A weather radar's resolution volume is generally determined by beam and pulse widths. Recently, a single-antenna interferometric technique was developed to measure cross-beam and radial wind using angular and range interferometry. Here it is shown that the interferometry technique can refine radar resolution. The technique for resolution refinement is based on the fact that the cross-correlation function of signals from two resolution volumes is contributed only by scatterers in the shared volume. Therefore all the radar measureables derived from cross-correlation estimates have a resolution of the shared volume size rather than a radar resolution volume size. The performance of the technique depends on the shape of beam/range-weighting functions and relative sample error. A sharp change at the edges of the weighting function and low sidelobes are important for resolution refinement using weather radar interferometry.
 Weather radar's cross- and along-range resolutions are determined by antenna size and transmitted pulse width/receiver bandwidth respectively [Doviak and Zrnic, 1993, section 4.4.4]. Improving angular resolution by increasing antenna size is limited by cost constraints, whereas improving range resolution by increasing bandwidth and decreasing pulse width is often limited by Federal Communications Commission (FCC) regulations.
 Existing angular resolution refinement techniques include coherent processing methods such as the monopulse radar, Doppler Beam Sharpening (DBS), developed by Wiley in the early 1950s [Ulaby et al., 1982, section 9–2], and an incoherent deconvolution method [Andrews and Hunt, 1977; Magain et al., 1998]. The monopulse radar improves the resolution of a discrete (point) scatterer's location within the radar's resolution volume V6 [Doviak and Zrnic, 1993, section 4.4.4]. DBS, a Synthetic Aperture Radar (SAR) method, applies to a distribution of scatters fixed on a surface, but obliquely observed with an elevated moving antenna. The deconvolution method applies to a volumetric distribution of scatterers and improves resolution by separating the effects of a known antenna pattern on measured gradients of the reflectivity field. However, deconvolution is an ill-posed problem, and has no unique solution, especially in the presence of noise [Magain et al., 1998]. Furthermore, sampled data can be perfectly deconvolved only if the sampling theorem, applied to the true reflectivity field, is not violated. Therefore various optimization techniques, such as a compensating filter [Andrews and Hunt, 1977; Palmer et al., 1998; Yu and Palmer, 2001], are used to improve the result. However, the various optimization techniques do not work well for radar observations of weather in the presence of noise [Sadjadi, 2000].
 Weather Radar Interferometry (WRI) is a method whereby the cross correlation of two or more time series of weather signals from a pair or more of overlapping V6s is used to extract meteorological information. For example, WRI is used to cross-correlate signals from overlapped bistatic V6s [Doviak and Zrnic, 1993, section 11.3] to measure cross-beam wind [Briggs et al., 1950; Doviak et al., 1996]. Zhang et al. [2003b] have extended the WRI technique to measure the cross-beam and radial wind components using overlapped resolution volumes of a monostatic radar. Herein WRI is applied to signals from overlapped V6s to refine weather radar resolution without increasing bandwidth or antenna diameter. WRI requires relatively high correlation and many samples of signals from overlapped V6s; this in turn requires radar with beam agility. The rapid and flexible scan capability of phased array antennas, such as that used by the National Weather Radar Testbed (NWRT) being developed at National Severe Storms Laboratory (NSSL), could provide this required capability, and opens opportunities to verify the theories for refinements in resolution and measurements of cross-beam winds using WRI.
 A conceptual description of the WRI technique for resolution refinement is presented in section 2. Beam splitting to further sharpen the beam is described in section 3. Numerical simulations are presented in section 4 to demonstrate how Angular Interferometry (AI) increases angular resolution and Range Interferometry (RI) improves range resolution. Finally, the feasibility for practical application is discussed in the summary.
2. Conceptual Description and Formulation
 Weather radars often measure hydrometeor characteristics, such as reflectivity factor, Z, and Doppler velocity v, through estimation of the temporal autocorrelation function obtained from a time series of weather signal samples spaced a Pulse Repetition Time (PRT) apart [Doviak and Zrnic, 1993, chapter 6]. Strictly speaking, measurements of hydrometeor characteristics are obtained from estimates of the angular autocorrelation function with sample spacing determined by the antenna's speed of rotation and the PRT. If the density of scatterers is homogeneous over the angular displacement of the beam, the angular autocorrelation can be treated as a temporal autocorrelation from a volume having an angular width larger than the beam width [Doviak and Zrnic, 1993, section 7.8].
