## 1. Introduction

Detailed microphysical insights from weather radar systems are in demand within operational weather agencies and for the evaluation of numerical weather and climate prediction models, including the construction of model forcing datasets. An emphasis has been on polarimetric radar technology, as highlighted most recently by the upgrade of the U.S. National Weather Service (NWS) Weather Surveillance Radar-1988 Doppler (WSR-88D) network to polarimetric capabilities. Polarimetric systems promise more robust precipitation products over conventional weather radar systems with reduced uncertainty for applications including echo classification, rainfall rate estimation, and drop size distribution (DSD) retrievals.

Hydrological product improvements using polarimetric methods are closely tied to the inclusion of the differential phase Ψ_{DP} and its range derivate *K*_{DP}, which are less sensitive to hail contamination and variation of DSDs, immune to attenuation in rain, radar miscalibration, and partial beam blockages (e.g., Zrnić and Ryzhkov 1996). Measurements of *K*_{DP} as inputs to radar algorithms demonstrate improved accuracy and usefulness of radar rainfall accumulations (e.g., Ryzhkov et al. 2005b; Wang and Chandrasekar 2010; Gourley et al. 2010). The *K*_{DP} measurements also allow new opportunities for radar reflectivity factor *Z* and differential reflectivity *Z*_{DR} quality control and relative calibration monitoring using self-consistency and natural media expectations (intrinsic properties of rain, e.g., Goddard et al. 1994; Scarchilli et al. 1996; Smyth and Illingworth 1998; Bringi and Chandrasekar 2001; Vivekanandan et al. 2003; Ryzhkov et al. 2005a; Giangrande and Ryzhkov 2005).

Despite the attractive qualities of polarimetric phase measurements, the usefulness of propagation differential phase measurements Φ_{DP} at operational weather radar wavelengths is lessened as a consequence of measurement noise, unreliability in lighter rainfall and artifacts stemming from nonuniform beam filling (NBF). These NBF artifacts can be associated with the (vertical, horizontal) gradients of Φ_{DP} and *Z* in the vicinity of stronger convective cells and in response to changes in the vertical profile of reflectivity/differential phase near the melting layer (e.g., Ryzhkov and Zrnić 1998; Ryzhkov 2007). Even if differential phase insights are selectively interjected under advantageous regimes, phase-based rainfall relations commonly demonstrate undesirable characteristics (smearing, speckling) and unphysical behaviors (negative rainfall rates, intrinsic negative behaviors) in the situations of supposed strength (hail and/or melting layer; e.g., Giangrande and Ryzhkov 2008). The challenges are exacerbated at shorter wavelengths, wherein the expectation for an increased (potentially, more reliably sampled) total differential phase shift through similar rain conditions can be compromised by significant contributions from a nonzero differential backscatter phase shift δ (herein, we adopt the convention; total measured Ψ_{DP} = Φ_{DP} + δ). These problems point to the necessity for robust phase-processing methods that attempt to deconvole propagation Φ_{DP} and backscatter δ phase components, as these components represent significant independent insights into precipitation microphysics.

Several methods for Ψ_{DP}, Φ_{DP}, and/or *K*_{DP} processing have been proposed to handle challenges associated with the aforementioned issues (e.g., Hubbert et al. 1993; Hubbert and Bringi 1995; Bringi and Chandrasekar 2001; Ryzhkov et al. 2005b; Wang and Chandrasekar 2009; Otto and Russchenberg 2011). To be sure, there is no standard processing or universally adopted approach within the radar community. It is an error to assume existing methods as interchangeable and that processing details do not impact subsequent hydrological products. Recent examples of this sentiment include Cifelli et al. (2011), wherein the National Severe Storms Laboratory (NSSL) *K*_{DP} processing and rainfall methods, prototypes for the operational NWS WSR-88D standard, demonstrate functional however nonoptimal performance as compared to methods that include *K*_{DP} inputs following Wang and Chandrasekar (2009). As computing power increases, the ease for incorporating additional computational complexity to processing routines should further improve the reliability of fundamental polarimetric phase measurements. For example, several recent studies have already proposed incorporation of polarimetric self-consistency behaviors and realistic physical constraints (e.g., Goddard et al. 1994; Scarchilli et al. 1996) into more complex or nonlinear schemes within research and operational settings (e.g., Otto and Russchenberg 2011; Schneebeli and Berne 2012; Maesaka et al. 2012).

