECH , J. Rémillard, E. Luke, and W. Szyrmer, 2011: Cloud radar Doler ectra in drizzling tratiform cloud: 1. Forward modeling and remote ening alication. J. Geohy. Re., 116, D13201, doi: /2010jd Lyon, R. G., 2004: Undertanding Digital Signal Proceing. 2nd ed. Prentice Hall, 663., 2011: Reducing FFT calloing lo error without multilication. IEEE Signal Proce. Mag., 28, , doi: / MSP Marhall, J. S., and W. Hitchfeld, 193: Interretation of the fluctuating echo from randomly ditributed catterer. Part 1. Can. J. Phy., 31, , doi: / Mead, J. B., 2010: MMCR calibration tudy. U.S. Det. of Energy Tech. Re. DOE/SC-ARM/TR-088, 13. [Available online at htt:// doe-c-arm-tr-088.df.]