STAGGERED PRT WITH GROUND CLUTTER FILTERING AND OVERLAID ECHO RECOVERY FOR DUAL POLARIZATION

Transcription

1 STAGGERED PRT WITH GROUND CLUTTER FILTERING AND OVERLAID ECHO RECOVERY FOR DUAL POLARIZATION ALGORITH DESCRIPTION by Sebastián Torres, David Warde and Beatriz Gallardo-Hernando National Severe Storms Laboratory July 00 PREFACE This document extends the previous Staggered PRT algorithm description from July 009 by including dual polarization sequences and the calculation of polarimetric variables. This algorithm description includes a high-level description with the overall processing logic followed by a detailed explanation of each pre-computation and processing step. ost of the July 009 algorithm steps are now repeated for both H- and V-channels calculation. Unlike them, the SACHI filter has been modified preserving the phase value in order to allow the calculation of the polarimetric variables. Now, the output of the SACHI filter produces autocorrelations for both H- and V-channels and also the cross-correlation between them. To ease implementation and reduce ambiguity, most of the steps in the SACHI algorithm are described in algorithmic form. The DC removal ground clutter filter has been retained to operate on those range gates where only long-prt data is available and ground clutter filtering is needed and now it includes both channels. As in the July 009 description, the algorithm is able to handle overlaid echoes, extending the recovery of Doppler moments to the unambiguous range of the long PRT. oment-specific overlaid power thresholds are used to identify recoverable data and flag unrecoverable Doppler moments. In this version of the algorithm, ground clutter is also assumed to be within the unambiguous range of the short PRT.

2 ASSUPTIONS ) The transmission sequence alternates two pulse repetition times (PRT) as: T, T, T, T for a total of pulses. ) The PRT ratio T /T = /3, where m =, n = 3 and T T = T u. 3) All range gates are available and there is a perfect alignment of range gates between the two PRTs (i.e., a given range gate represents the same resolution volume in space for every transmitted pulse). Also, the number of range gates for each PRT is: N = T / s and N = T / s, where s is the sampling period. 4) There are no significant echoes beyond the maximum unambiguous range corresponding to T (r a ). 5) There is no significant ground clutter beyond the maximum unambiguous range corresponding to T (r a ). 6) The number of staggered PRT samples per range gate () is even. 7) The algorithm operates on a radial worth of data at a time. INPUTS ) Dual polarization complex time-series data: V H (n, m) = I H (n, m) + jq H (n, m), V V (n, m) = I V (n, m) + jq V (n, m), where subscripts H and V denote horizontal and vertical polarization, 0 < n < N for even m, 0 < n < N for odd m and 0 < m <. Note that n indexes the range gates and m the sweeps (or pulses). ) Associated metadata: λ is the radar wavelength in meters N H is the noise power in linear units for the horizontal channel N V is the noise power in linear units for the vertical channel dbz0 is the system calibration constant in db ATOS is the elevation-dependent atmospheric attenuation in db/km R is the spacing between range gates in km (R = c s /) T Z is the signal-to-noise ratio threshold for reflectivity in db T V is the signal-to-noise ratio threshold for velocity in db T W is the signal-to-noise ratio threshold for spectrum width in db T OV is the velocity overlaid threshold in db (Note: recommended value is 0 db) T OW is the spectrum width overlaid threshold in db (Note: recommended value is 0 db) 3) Data window: d (m), where 0 < m < 5/. Note that d does not need to be normalized or scaled in any way. A tapered data window such as the Blackman window is recommended for best performance of the SACHI ground clutter filter. Otherwise, rectangular window (i.e., no window) should be applied. 4) Ground clutter filter bypass map: B(n), where n indexes the range bins with the same resolution as the time-series data along a radial, and the map corresponds to the elevation and azimuth of the radial being processed. B is 0 if clutter filtering is required and otherwise. In this algorithm, the clutter map is ignored beyond the unambiguous range corresponding to the short PRT where clutter is assumed not to be present.