The U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Program (Ackerman and Stokes 2003) recently installed several scanning weather radar networks operating at C-band (5 cm) and X-band (3 cm) wavelengths. For these systems, there was immediate demand for open/free source radar processing codes and community data formats that encourage use and visualization of DOE ARM radar datasets. As an example of an early ARM radar processing activity, this study includes the results for an application of well-known linear programming (LP) concepts (or linear optimization) to the challenge of radar differential phase estimation. This activity was initiated not only as a functional solution to differential phase processing at ARM wavelengths but also as a proof-of-concept motivation for similar physical-based LP retrieval applications in the future. To our knowledge, LP concepts have not been attempted for radar data processing or physical retrievals using radar data, with most LP applications limited to decision making and resource planning in the context of atmospheric sciences (e.g., Helbush 1968; Minciardi et al. 2003). The present study attempts to construct an LP method that shares in the desirable measurement properties sought by existing polarimetric phase-processing techniques, including (i) an ability to maintain monotonic (nonnegative *K*_{DP}, e.g., Vivekanandan et al. 2001) behaviors in rain and to isolate Φ_{DP} from δ contributions (extract a monotonic profile), (ii) producing unbiased *K*_{DP} estimates, (iii) adopting polarimetric self-consistency constraints to boost performance in rain regions, and (iv) the potential to resource external datastreams or additional constraints for later improvements. This suggests that our initial methods should share in several benefits of existing nonlinear variational methods that have already been proposed (e.g., Schneebeli and Berne 2012; Maesaka et al. 2012), but herein only a simplified version is offered that is still flexible to handle future modification.

This study is organized as follows. Section 2 describes the linear programming solution to a problem of differential phase processing. Section 3 presents the results of our application of LP methods for a simulated radar dataset at S band (10 cm) and examples collected from the ARM C-band [C-band scanning ARM preciptation radar (CSAPR)] and X-band [X-band scanning ARM precipitation radar (XSAPR)] radar systems in challenging convective storm environments around north-central Oklahoma. Phase processing for this study is benchmarked as compared to NWS WSR-88D phase-processing standards based on NSSL concepts. These results are discussed in section 4 and include additional comments on potential advantages and limitations of LP methods as compared to a wider array of processing options.

## 2. Application of LP to estimate differential propagation phase

This section defines an LP problem associated with polarimetric differential phase processing. Our goal is to demonstrate a reasonable LP retrieval for the Φ_{DP} component (having a range derivative *K*_{DP}), which implies that one may estimate δ as a residual of this retrieval and the actual Ψ_{DP}. For the example LP formulation that follows, a few basic assumptions have been adopted that are similar to many previous processing efforts.

A monotonic behavior is assumed for propagation phase Φ_{DP} profiles in rain (nonnegative *K*_{DP}). This assumption should be valid below the melting layer and along radials unaffected by NBF (strong storm or melting layer gradients) or significant artifacts from larger hail (intrinsic negative differential phase behaviors). Invalid data associated with ground clutter, second-trip echo, nonmeteorological, and mixed phase hydrometeors (e.g., melting layer) should be identified so as to adjust (zero out) weighting factors or for possible removal. These problematic gates are identified using polarimetric threshold criteria (often, highly radar specific) for the cross-correlation coefficient *ρ*_{HV} or the standard deviation of Ψ_{DP}. Total Ψ_{DP} profiles in this LP formulation are also expected unfolded and having nonnegative values.

Profiles of Ψ_{DP} associated with substantial NBF are challenging for any processing method or eventual *K*_{DP} estimation that screens Ψ_{DP} according to *ρ*_{HV} thresholds. While not common, the Ψ_{DP} profiles in some extreme instances may become too contaminated for accurate differential phase processing. Since *Z* fields are less susceptible to NBF (e.g., Ryzhkov 2007), hydrological products in these regions may only be possible using *Z*-based or attenuation-based methods.