 WRI for resolution refinement is based on the fact that the cross-correlation function of signals from two distinctly spaced but overlapping resolution volumes has, on average, contributions principally from those scatterers in the shared or overlapped volume. Consider two radar resolution volumes, V6(1) and V6(2), that are partially overlapped as shown in Figure 1, and N scatterers spread uniformly over the entire domain, but K of those are within the shared region. The weather signal is the sum of signals returned from each scatterer. Thus the weather signal voltage V(01, t1) for V6(1) is proportional to [Doviak and Zrnic, 1993, section 4.2],
and for V6(2) it is
where A1n is the prefilter, echo amplitude of the nth scatterer located at n(t1) at time t1, 01 is the location of the center of V6(1), and W1n is a range-dependent weight, a function of the transmitted pulse width and the receiver filter's bandwidth. Similar definitions apply to (2). A1n is proportional to the normalized radiated power density pattern f2(θ − θ01, ϕ − ϕ01) of the antenna (f2(θ, ϕ) is also the two-way pattern gain function of the electric field) as well as the backscattering cross section of the scatterer. Because the range extent of V6 is usually small compared to r0 the small changes in the weighting function W(r) due to the 1/r2 factor can be ignored.
 To apply WRI to refine angular resolution it is assumed that the location of V6 is alternately switched between V6(1) and V6(2) in a Pulse Repetition Time (PRT ≡ Ts). There is no need to account for changes in A1n due to changes in scatterer location because hydrometeors move, during a PRT, a short distance compared to the typical size of V6. However, the phase term can have considerable change because scatterers can move an appreciable fraction of the wavelength during a PRT. Nevertheless, to illustrate the technique and to keep the development simple, differential phase shifts during the PRT are assumed to be negligible (i.e., it is assumed that ∣rn(t1 + PRT) − rn(t1)∣ ≪ λ/4π for all n). In other words, it is assumed that the PRT ≪ τc, the correlation time.
 Typically, weather signals can be considered to be statistically stationary during the data acquisition or dwell time Td. Thus the cross-correlation function associated with signals from the two spaced resolution volumes can be written as
where brackets indicate time or ensemble averaging, and τ is sample lag spacing τ ≡ mPRT ≡ mTs. Therefore substituting from (1) and (2),
where 1n ≡ (θn − θ01, ϕn − ϕ01) is the angular position of the nth scatterer from the center of V6(1). Without losing the essence in explaining the WRI method, assume that the scatterers' cross sections are given unit weight if inside V6, but zero weight if outside. Thus the double sum in (3b) reduces to a single sum over the K scatterers within the shared volume, and (3b) reduces to
There is no contribution to C12(τ) from scatterers in the unshared region because, in this simplified example, either f2(1n)W(rn, r01) or f2(2n)W(rn, r02) is zero there. If two resolution volumes are displaced in angle (i.e., to refine angular resolution), the smallest τ could be is the PRT, and the beam would alternate between two closely spaced directions θd (Figure 1) every PRT. If the PRT is short (i.e., Ts ≪ τc) so that echoes from the two V6s are highly correlated, ∣C12(Ts)∣ ≈ ∣C12(0)∣ is proportional to the reflectivity within the shared volume. If the PRT is not short compared to τc, then the reflectivity would be obtained by interpolation to ∣C12(0)∣.
 Henceforth, to simplify our numerical simulations, it is assumed as stated earlier that Ts ≪ τc.
 Therefore measurements Z, v, etc. can be derived from ∣C12(Ts)∣ and are expressed as
with a resolution of the shared volume V. Conventionally, Z, v, etc. are estimated from the autocorrelation function [Doviak and Zrnic, 1993], and the resolution is given by the size of V6.
 To refine range resolution, assume two resolution volumes displaced radially rd (determined by range gate spacing), and assume rd is less than the range resolution r6 determined by the transmitted pulse width, τp, and the 6 dB bandwidth (B6) of the receiver [Doviak and Zrnic, 1993, section 4.4.3]. If B6 ≥ τp−1, only scatterers inside the shared volume contribute significantly to ∣C12(rd)∣.
 Using the above simplified example, it is seen how WRI can be used to refine resolution. In practice, however, the implementation is not as simple. Three main difficulties that limit the application of WRI for resolution improvement are as follows.