3 OUTPUTS ) Reflectivity, Doppler velocity, and spectrum width calculated from H channel data: Z (n) for 0 < n < N, v (n) and w (n) for 0 < n < N. ) Differential reflectivity, differential phase and correlation coefficient calculated from H and V channel data: Z DR (n) for 0 < n < N, DP (n) for 0 < n < N, HV (n) for 0 < n < N. 3) Signal-to-noise ratio and overlaid censoring flags*: NS Z (n), NS V (n) and NS W (n) for 0 < n < N, OV V (n) and OV W (n) for 0 < n < N. * NS Z (n) is used for censoring Z DR (n), DP (n) and HV (n). FUNCTIONS AND CONVENTIONS ) Returns the absolute value of a complex number or the absolute value of each element of a matrix of complex numbers. ) arg Returns the principal phase angle of the input complex number in radians. The algorithm is written to accommodate this phase in the interval [0, ) or [, ). 3) arg min k Returns the index k to the element in the input vector that has the minimum value. 4) diag Returns a square matrix with the input vector along the principal diagonal (row index = column index) of the matrix and all other elements not on the principal diagonal equal to zero. The number of rows (columns) of the matrix is equal to the number of elements in the vector. 5) ceiling Returns the smallest integer value not less than the input number. 6) floor Returns the largest integer value not greater than the input number. 7) round Returns the nearest integer to the input number. 8) max Returns the maximum value among the input numbers. 9) Italicized names are used to denote scalars (e.g., Noise). 0) Bolded names are used to denote vectors or matrices (e.g., A). Italicized names with indexing in parentheses are used to denote elements of a vector or matrix [e.g., A(i,j)]. ) * Denotes complex conjugate. ) T Denotes matrix transpose. 3) j Denotes the imaginary unit. 3

4 HIGH-LEVEL ALGORITH DESCRIPTION If first run of SPRT algorithm ) Pre-computation of velocity dealiasing rules ) Pre-computation of -independent SACHI filter parameters If the number of samples () changed 3) Pre-computation of window parameters 4) Pre-computation of -dependent SACHI filter parameters For each range bin n, where 0 < n < N If n > N 5) Short-PRT Segment-III Data Reconstruction If B(n) = 0 AND n < N 6) SACHI Clutter Filtering (Segment-I/II gate with segment-i/ii clutter) If n > N AND B(n N ) = 0 7) DC Removal Clutter Filtering (Segment-III gate with segment-i clutter) 8) No Clutter Filtering 9) Power and correlation computations for each PRT 0) Combined power and cross-correlation computation ) Strong point clutter canceling For each range bin n, where 0 < n < N ) Signal power computation 3) Reflectivity computation 4) Velocity computation 5) Spectrum width computation 6) Differential reflectivity computation 7) Differential phase computation 8) Cross-correlation coefficient computation 9) Determination of significant returns for reflectivity and polarimetric variables 0) Determination of significant returns for velocity ) Determination of significant returns for spectrum width For each range bin n, where 0 < n < N ) Determination of overlaid returns for velocity and spectrum width 4

5 STEP-BY-STEP ALGORITH DESCRIPTION ) Pre-computation of velocity dealiasing rules This method is described in the paper Design, Implementation, and Demonstration of a Staggered PRT Algorithm for the WSR-88D by Torres et al. (004). Herein, VDA c are the normalized velocity difference transfer function (VDTF) constant values and VDA p are the normalized number of Nyquist co-intervals for dealiasing. A set of velocity dealiasing rules is pre-computed at the initiation of the SPRT algorithm as follows: (Compute type-i and II positive VDTF discontinuity points. m and n are the integers in the PRT ratio) p = 0 While p + < m D (p) = (p + )/ m TYPE (p) = p = p + q = 0 While q + < n D (q) = (q + )/ n TYPE (q) = q = q + (Create TYPE by combining and sorting both sets of discontinuity points) Concatenate D and D to create D with p + q elements. Concatenate TYPE and TYPE to create TYPE with p + q elements. Sort TYPE in a slave mode using D as the master. (Compute VDTF constants and dealiasing factors for non-negative discontinuity points) VDA c (p + q) = 0 VDA p (p + q) = 0 For 0 < k < p + q If TYPE (k) = VDA c (p + q + k + ) = VDA c (p + q + k) / m VDA p (p + q + k + ) = VDA p (p + q + k) + / m VDA c (p + q + k + ) = VDA c (p + q + k) + / n VDA p (p + q + k + ) = VDA p (p + q + k) (Compute VDTF constants and dealiasing factors for negative discontinuity points) For (p + q) < k < 0 VDA c (p + q + k) = VDA c (p + q k) VDA p (p + q + k) = VDA p (p + q k) (Note that since the PRT ratio does not change, these vectors can be hard-coded in a real-time implementation of the SPRT algorithm.) 5