### a. Minimizing the L1 norm

*n*. To minimize the

*L*

_{1}norm, we have to minimizeThe standard approach to handle the absolute value (Kiountouzis 1973; Portnoy and Koenker 1997) is to allow either sign possibility for

*x*−

_{i}*b*by doubling the number of inequality constraints to two for each variable

_{i}*x*:The inequalities in row

_{i}*i*cover the two possibilities: (i)

*L*

_{1}is equivalent to the minimization of

*n*-term cost function

*L*

_{1}norm, Eq. (1), can be achieved using LP by doubling the number of inequality constraints and minimizing the cost function:

**x**=

**b**.

*n*× 2

*n*matrix

_{n}is the

*n*×

*n*identity matrix, and the length 2

*n*vectors

**x**and

**b**are

### b. Monotonicity constraint

*K*

_{DP}determination requires abstracting reliable derivative estimates from noisy radar data, and in this first presentation of the LP–

*K*

_{DP}approach, we limit filter testing to the members from the SG class. The selection of filter length will be discussed later in this section. Monotonicity is enforced through the requirement that the derivative, so defined, be everywhere nonnegative. To avoid edges, the filtering is initiated at the radial position corresponding to

*x*

_{m}_{+1}and ended at

*x*

_{n}_{-m}, where 2

*m*+1 is the filter length. Thus, monotonicity according to the five-point filter adds a total of

*n*−4 derivative constraints to the 2

*n L*

_{1}norm minimization constraints already present. The new constraints areThe resulting 3

*n*− 4 × 2

*n*row augmentation of

_{n−4,n}is an (

*n*− 4) ×

*n*matrix of zeros and

_{n−4,n}is the like-size matrix of monotonicity constraints. Note from their position in Eq. (5) that inequalities in Eq. (4) operate only on the

**x**part of

**x**is again of length

*n*,

**0**is a vector of zeros of length

*n*− 4, andHere, 0

_{j}represents a zero in column

*j*. Vector

**b**is also augmented by the addition of 0 to its column, that is, one added zero for each entry on the right-hand side of inequalities in Eq. (4). The totality of constraints is then

**b**

_{AUG}is the augmented version of

**b**.

### c. Setting the primal and dual LP problems and their solution

*P*) and its dual (

*D*):The centered dot (⋅) is the dot product. See Bazaraa et al. (2010, chapter 6) for a clear description of the relationship between primal and dual problems in LP. For the primal

*P*, we require

**x**(differential phase) to be nonnegative, as indicated above. The length 2

*n*vector of cost coefficients,

**z**, as described above. A useful feature of LP is that

**c**is easily modified to handle a nonuniform distribution of weights. Figure 1 shows an arbitrary radial of 5-cm CSAPR data with gaps (blue trace). Here, the cost function was modified by setting unit weights to zero in regions of “missing” data so as to give zero weight to those regions. Modified cost coefficients 1 −

*n*are indicated by the black dots in Fig. 1. The red trace in the figure marks the optimal solution for

**x**from problem

*P*. One immediately notices a significant level of noise reduction compared to the original data (blue trace). The modified cost function allows the optimal LP solution to match the original data across gaps without penalty. Another important property of the optimal solution is that it is “global,” in the sense that it is computed at once for the entire radial trace. This global property makes the LP optimized solution less pervious to local variations in the data, like the substantial peak just beyond the range bin of 600 most likely associated with δ.

For the problems we have examined, both *P* and *D* have optimal solutions that can be generated simultaneously using the simplex method (see below). The fundamental theorem of linear programming then states these must have equal objectives (i.e., *P* maximizes the objective for problem *D* (Bazaraa et al. 2010, chapter 6). This minimax property provides a quick check on whether an optimal solution has been found. The array elements of **w** also provide cost sensitivities to changes in the constraints of *P*. These and other useful properties of the dual form can be explored in future studies.