2.1. Limited Spatial and Temporal Bandwidths
 The weighting function, proportional to f2()W(r), does not have a rectangular form with zero weighting outside V6. A sharp change at the edges of the weighting function and low sidelobes are important for resolution refinement using WRI. For a simple illustration, consider two one-dimensional weighting functions (e.g., the one-way antenna power gain function proportional to g(X − X0)) having a rectangular shape with centers separated by 1.0, and the product of these weighting functions (i.e., g(X + 0.5)g(X − 0.5)) as shown in Figure 2a. The product of the two weighting functions gives the shared region having half the width of the individual weighting functions.
 The rectangular function can be a good approximation for the range-weighting function if the transmitted pulse of width τp is rectangular, and B6 ≫ τp−1. However, a rectangular function is not a good representation of the antenna gain function. Therefore consider the usually assumed Gaussian shaped antenna patterns as shown in Figure 2b. It can be shown that the product of the two displaced gain patterns has the same width as each of the two individual gain functions squared! In the other words, cross-correlation estimates do not improve the resolution if the weighting function has a Gaussian form. The shape of the gain function and sidelobe levels, however, can be controlled by appropriate excitation of the array elements as shown in the next section. On the other hand, there is an inverse relation between low sidelobe levels and sharp edges of the antenna gain function; that is, narrow beams having sharper edges are usually obtained with uniformly excited elements, but at the cost of higher sidelobe levels.
2.2. Finite Sample Error
 Because perfect estimates of the correlation functions can only be obtained with an infinite number of samples (assuming the weather is statistically stationary), errors are incurred when a finite number of samples are processed. However, these relative errors (actually variance of the estimates) are larger for cross-correlation estimates than for autocorrelation estimates as is now shown. It is known that signal power estimates, (i.e., the autocorrelation function estimate at zero lag) have a relative Standard Deviation SD given by [Doviak and Zrnic, 1993]
whereas the relative SD for estimates of the cross-correlation function magnitude is [Zhang et al., 2003a]
where MI is the number of independent samples. Obviously, (7) is larger than (6) because S/∣C12∣ > 1 where S is the signal power from scatterers including those outside the shared volume. Scatterers outside the shared volume do not contribute the expected cross-correlation magnitude, but do contribute to S and thus to the SD of the cross-correlation estimates. In other words, signals from scatterers outside the shared volume are unwanted causing a larger relative standard deviation. Echoes from scatterers outside the shared resolution volume form two relatively independent streams of weather signals. One stream is associated with the scatterers weighted with f2(1n)W(rn, r01) ≡ W1, and a second stream is associated with echoes from the same scatterers but weighted with the function f2(2n)W(rn, r02) ≡ W2. Because W1 and W2 significantly weight different regions of scatterers, and because the scatterers' locations are independent of one another, the two echo streams will be relatively independent. The signal level associated with these two streams, however, is of comparable level to the signal coming from the shared volume. For example, if the weighting function is rectangular and half of the function is overlapped and the scatterers are uniformly distributed, the RMS level of the signal associated with scatterers outside the shared volume would be equivalent to the RMS level of signals coming from the shared volume. Nevertheless, the increased SD of the cross-correlation estimates can be diminished by increasing the dwell time, or equivalently MI.
2.3. Relative Noise Power
 WRI can have a lower effective signal-to-noise ratio (SNR) because ∣C12∣ < S. It is true that the noise contribution to the variance of the cross-correlation estimates is smaller than that to autocorrelation estimates as shown by (12) and (13) of Zhang et al. . However, the first-order noise term decreases from (S + ∣C11(2τ)∣) to , and the second-order term decreases from (1 + δ(τ)) to . However, such decreases in the noise terms are limited by the reduced cross-correlation magnitude. For example, if S/∣C12∣ > 2 (i.e., less than one half the resolution volume shared), the WRI has a lower SNR compared to the SNR associated with auto-correlation processing.