6 ) Pre-computation of -independent SACHI filter parameters This method is described in NSSL Signal Design and Processing Techniques for WSR-88D Ambiguity Resolution (Report 3, Report 9 and Report ). The SACHI filter parameters could be pre-computed at the initiation of the SPRT algorithm as follows: (Create 5-by-5 convolution matrix, C r ) C(0) C(4) C(3) C() C() C() C(0) C(4) C(3) C() C C() C() C(0) C(4) C(3) r Cr, Cr, Cr,3 Cr,4 C r,5, C(3) C() C() C(0) C(4) C(4) C(3) C() C() C(0) 4 Ck cn j nk ; for 0 < k < 5 and c = [, 0,, 0, 0], and C r,k is the k-th column of C r. 0 n0 where ( ) ( )exp /5 (Calculate magnitude deconvolution matrix, C md ) (Note: The following formulas are written in matrix algebra notation with the conventions described above) C md Cmd, C, md C C md, r Cmd, C md, , where C md,k is the k-th row of C md. (Calculate matrices C f and C f using st and 5 th columns of C r ) j j j j j j j j0.05 C f C r, C *T r, 0.68 j j j j j j j j j j j j j j j j j j j j0.90 C f C r,5 C *T r, j j j j j j j j j j j j where *T stands for the matrix conjugate transpose (a.k.a. Hermitian) operation. (Calculate the correction coefficients ξ and ξ 3 for correction vector X) k ; k =, 3. * T C C C C C md, r, k r, r, k r,.056 and

7 (Note: since the PRT ratio does not change, these matrices and coefficients can be hard-coded in a real-time implementation of the SPRT algorithm. The numbers provided here are for reference purposes only; the highest precision available is recommended for hard-coding these numbers.) 3) Pre-computation of window parameters (Calculate the extended number of coefficients) x = 5 / (Calculate the number of pulse pairs) p = / (Calculate normalized window d for un-normalized window function d' with x points) x dm ( ) d( m) d( m) ; 0 < m < x. x m0 (Calculate window correction factor for lag-) x dc d( m) d( m) x m0 4) Pre-computation of -dependent SACHI filter parameters (Compute correction vector, X) For 0 k < ceiling( p /) X (k) = For ceiling( p /) k < ceiling( p /) + p X (k) = ξ For ceiling( p /) + p k < ceiling( p /) + 3 p X (k) = ξ 3 For ceiling( p /) + 3 p k < ceiling( p /) + 4 p X (k) = ξ For ceiling( p /) + 4 p k < x X (k) = 7