### d. Derivative and smoothing filters

*L*= 2

*m*+ 1 SG derivative filters applied in this study are antisymmetric filters of the formProperties of these filters that we will require later are (Madden 1978)In applying LP to the analysis of radar data, we require of the optimized solution that it satisfy the nonnegativity constraintsfor all

*i*. Even with these constraints in place, however, the processed signal can still oscillate (see, e.g., the red traces in Figs. 1 and 2). To understand what is happening, consider the simple three-point filter

**d**

_{3}= {−0.5, 0.0, 0.5}. For this case the inequalities in Eq. (9) require only that the odd-numbered element and even-numbered element subsequences of

**x**be separately monotonic. There is no requirement that neighboring differences,

*x*(

*i*+ 1) −

*x*(

*i*), be nonnegative. In other words, the processed signal can still oscillate so long as

*x*(

*i*+ 2) −

*x*(

*i*) ≥ 0 for

*i*when it is even or odd. Figure 2 shows similar oscillations resulting from the five-point filter LP solution more clearly in an expanded version of Fig. 1 (red trace). Indeed, the LP solution to

*L*

_{1}norm minimization makes use of such oscillations as it tries to match noise present in the original data so as to minimize the cost function. We will refer to an LP-optimized signal that satisfies constraints

*D*(

*i*) ≥ 0 for all

*i*as being “weakly monotonic.” Although its filter derivative is strictly nonnegative, such a weakly monotonic trace can still exhibit oscillations on scales shorter than the filter length.

For any filter, filter width should be roughly twice the full width at half height of the smallest feature one is attempting to resolve (Enke and Nieman 1976). For the typical operational and ARM weather radar gate spacing of ~100 m, this implies that a five-point filter is attempting to resolve features larger than 0.5 km. Longer derivative filters in the LP allow more leeway for oscillation; instead of smoothing the optimized solution, more oscillations manifest at subfilter-length scales with an increase in filter length. Figure 3 shows results from using the 25-point derivative filter from Madden [1978, Table I, Eq. (III)] for *m* = 12. Notice in the gradually increasing region of differential phase, oscillations in the optimized trace (red) track noise fluctuations in the measurements (blue) almost perfectly. In this region, the derivative constraints tend to be inactive (the derivative is positive anyway), allowing *L*_{1} minimization to take over. Elements of the dual solution vector **w** tend to be zero in this region as well—illustrating a well-known property of the primal–dual relationship known in LP as the complementary slackness condition.

**d**

_{3}defined as above, the smoothing filter is

**s**

_{3}= {0.25, 0.5, 0.25}. We now obtain these smoothing filters for the general case:Consider the action of

**s**

_{n}on two adjacent subsequences from the LP-processed sequence

**x**:We obtain the smoothing coefficients {

*s*} from the derivative filter coefficients {

_{i}*d*} using the conditionwhere the final inequality from Eq. (12) is the starting point and follows from the inequality listed in Eq. (9), and is a simple weighting of the derivatives evaluated at the endpoints of the interval under consideration. Equating coefficients for each of the

_{i}**x**(

*i*) and back substitution givesThe second equality of

*s*

_{0}makes use of the first property of Eq. (8). Finally, it is not difficult to show from Eq. (13) and the second property of Eq. (8) that the sum of the

*n*smoothing filter coefficients is unity:The smoothing filters developed here are very similar to the standard SG smoothing filters (Madden 1978). They have the same normalization and same shape, which is parabolic for the second-order polynomial cases we show, but they are somewhat broader and shortened in vertical extent. Differences are to be expected, as the standard SG smoothers are defined to operate at a single discrete point, located at the center of the filter, whereas ours [cf. Eq. (12)] involves differences and averaging over an interval of two neighboring points.

For the five-point derivative filter used in this study, *d*_{2} = 0.2, *d*_{1} = 0.1, *d*_{0} = 0, *d*_{−1} = −0.1, and *d*_{−2} = −0.2. Substitution into Eq. (13) gives the corresponding smoothing filter corresponding to **d**_{5} as **s**_{5} = {0.1, 0.25, 0.3, 0.25, 0.1}. The green trace of Fig. 2 results from applying this smoothing filter to the red trace (LP solution) and shows complete removal of the oscillations. We will refer to the smoothed trace as exhibiting a “strong monotonicity” condition, whereby monotonicity is preserved at all scales down to the smallest scale of the data spacing.