3. Beam Splitting
 As shown in the previous section, sharp edges of the antenna gain function are desirable in using WRI to refine angular resolution. However, a rectangular pattern cannot be obtained with an antenna of finite size; that requires a current distribution having the form sin(x)/x vs x:[−∞, +∞]. The narrowest beam and sharpest edges are obtained if the current distribution is uniform across a finite array. Nevertheless a pair of sharper beams, each with higher resolution than a uniformly illuminated aperture, can be obtained by splitting the main lobe, a technique similar to that used in the formation of the receiving beam of monopulse radar like the AN/SPY-1. The NWRT being developed at NSSL uses an AN/SPY-1 phased array antenna (Figure 3a). Figure 3b shows the antenna pattern for the monopulse sum (i.e., the transmitted radiation pattern), and Figure 3c shows the monopulse elevation difference pattern used for reception. As can be seen, the individual beams of the difference pattern (i.e., split beams) are about 25% narrower than that of a sum pattern. Considering the cost of an antenna is at least proportional to the square of its diameter, a 25% improvement in the angular resolution could produce a cost savings of nearly 50% or more.
 The gain pattern is determined by antenna size and distribution of the current's magnitude and phase across the aperture. To illustrate the improvement of angular resolution provided by beam splitting we consider, for sake of simplicity, one-dimensional arrays and the following three current distributions:(1) uniform magnitude and phase
(2) uniform magnitude but antisymmetric phase distribution
and (3) a linear distribution of current
The electric field–weighting function f(θ) is the Fourier transform of the current distribution [Ishimaru, 1991, section 9–4], that is,
where θ is the elevation angle, and the beam axis is assumed to be at θ01 = 0°. Substitution of (8)–(10) into (11) yields the three corresponding field patterns:
Figure 4a shows the current distributions, and Figure 4b shows the corresponding antenna power patterns normalized by (I0D)2 (i.e., ∣f1(n)(θ)∣2 = ∣f1(θ)∣2/(I0D)2) for an antenna size of 3.66 m at a frequency of 3.2 GHz (i.e., the NWRT parameters). With opposite phase of the excitation current on each half of the antenna, the one-way power density pattern is split into two beams as shown by ∣f2(n)(θ)∣2 and ∣f3(n)(θ)∣2. The one-way half-power beam widths are 1.30°, 1.18°, and 1.01°, respectively. To achieve a 1° beamwidth with uniform illumination, the antenna diameter would have to be increased by about 30%, a costly solution. However, the sidelobes levels for ∣f32(θ)∣ are significantly higher. For example, the normalized first sidelobe levels are: −13.3, −10.6, and −8.3 dB, respectively.
4. Numerical Simulations
 In this section, we describe numerical simulations of wave scattering that are used to verify WRI for resolution refinement. In these simulations, the medium is modeled by a collection of randomly distributed scatterers. Again, for the sake of illustration, we consider a 1-D power density pattern.
4.1. Interferometry to Refine Angular Resolution
 Consider two layers of randomly distributed scatterers at a range of 50 km and separated by 800 m. The WRI technique is applied to a pair of overlapped antenna patterns, one pair for f12(θ), and the other pair for the split beam pattern f32(θ). WRI to refine angular resolution with split beams is achieved by alternately shifting beam directions such that only one of the split beams is overlapped with the other, whereas angular refinement with f12(θ) is obtained by over lapping the pair of main lobes. Range weighting is uniform over a 100 m interval, and within this range interval there are N = 50 scatterers in each layer having a thickness of 10 m. The aim of refining the angular resolution using WRI is to resolve the two layers. At the 50 km range, the angular separation θsc of the two scattering layers is 0.92°. The weather signal for each beam position of the pair can be expressed as
where the center of V6 is at θ = θ01 for the first beam position, and θ = θ02 for the second beam position (Figure 5), and θn is the angular location of the nth scatterer. Time series data pairs are obtained by randomly changing the scatterers' positions after alternately sampling the media with the two different beam directions; a total of two hundred pairs are realized to obtain M = 200 = MI independent estimates of C12(0); in real radar applications, the zero-lag cross-correlation function C12(0) can be approximated by the first-lag cross-correlation function, that is, C12(0) ≈ C12(Ts), if Ts ≪ τc, or through interpolation. Both the single-beam pattern f12(θ) and the split beam pattern f32(θ) are used in the simulations. The cross-correlation function is then estimated from the time series data by averaging over M realizations to obtain
where θc = (θ01 + θ02)/2 is the mean angular location, and θd = θ02 − θ01 is the angular separation of the two beam directions. When θd = 0, 12 = 11.