8 PROCESSING STEPS 5) Short-PRT Segment-III Data Reconstruction Long-PRT Segment-I data is used as a proxy for short-prt segment-iii data For 0 m < p V H (n, m) = V H (n N, m + ) V V (n, m) = V V (n N, m + ) 6) SACHI Clutter Filtering The SACHI filter algorithm is used when clutter filtering is required inside the maximum unambiguous range corresponding to T (r a ). (Form derived time series, V Hd and V Vd, from input time series V H and V V ) For 0 m < p V Hd (5m) = V H (n, m) V Hd (5m + ) = 0 V Hd (5m + ) = V H (n, m + ) V Hd (5m + 3) = 0 V Hd (5m + 4) = 0 V Vd (5m) = V V (n, m) V Vd (5m + ) = 0 V Vd (5m + ) = V V (n, m + ) V Vd (5m + 3) = 0 V Vd (5m + 4) = 0 (Compute DFT of windowed extended time series power compensated for added zeroes) 5 x F H (k) V Hd (m)d(m)exp( jkm / x ) x m0 ; k = 0,, x. F V (k) 5 x V Vd (m)d(m)exp( jkm / x ) x m0 ; k = 0,, x. (Determine clutter filter width parameter, q) (Use GAP to return the number of coefficients identified as clutter, GAP Hcoef and GAP Vcoeff. Pass to GAP the 5 th of the Doppler spectrum containing the main clutter replica; i.e., F {H,V} (0),, F {H,V} [ceiling( p /) ], F {H,V} [ x floor( p /)],, F {H,V} ( x ) }; initialize GAP for spectra with v a /5, and get the number of coefficients identified as clutter to estimate q for both H and V channels) q H = floor [(GAP Hcoef + )/] q V = floor [(GAP Vcoef + )/] (Use the largest q to create the clutter filter vectors for polarimetric variable calculation) q = max(q H, q V ) 8

9 (Create clutter filter vectors I f,i f ) For 0 k < p If k < q I f (k) = I f (k) = 0 If k p q I f (k) = 0 I f (k) = 0 I f (k) = 0 I f (k) = (Row-wise re-arrange F H and F V into 5-by- p matrices, F Hr and F Vr ) For 0 k < p F Hr (0, k) = F H (k) F Hr (, k) = F H (k + p ) F Hr (, k) = F H (k + p ) F Hr (3, k) = F H (k + 3 p ) F Hr (4, k) = F H (k + 4 p ) F Vr (0, k) = F V (k) F Vr (, k) = F V (k + p ) F Vr (, k) = F V (k + p ) F Vr (3, k) = F V (k + 3 p ) F Vr (4, k) = F V (k + 4 p ) (Compute the clutter filtered spectrum matrices, F Hf and F Vf ) (Note: The following formulas are written in matrix algebra notation. Complex-matrix multiplications can be implemented using four real-matrix multiplications as: AB = (A r + ja i )(B r + jb i ) = (A r B r A i B i ) + j(a r B i + A i B r ) ) F Hf F Hr C fl F Hr diagi f ' C f F Hr diag I f ' F Vf F Vr C fl F Vr diagi f ' C f F Vr diag I f ' (Row-wise unfold F Hf and F Vf into F Hdf and F Vdf ) For 0 k < p F Hdf (k) = F Hf (0, k) F Hdf (k + p ) = F Hf (, k) F Hdf (k + p ) = F Hf (, k) F Hdf (k + 3 p ) = F Hf (3, k) F Hdf (k + 4 p ) = F Hf (4, k) F Vdf (k) = F Vf (0, k) F Vdf (k + p ) = F Vf (, k) 9

10 F Vdf (k + p ) = F Vf (, k) F Vdf (k + 3 p ) = F Vf (3, k) F Vdf (k + 4 p ) = F Vf (4, k) (Compute mean power for both channels, P H and P V, and cross-correlation at lag 0, R HV (0)) x P' H (n) F Hdf k k 0 x P V (n) F Vdf k R HV (n) k0 x * F Hdf k 0 F k Vdf k (Proceed with the conventional SACHI Clutter Filtering, only H-channel data) q = q H If q < q (Create clutter filter vectors I f, I f ) For 0 k < p If k < q I f (k) = I f (k) = 0 If k p q I f (k) = 0 I f (k) = 0 I f (k) = 0 I f (k) = F f F Hr C fl F Hr diagi f C f F Hr diagi f F f F Hf (Create clutter filter vectors, I, and I ) For 0 k < p If k < q I (k) = 0 I (k + p ) = 0 I (k + p ) = 0 0