Figure 2 also shows the result of applying the derivative filter **d**_{5} to the smoothed green trace, thereby smoothing the derivative by removing the influence of noise-matching oscillations in the LP derivative in the same way that the smoothing filter removes these oscillations from *x*(*i*). Smoothing, as a postprocessing step applied to the LP solution, allows the benefits of using longer derivative and smoothing filters to be realized. Figure 3 shows similar results for the 25-point filter. Again, Eqs. (13) have been used to develop a smoothing filter, **s**_{25}, that when applied to the weakly monotonic red trace yields the strongly monotonic green one. Application of **d**_{25} to the green trace yields the correspondingly smoothed derivative field. Note that these derivatives are proportional to (since the range bins are arbitrary length) *K*_{DP} in degrees per kilometer.

### e. Self-consistency constraints

One adjustment to improve performance of the optimal LP solution is to incorporate additional physical (or radar self-consistency) constraints. We introduce one modification here—that is, to modify the right-hand-side constraints in *P*: **b** ≥ 0. These constraints ensure monotonicity, but it follows that modified values may interject additional insights from reliable polarimetric radar fields to reinforce whether derivatives (*K*_{DP}) receive additional nonzero behaviors (**b** ≥ a positive nonzero value). We suggest a “steering” of *K*_{DP} in a self-consistent manner associated with locations of higher *Z*, as in a form *b _{i}* ≥

*K*

_{DP}(

*Z*) =

_{i}*aZ*. Here, the power-law coefficients can be modified to provide only weak constraints, or to better match theoretical expectations. Steering with additional weighting factors was found to quickly improve

^{b}*K*

_{DP}field behaviors to more closely resemble only those cell patterns found in

*Z*or to ignore low

*Z*regions also having low cross-correlation coefficient

*ρ*

_{HV}(e.g., redundancy checks for insects, ground clutter echo). At shorter wavelengths and as a consequence of attenuation in rain,

*Z*fields may also require additional processing to include a baseline linear correction for attenuation in rain that capitalizes on a first-pass heavily smoothed differential phase profile (or iterative processing). Note that in situations having possible hail/mixed phase contamination, care must be taken to mitigate the associated influence on

*Z*measurements (e.g.,

*Z*threshold/cap). Here, we suggest the influence weighting of

*Z*may be capped at 50–53 dB

*Z*for the possible presence of hail.

### f. Solutions for the LP problem and computational considerations

Numerical solutions to the LP problem are obtained using the simplex method of Dantzig (1963). These methods are available in many popular software packages and within the open/free source community. Recent algorithmic approaches to LP, including interior point methods (e.g., Bazaraa et al. 2010, chapter 8; Portnoy and Koenker 1997), known to greatly improve computational speed for certain types of problems, were not tested for their computational speed at the time of this writing. It is also noted that computational speed for numerical solutions to LP problems, including the simplex method, is contingent on the number of points in the data arrays. Computational speed behavior was found superlinear for this study. An ability to perform data array reduction (or applications on shorter-wavelength systems that typically imply fewer range gates) can greatly reduce computational time. Parallelization is an option for the processing of LP methods as described, since all operations are performed along single radial arrays.

## 3. Results

In this section, we provide CSAPR and XSAPR radar observations following LP–*K*_{DP} estimation (LP − *K*_{DP}) as compared to observed polarimetric fields and *K*_{DP} processing following NSSL-type smoothing methodology (NL − *K*_{DP}, e.g., Ryzhkov et al. 2005b). The ARM radars provide an interesting challenge since X- and C-band datasets exhibit strong convective storm cores with potential δ behaviors not observed at operational NWS WSR-88D wavelengths. Since NSSL methods were not originally intended for application at shorter ARM wavelengths, the authors have attempted to faithfully adjust these methods and associated thresholds for the ARM systems. The particular ARM radar systems for use in this study are located near Lamont, Oklahoma. ARM radars feature radar gate spacing of approximately 100 m and a beamwidth of 1°. Although advanced clutter suppression is planned for the ARM radar systems (e.g., Bharadwaj et al. 2010), existing ARM datasets require additional processing to identify echoes contaminated with insect and ground clutter from hydrometeor-linked differential phase measurements. This is accomplished using simple cutoffs to avoid regions having a lower cross-correlation coefficient and higher along-radial standard deviations of differential phase (e.g., Wang and Chandrasekar 2009). For the ARM X-band radar systems at the time of this study, the data quality was found at the lower end of research-quality standards as compared to previous differential phase measurement expectations at X band (e.g., Park et al. 2005; Matrosov et al. 2006).