 The simulation results for the correlation coefficient estimates plotted as a function of θc are shown in Figure 6. Figure 6a shows the results for a single-beam pattern f12(θ) with beam displacement θd as a parameter. Using autocorrelation estimates (i.e., θd = 0; WRI is not used), the two layers of the scatterers can hardly be distinguished. However, using WRI and the cross-correlation estimates with θd ≠ 0 the two scattering groups are better resolved. However, the cross correlation is lower for larger separation, which means a lower SNR, a larger relative sampling error, and a higher sidelobe level.
Figure 6b gives a similar presentation for the split beam pattern f32(θ). The autocorrelation estimates give ghost images as each layer of scatterers are seen by the two main lobes of the split beam. On the other hand, the cross-correlation estimator with beam displacement θd = 2° clearly distinguishes the two layers of scatterers while maintaining the same level of correlation magnitude.
4.2. Interferometry to Refine Range Resolution
 To demonstrate refinement of range resolution using WRI, consider two slabs, each with a thickness of 10 m, and placed at different ranges separated by 60 m as shown in Figure 7a. Each slab contains randomly distributed scatterers. The weather signal is calculated from
where the resolution volume center is r, rn is the range to the nth scatterer, W(r) is the range-weighting function assumed to have a rectangular form with a resolution of 100 m. Then, the range cross-correlation function is estimated from the time series data V(r01, t), V(r02, t) by averaging over M realizations as
where rc = (r01 + r02)/2 is the range to the center of the shared volume, and rd = r02 − r01 is the range separation of the two resolution volumes. When rd = 0, 12 becomes autocorrelation function estimate 11.
 The simulation results of correlation coefficient estimates are plotted in Figure 7b as a function of rc. Autocorrelation estimation cannot separate the two groups of scatterers, but WRI using the cross correlation estimated with a range separation of 75 m clearly distinguishes the two slabs of scatterers while it maintains a substantial level of correlation magnitude.
5. Summary and Discussions
 In this paper, we apply Weather Radar Interferometry (WRI) to refine weather radar resolution. The resolution refinement technique is described using the theory of wave scattering from a random medium. The WRI technique for resolution refinement is based on the fact that the most significant contribution to the cross-correlation function, of signals from two overlapping resolution volumes V6(1) and V6(2), is from scatterers in the shared volume. The shared volume for beam width refinement must be obtained by rapidly alternating, from transmitted pulse to transmitted pulse, the direction of the beam, and cross correlating the signals from the two overlapped beam directions.
 The difficulties for the practical application of WRI to refine resolution are: (1) a larger relative sampling error, (2) the requirement of a sharp change at the edges of the angular/range-weighting functions, (3) lower signal-to-noise ratio (SNR) than that associated with the commonly used autocorrelation techniques, and (4) higher sidelobe levels, especially for angular resolution refinement. The sampling error may be reduced using a larger number of independent samples. A sharp edge of the principal lobe of the antenna's radiation pattern can be achieved by splitting the beam as discussed in section 3. The lower SNR is difficult to overcome because the added noise originates from the scatterers outside the shared volume and thus the added noise increases in proportion to the level of signal from the shared volume. Worse yet, if the shared volume is made smaller to increase the resolution, the added noise increases which further increases the variance of the estimates of the cross-correlation function and, in turn, the estimates of reflectivity and Doppler velocity in the shared volume. Furthermore, the number of independent samples that can be used to reduce the added noise is typically much less than that number for receiver noise. The deleterious effects due to increased side lobe levels might be suppressed by using the switched antenna pattern technique described by Sachidananda et al. .
 Numerical simulations were used to verify that range and angular resolutions of weather radars can be improved by cross-correlating signals from resolution volumes displaced in range and angle. The numerical results were based on single-lag estimates. They can be further improved by optimally combining multilag cross-correlation function estimates as shown by Backus and Gilbert  and T. Yu et al. (Resolution enhancement using range oversampling, submitted to Journal of Atmospheric and Oceanic Technology, 2004). By shifting the shared resolution volume, WRI can also be used to increase the number of independent samples. This latter application should be compared to the whitening transformation described by Torres and Zrnic . Although these preliminary results demonstrate resolution refinement for weather radar, further detailed studies (e.g., optimal inversion) are required to better define the limitations of the technique.
 The authors greatly appreciate helpful discussions with Dusan S. Zrnic, J. Vivekanandan, Jeff Keeler, Edward A. Brandes, Robert J. Serafin, and Akira Ishimaru and the support provided by NCAR, NOAA, and the University of Oklahoma.