11 I (k + 3 p ) = 0 I (k + 4 p ) = 0 I (k) = I (k + p ) = I (k + p ) = I (k + 3 p ) = I (k + 4 p ) = If k p q I (k) = I (k + p ) = I (k + p ) = I (k + 3 p ) = I (k + 4 p ) = I (k) = 0 I (k + p ) = 0 I (k + p ) = 0 I (k + 3 p ) = 0 I (k + 4 p ) = 0 I (k) = 0 I (k + p ) = 0 I (k + p ) = 0 I (k + 3 p ) = 0 I (k + 4 p ) = 0 I (k) = I (k + p ) = I (k + p ) = I (k + 3 p ) = I (k + 4 p ) = (agnitude deconvolved matrix, F d ) F d =Cmd F f (Row-wise unfold F d into F df ) For 0 k < p F df (k) = F d (0, k) F df (k + p ) = F d (, k) F df (k + p ) = F d (, k) F df (k + 3 p ) = F d (3, k) F df (k + 4 p ) = F d (4, k) (Compute the lag- autocorrelation, R df ) R x df Fdf ( k) exp j k / x d c k 0 (Compute vector I v with / ones centered on arg(r df ))

12 (Round to the nearest spectral coefficient. Choose symmetric window of coefficients around it) arg x R df k 0 round df If k 0df < 0 k 0df = k 0df + x If k 0df x k 0df = k 0df x k df k 0df floor( /4) If k df < 0 k df = k df + x k df k 0df ceiling( /4) If k df x k df = k df x (k 0df is the coefficient corresponding to arg(r df ), k df and k df specify the extent of p spectral coefficients centered on the mean velocity. If k df < k df, the ones span from k df to k df ; otherwise, the ones will span from k df to x, and 0 to k df ) If k df < k df For 0 k < x If k < k df OR k > k df I v (k) = 0 I v (k) = For 0 k < x If k < k df AND k > k df I v (k) = 0 I v (k) = (Interpolate the elements for the region around zero velocity in F df with linearly interpolated values from S and S ) If q > 0 S F () q df S Fdf ( x q) For 0 k < x If k < q F i k S S S If k > x q F i k S S S q k / q q k x / / q /

13 Fi( k) Fdf ( k) (Don t interpolate if not needed) For 0 k < x Fi( k) Fdf ( k) (Compute the corrected spectrum, F c ) For 0 k < x F c (k) = F i (k) I (k) + F i (k) I (k) I v (k) X (k) (Compute vector I c with ones where there s a non-zero spectral component in vector F c ) For 0 k < x I c (k) = I (k) + I (k) I v (k) (Compute the mean power, P c, and autocorrelation at lag T u, R c, using F c ) P x F ( k ) c k 0 c R F k j k x c c( ) exp x dc k 0 (Retain only coefficients centered on velocity based on R c and delete the rest from F c and I c ) x arg R c k 0c round If k 0c < 0 k 0c = k 0c + x If k 0c x k 0c = k 0c x k c = k 0c p If k c < 0 k c = k c + x k c = k 0c + p If k c x k c = k c x If k c < k c For 0 k < x If k < k c OR k > k c F m (k) = 0 I m (k) = 0 3

14 F m (k) = F c (k) I m (k) = I c (k) For 0 k < x If k < k c AND k > k c F m (k) = 0 I m (k) = 0 F m (k) = F c (k) I m (k) = I c (k) (Compute the modified mean power, P m, and autocorrelation at lag T u, R m, using F m ) x Pm Fm( k) k 0 R F k j k x m m( ) exp x dc k 0 (Compute noise correction factors) N x c Ic( k) x k 0 N x m Im( k) x k 0 (Compute overlaid power correction if in segment I) If n < N N S ov p V H n N,m p m0 If S ov < 0 S ov = 0 S ov = 0 Noise (Correct powers to remove overlaid contamination adjusted for each spectrum) P m = P m N m S ov If P m < 0 P m = 0 P c = P c N c S ov If P c < 0 P c = 0 4