The results of a simulated convective storm radar dataset at S band (10 cm) are also included to further evaluate the performance of the LP–*K*_{DP} and NL–*K*_{DP} methods. Here, simulated radar fields provide an opportunity to test the LP formulation under ideal but NWS operationally relevant conditions. At these wavelengths, we will assume Ψ_{DP} is associated only with Φ_{DP} and fluctuation noise, having δ = 0. The simulator does not account for profile behaviors associated with NBF, as has been previously discussed at S band by Ryzhkov and Zrnić (1998). The radar simulator is based on Advanced Regional Prediction System (ARPS) model outputs (Xue et al. 2000; Jung et al. 2008a,b) having a conventional radar emulation for a thunderstorm simulation as viewed at radar grazing angles and previously outlined in Cheong et al. (2008). The simulation capitalizes on Kessler-type warm rain microphysics (exponential DSD assumption) for a 20 May 1997 Del City, Oklahoma, sounding environment [as from Ray et al. (1981)] with the radar simulator expanded to include realistic polarimetric fields at S band following Lei et al. (2009).

### a. Performance with simulated storm outputs

Figure 4 includes simulated plan position indicator (PPI) images of the model output *Z* field, as well as a *K*_{DP} from the intrinsic radar model output, the NSSL-based and LP-based processing methodologies. Because the radar model outputs a *K*_{DP} field that may be considered a “truth” field, we have separately plotted the scatterplot of the LP–*K*_{DP} and NL–*K*_{DP} performances as compared to model truth *K*_{DP} values (Fig. 5). Note that from these S-band model examples, *K*_{DP} values at maximum are allowed to approach 7° km^{−1} having *Z* exceeding 60 dB*Z*. From PPI imagery in Fig. 4, all *K*_{DP} fields typically demonstrate desirable (nonnegative) results that match storm hook echolike behavior found in the *Z* field. This result is not surprising, since NBF is not allowed in our model and δ is negligible. Thus, fluctuation noise in Ψ_{DP} tends to be readily removed with smoothing for simulation fields and in regions of high *Z*. Negative *K*_{DP} data regions after processing are limited and located only at the periphery of the convective cell in low *Z* regions.

The suggested LP-based method enforces nonnegative *K*_{DP}. For such nonnegative *K*_{DP} fields, there was a concern of possible positive bias that might have been inadvertently interjected to ensure this behavior. Scatterplots in Fig. 5 reflect that there is no significant mean bias found between the processed *K*_{DP} fields. For regions of *Z* > 40 dB*Z*, the mean bias between the model and LP–*K*_{DP} field is found to be 0.1° km^{−1} (overestimate). NSSL methods for *Z* > 40 dB*Z* reflect a nine-gate smoothing and K_{DP} calculation procedure (assumed slightly more smeared than LP efforts), exhibits a 0.18° km^{−1} mean bias (overestimate). For typical values of *K*_{DP} for that interval of *Z*, this relative error is to within 10%. The most visible discrepancy between the methods occurs at *Z* < 40 dB*Z*, where it should be stated that the NSSL *K*_{DP} is processed using a longer smoothing window (25 gates) and *K*_{DP} is generally assumed less reliable for operational radar applications at S band. Note that the standard deviations of the LP–*K*_{DP} and NL–*K*_{DP} results are improved as compared to the *K*_{DP} field from the simulator, but especially in the case of the LP method. These results support an initial goal of this study to suggest LP methods as suitable for radar phase processing without imposing a bias.