15 (Compute spectrum width power ratio adjustment) S m = P m N m Noise If S m < 0 S m = 0 If S m > 0 Rm Padj S P adj = 0 (Compute signal power) S c = P c N c Noise If S c < 0 S c = 0 m (Compute short PRT autocorrelation at lag T ) R H (n) S c P 4 adj exp jarg R c (Compute long PRT autocorrelation at lag T ) R H (n) S H P 9 adj exp j3arg R c (Adjust signal power to include noise) P H (n) = S c + Noise (Note that the outputs of SACHI are P H (n), P V (n), R HV (n), P H (n), R H (n) and R H (n)) 7) DC Removal Clutter Filtering (Segment-III gate with segment-i clutter) This DC Removal clutter filtering algorithm removes the mean (DC) component of the short-prt segment-iii gates in those locations where the site-dependent clutter filter bypass map B indicates the need for clutter within segment I. (Calculate the mean of the even pulses.) p VHm VH ( n, m) p m0 p VVm VV ( n, m) p m0 (Subtract mean from even pulses.) For 0 m < p V HF (m) = V H (n, m) V Hm V HF (m + ) = V H (n, m + ) V VF (m) = V V (n, m) V Vm V VF (m + ) = V V (n, m + ) 5

16 8) No Clutter Filtering For 0 < m < V HF (m) = V H (n, m) V VF (m) = V V (n, m) 9) Power and correlation computations for each PRT If n < N (Compute power from even pulses, if available) p PH VHF( m) p m0 p PV VVF( m) p m0 (Compute cross-correlation from even pulses, if available) p * RHV( n) VHF ( m) VVF ( m) p m0 (Compute power from odd pulses) p PH VHF(m) p m0 p PV VVF(m) p m0 (Compute cross-correlation from odd pulses) p * HV ( ) HF ( ) VF ( ) p m0 R n V m V m (Compute lag- correlations from all pulses from H channel) p * H( ) HF( ) HF( ) p m0 R n V m V m p * H ( ) HF( ) HF( ) p m0 R n V m V m 0) Combined power and cross-correlation computation To use as much information as possible, data are extracted from the two power arrays with different rules for each of the three segments depicted in Figure. For segment, data are extracted only from P, since P may be contaminated on those range bins with overlaid powers. An average of P and P is extracted for segment, given that both power vectors are clean there. Finally, segment data are obtained from P. In algorithmic form: 6

17 If n < N N (Segment I) PH( n) PH PV( n) PV If n < N (Segment II) P ( n) P P H H H V ( ) V V P n P P (Segment III) PH( n) PH P ( n) P V P' H (n) P H V T T Fig.. Signal powers in the staggered PRT algorithm. Roman numerals indicate segment numbers. The same rules apply for the cross-correlation computation. If n < N N (Segment I) RHV ( n) RHV If n < N (Segment II) R HV (n) R HV R HV (Segment III) RHV ( n) RHV ) Strong point clutter canceling Processing is as in the current system. Strong-point clutter canceling is applied to P H, P H, R H and R H based on radial power continuity in P H. For the remainder of the algorithm it is assumed that the outputs of this step are P H, P H, R H and R H. ) Signal power computation If P H (n) < N H S H = 0 S H = P H (n) N H If P H (n) < N H 7

18 S H = 0 S H = P H (n) N H If P V (n) < N V S V = 0 S V = P V (n) N V 3) Reflectivity computation (Range in km) R = nr + R/ (Reflectivity in dbz. log 0 is the base-0 logarithm) If S H > 0 Z (n) = 0log 0 (S H) + dbz0 + R ATOS + 0log 0 (R) 0log 0 (N H), Z(n) should be set to the smallest possible reflectivity value 4) Velocity computation (Compute Doppler velocities for each PRT using the corresponding correlation estimates) v arg RH ( n) 4 T v arg R ( n) H 4 T (Compute extended Nyquist velocity) v a T (Dealias velocity using pre-computed rules) l arg min v v VDA c (k)v a k v(n) v v a VDA p (l) (Prevent dealiased velocities outside of the extended Nyquist co-interval) If v(n) > v a vn ( ) vn ( ) v a If v(n) < v a vn ( ) vn ( ) v a 5) Spectrum width computation 8