### b. Performance on observed ARM radar datasets

*R*(

*Z*),

*R*(NL-

*K*

_{DP}),

*R*(LP-

*K*

_{DP}) and scatterplots of

*K*

_{DP}versus

*Z*for events viewed by the CSAPR (Fig. 6) and XSAPR (Fig. 7) radar systems, respectively. The

*R*operation denotes that

*Z*and

*K*

_{DP}values have been converted to rainfall rate fields (mm h

^{−1}) for ease in cross comparisons between the panel images. Rainfall relation operators follow standard rainfall and Oklahoma-specific relations found in the literature (e.g., Bringi and Chandrasekar 2001; Gu et al. 2011). For the CSAPR, we use a matched Oklahoma rainfall rate

*R*(

*K*

_{DP}) expression:and for the XSAPR,Note, the NSSL-based

*K*

_{DP}estimates may be negative as a consequence of fixed radial smoothing windows and these situations are handled by associating the negative

*K*

_{DP}values with negative rainfall rates comparable to positive counterparts (e.g., Ryzhkov et al. 2005a,b). For these plots,

*Z*has been corrected for attenuation in rain following the methods of Gu et al. (2011) and using the LP-processed differential phase profiles.

Figure 6 highlights an impressive squall-line event that developed over north-central Oklahoma on 20 May 2011, with values of accumulated Ψ_{DP} exceeding 300° along several radials prior to complete extinction (Fig. 8). Figure 7 illustrates an isolated severe convective event that took place on 23 May 2011 and includes several storms meeting NWS severe thunderstorm criteria for significant hail (>2 cm) and tornadic activity. Associated differential phase and *Z* radials for Fig. 7 are located in Fig. 9.

As in the previous simulated radar dataset convective example at S band, there is an apparent significant benefit for the CSAPR and XSAPR datasets when switching to the LP–*K*_{DP} formulation as compared to NL–K_{DP}. It should be noted that both *K*_{DP}-based fields from the CSAPR event in Fig. 6 offer significant value for rainfall estimation with noticeably attenuated regions for *Z* behind the squall line. Note that scatterplot behaviors in Figs. 6 and 7 still imply *Z* attenuation corrections did not always completely recover these fields.

The performance of *R*(LP−*K*_{DP}) is significantly cleaner for the XSAPR examples in Fig. 7 and in associated Φ_{DP} profile retrievals as in Fig. 9, where the greatest benefit to these methods is argued. The 23 May 2011 event was a significant convective storm outbreak example with hail contamination (suggested also by radials in Fig. 9 featuring cores exceeding 55–60 dB*Z*). In addition to larger δ contributions near strong cores, extended nonmonotonic Ψ_{DP} behaviors are more frequently observed at X band, as NBF is proportional to the gradients of differential phase (e.g., Ryzhkov 2007). We suggest that the oscillations of Ψ_{DP} associated with fluctuation noise, NBF, and δ in the vicinity of these convective cells are not as well handled by the fixed NSSL smoothing procedures that also attempt to rely on a *ρ*_{HV} threshold to isolate and/or interpolate over exceptionally problematic/hail regions. This is a particularly challenging case, since the low *ρ*_{HV} ~ 0.9 threshold is not sufficient to properly identify NBF- and/or δ-prone regions. From Fig. 7, NL–*K*_{DP} and associated *R*(NL–*K*_{DP}) fields are shown in reasonable agreement for high *Z* > 40 dB*Z* regions having stronger polarimetric phase measurement contrasts. However, these regions exhibit extended negative *K*_{DP} regions (decreasing estimated Φ_{DP} along radials for several kilometers) in regions having *Z* > 40 dB*Z* that are apparently better handled by LP–K_{DP} method examples. Profile slopes from the offered LP formulation through stronger storm core regions (Fig. 9) are consistent and typically do not exceed 7°–10° km^{−1}. These slopes correspond to rainfall rates on the order of 70–100 mm h^{−1} at XSAPR using the expressions such as in Eq. (16).

## 4. Discussion and conclusions

Polarimetric radar differential phase and its range derivative *K*_{DP} are of interest to several hydrological applications, including rainfall retrievals and drop size distribution insight. Reducing the uncertainty and unphysical behaviors often found for these fields can have a significant impact on the quality of base polarimetric products of subsequent importance for the forcing of models and/or model evaluation. This study offers a unique approach for differential phase processing that capitalizes on LP ideas and conventional methods for solving LP problems. The apparent success of this method is encouraging for subsequent radar applications and improvements of other physical retrieval problems (radar or otherwise).

Within this study, the LP approach was simple yet designed to replicate many desirable qualities of Φ_{DP} and *K*_{DP} behaviors sought by other currently available radar processing methods. As compared to standard published methods, the strength of these LP methods is an ability to enforce monotonic behaviors if desired. If the original data array behaves according to the constraint conditions, then there is no requirement that the data be altered (smoothed). Additional constraints, including polarimetric self-consistency, are apparently not difficult to explore in future iterations of these methods. Our simple “bounding” of this phase problem using (typically) more robust *Z* field constraints improves the visual appearance of the *K*_{DP} fields. As with more complex and/or published nonlinear variational methods, there is ultimately a clear trade-off between method flexibility when applying realistic constraints and the additional computational expense for LP methods that scales with the number of data and constraint equations. Most computational costs were not found prohibitive for the DOE ARM computing framework, although ARM's emphasis is on algorithm reliability and product quality.

In this LP approach, the monotonicity constraint is always maintained. Specifically, the optimal solution includes a small five-point derivative filter length over which *K*_{DP} is nonnegative. This implies that the solution Φ_{DP} values at subfilter lengths may still exhibit nonmonotonic fluctuation. Yet, these fluctuations can be handled by applying the optimal, offsetting smoothing filter as in section 2 (Figs. 1–3). The nonnegative property of our offered LP approach is not necessarily a behavior that is strongly enforced within other processing schemes. Recent exceptions include a nonlinear variational method wherein this behavior is guaranteed by applying a cost function to be minimized that includes a smoothing (low-pass filter) term with a tunable control parameter (e.g., Maesaka et al. 2012). For many fixed-length smoothing approaches, as is the case with the 9- and 25-gate options utilized in the NSSL method, these may be coupled to provide monotonic behaviors under certain conditions and at the longer wavelengths for which they were originally developed. Yet, Fig. 9 indicates that the current NSSL 25-gate “heavily” smoothed option (approximately 2–3-km window at ARM radars) is not sufficient to guarantee this behavior and that a longer filter (or iterative adjustment) would be required to consistently guarantee this behavior. In this way, even basic LP methods described in this manuscript suggest some benefit, since these methods maintain a small filter length and avoid improper smearing of differential phase data with range to achieve the monotonic behavior.

The Kalman filter approach recently proposed by Schneebeli and Berne (2012) appears quite elegant in design. The presented LP solutions for differential phase processing and improved phase-precipitation “mapping,” however, are not attempting a unified moment or simultaneous holistic polarimetric radar measurement improvement as in that approach. The current LP method descriptions in this study only consider *Z* moment behavior (which is corrected for attenuation in rain using linear methods; e.g., Bringi et al. 1990) as weak steering for nonzero *K*_{DP} to range gates of enhanced *Z*. Nevertheless, a primary concern for a more holistic approach in the Oklahoma ARM environment is the severity of the convective storm regimes (likelihood of hail, NBF) and the associated quality of polarimetric *Z*_{DR} measurements at those shorter ARM wavelengths. This tends to argue for capitalizing on “relative” (nonabsolute calibration) behaviors as a means to incorporate *Z* and *Z*_{DR} into LP scheme constraints [e.g., adapting constraint ideas as from Otto and Russchenberg (2011)]. However, these are topics of future effort and beyond the scope of the present demonstration of LP methods.

This paper has been authored by employees of Brookhaven Science Associates, LLC, under Contract DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the U.S. government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. government purposes. Doctor Giangrande's work is supported by the Climate Science for a Sustainable Energy Future project of the Earth System Modeling (ESM) program in the DOE Office of Science. Doctor McGraw's support is from the FASTER project (http://www.bnl.gov/faster/) supported by the DOE ESM program. The authors thank Drs. Alexander Ryzhkov (OU-NSSL), Michele Galletti (BNL), and Guifu Zhang (OU) for critical discussions throughout the writing process as well as the comments of several anonymous reviewers. Additional ARM radar dataset support and operational implementation were assisted by Dr. Scott Collis (ANL). A version of the LP-based method described by this manuscript and all associated datasets are freely available online (http://www.arm.gov) and as part of the open-source Python ARM Radar Toolkit.

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