19 The spectrum width estimator corresponds to the algorithm implemented in the legacy WSR-88D signal processor. If S H = 0 OR R ( ) 0 H n (Insert spectrum width of white noise) wn ( ) 4 3T If S R ( ) n H H (Insert spectrum width of a constant) wn ( ) 0 (Spectrum width computation. ln is the natural logarithm) S wn ( ) ln T R H( n) If wn ( ) 4 3T wn ( ) 4 3T 6) Differential reflectivity computation If S H > 0 AND S V > 0 If S H = 0 If S V = 0 ' S Z DR (n) 0 log H 0 S V Z DR (n) should be set to the smallest possible value Z DR (n) should be set to the highest possible value 7) Differential phase computation DP n arg RHV n 8) Cross-correlation coefficient computation If S H > 0 AND S V > 0 9

20 HV (n) R HV n ' S H S V HV (n) = 0 9) Determination of significant returns for reflectivity and polarimetric variables The non-significant return indicator array (NS Z ) is a binary array where 0 indicates significant and indicates non-significant. This array is also used for Z DR (n), DP (n) and hv (n). 0.T If SH NH 0 Z NS Z (n) = NS Z (n) = 0 0) Determination of significant returns for velocity The non-significant return indicator array (NS V ) is a binary array where 0 indicates significant and indicates non-significant 0.TV If SH NH 0 NS V (n) = NS V (n) = 0 ) Determination of significant returns for spectrum width The non-significant return indicator array (NS W ) is a binary array where 0 indicates significant and indicates non-significant 0. If S 0 T H NH W NS W (n) = NS W (n) = 0 ) Determination of overlaid returns for velocity and spectrum width Censoring of velocity and spectrum width data is only necessary in segments and. This is done by analyzing P in segment (P ) and P in segment P ) see Fig. ). The idea is to determine whether second trip signals mask first trip signals and vice versa. While such overlaid echoes appear in every other pulse and do not bias velocity estimates at those range locations, overlaid powers act as noise. Therefore, when overlaid powers are above a preset fraction of their non-overlaid counterparts, the corresponding velocity and spectrum width estimates exhibit very large errors and must be censored. The overlaid indicator arrays (OV V and OV W ) are binary arrays where 0 indicates not overlaid and indicates overlaid. If n < N N (Segment I: Range gates that may or may not have overlaid echoes) (Check power ratio using velocity threshold) 0

21 0.TOV If P H (n) > P H (n + N ) 0 OV V (n) = 0 (Power ratio not met, but consider non-significant returns as non-existent) If NS V (n + N ) = OV V (n) = 0 OV V (n) = (Check power ratio using width threshold) 0.TOW If P H (n) > P H (n + N ) 0 OV W (n) = 0 (Power ratio not met, but consider non-significant returns as non-existent) If NS W (n + N ) = OV W (n) = 0 OV W (n) = If n < N (Segment II: Range gates that, based on the assumptions, never have overlaid echoes) OV V (n) = 0 OV W (n) = 0 (Segment III: Range gates that may or may not have overlaid echoes) (Check power ratio using velocity threshold) 0.TOV If P H (n) > P H (n N ) 0 OV V (n) = 0 (Power ratio not met, but consider non-significant returns as non-existent) If NS V (n N ) = OV V (n) = 0 OV V (n) = (Check power ratio using width threshold) 0.TOW If P H (n) > P H (n N ) 0 OV W (n) = 0 (Power ratio not met, but consider non-significant returns as non-existent) If NS W (n N ) = OV W (n) = 0 OV W (n) = (Note that when processing the overlaid and significant return flags, the overlaid flags take a lower priority. That is, if a range bin is tagged as non significant and also as overlaid, the overlaid indication is ignored and the gate is treated as a nonsignificant return only; e.g., painted black as opposed to purple)