Supplement of Estimating radar precipitation in cold climates : the role of air temperature within a non-parametric framework

In cold climates, the form of precipitation (snow or rain or a mixture of snow and rain) results in uncertainty in radar precipitation estimation. Estimation often proceeds without distinguishing the state of precipitation which is known to impact the radar reflectivity – precipitation relationship. In the present study, we investigate the use of air temperature within a nonparametric predictive framework to improve radar precipitation estimation for cold climates. Compared to radar reflectivity gauge relationships, this approach uses gauge precipitation and air temperature observations to estimate radar precipitation. A 5 nonparametric predictive model is constructed with radar precipitation rate and air temperature as predictor variables, and gauge precipitation as an observed response using a k-nearest neighbour (k-nn) regression estimator. The relative importance of the two predictors is ascertained using an information theory-based rationale. Four years (2011-2015) of hourly radar precipitation rate from the Norwegian national radar network over the Oslo region, hourly gauged precipitation from 68 gauges, and gridded observational air temperature were used to formulate the predictive model and hence make our investigation possible. Gauged 10 precipitation data were corrected for wind induced catch error before using them as true observed response. The predictive model with air temperature as an added covariate reduces root mean squared error (RMSE) by up to 15 % compared to the model that uses radar precipitation rate as the sole predictor. More than 80 % of gauge locations in the study area showed improvement with the new method. Further, the associated impact of air temperature became insignificant at more than 85 % of gauge locations when the temperature was above 10◦ C, which indicates that the partial dependence of precipitation on air 15 temperature is most important for colder climates alone.


Introduction
Hydrological applications require accurate precipitation estimates at the catchment scale.Use of point precipitation gauges often proves inadequate in representing the spatio-temporal variability in the precipitation field (Beven, 2012;Kirchner, 2009).
Weather radars provide quantitative precipitation estimates over a large area with high spatial and temporal resolution.However, weather radars measure the precipitation rate indirectly, using the energy scattered back by hydrometeors in the volume illuminated by a transmitted electromagnetic beam (Villarini and Krajewski, 2010b).The backscattered energy is measured as reflectivity which is used to estimate precipitation.This measured reflectivity depends on many factors such as size, shape, or snow(S), mm h −1 ).Several studies (Battan, 1973;Marshall and Gunn, 1952;Sekhon and Srivastava, 1970) have investigated and then proposed different parameter sets (coefficients "a" and "b" in the power law equation relating reflectivity to precipitation) for rain and snow.The parameter set proposed by Sekhon and Srivastava (1970) has been used as a standard for snow, just as the work by Marshall and Palmer (1948) has been used widely for rain (Fassnacht et al., 2001;Saltikoff et al., 2015).
The Finish Meteorological Institute operationally uses their own equations for rain (Z = 316R 1.5 ) and snow (Z e = 100S 2 ) (Saltikoff et al., 2015).Here Z e represents the equivalent radar reflectivity factor of snow and it is different from Z because the radar signal processing uses the dielectric constant of liquid (water) instead of dielectric constant of solid (ice) for snow.Zhang et al. (2016) used the equation Z e = 75S 2 for the NEXt Generation Radar network (NEXRAD) in the United States which offers similarities to the equation by the Finish meteorological institute for snow.However, Saltikoff et al. (2000) reported that real time phase dependent adjustment of two different parameter sets does not improve the snowfall estimate significantly.
To account for varying precipitation phase (multiple snow types and mixture of snow and rain), many parameter sets could be required.Moreover, the precipitation phase changes rapidly even within the single winter storm and hence, operationally, switching between different parameter sets can be a challenging task (Koistinen et al., 2004;Saltikoff et al., 2015).
For the use of phase dependent reflectivity-precipitation (Z -R) relationship, the precipitation phase of the radar pixel must be estimated.Earlier, weather radar operations in cold climates switched between summer and winter Z -R relationships according to calendar date.However, this is obviously uncertain.As mentioned in the introduction, air temperature can be used to determine the phase (whether snow or rain) of the precipitation.The Finnish Meteorological Institute uses temperature and humidity observations from synoptic stations to estimate the precipitation phase and uses that information to apply a different parameter set for rain or snow (Koistinen et al., 2004;Saltikoff et al., 2015).Fassnacht et al. (2001) demonstrate the use of surface air temperature to estimate the fraction of snow content in mixed precipitation and use it to adjust the radar estimate for mixed precipitation.It is reported that these adjustments improve the accumulated snow estimates in Ontario, Canada.
Observations from dual polarised weather radars can also be used to classify precipitation phases (Ryzhkov and Zrnic, 1998).
However, many radars use a single polarity and moreover, even from dual polarised radars, data on phase information are not readily available to end users to help refine their estimation algorithms.Operational use of dual polarised radars in hydrometeor classification has progressed significantly; however, the classification for high latitude winter storms is still challenging (Chandrasekar et al., 2013).

Nonparametric Radar rainfall estimates
Parametric (or regression type) and nonparametric approaches have been used to build predictive models for a range of applications.When sufficient data are available, nonparametric approaches are efficient alternatives for specifying an underlying model as compared to parametric approaches.Nearest neighbour and kernel density estimation are amongst the most commonly used nonparametric methods.The simplicity of nonparametric approaches have made them attractive for use in hydrology and other sciences (Mehrotra and Sharma, 2006).A key advantage of nonparametric approaches is that less rigid assumptions about the distribution of the observed data are needed (Silverman, 1986) and hence no major assumptions about the process being Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.modelled are required to construct the complete predictive system (Sharma and Mehrotra, 2014).Due to the availability of sufficient radar precipitation rate observations, nonparametric methods provide an attractive basis for assessing the hypotheses posed here.Ciach et al. (2007) used nonparametric kernel regression to model radar rainfall uncertainty.They described the relation between true rainfall and radar-rainfall as the product of a systematic distortion function along with a random component and presented procedures to identify the two components.The distortion function could account for systematic biases which can be mathematically defined as a conditional expectation function, while the random component accounts for random errors in radar rainfall estimation.Villarini et al. (2008) estimated the conditional expectation function (distortion function) using both nonparametric (similar to Villarini et al. (2008)) and copula-based methods and compared the difference in performance between the two approaches using different quality metrics.It was found that performance of the nonparametric method was comparable with the copula-regression estimate and even outperformed when Nash Sutcliffe Efficiency (NSE) was used as a quality metric.The strength of nonparametric approaches is the ability to adapt to the data locally and the weakness is that the method is sensitive to outliers and to large variability of data at the smallest (sub hourly) time scales.Hasan et al. (2016b) used a kernel based nonparametric method for radar rainfall estimation.In their approach, expected ground rainfall was estimated for a given reflectivity using a kernel-based conditional probability distribution.However, none of the methods above considered an additional covariate as air temperature as proposed in this study.

Methodology
This section describes the methods used to formulate a nonparametric predictive model with incident air temperature and radar precipitation rate as the two predictors for radar precipitation in cold climates.A description of how the incident air temperature is incorporated as a covariate in the nonparametric radar precipitation estimation approach is presented next.

Radar precipitation estimation
The proposed radar precipitation estimation algorithm consists of two steps.The first step quantifies the partial dependence of precipitation on radar precipitation rate and incident air temperature.The second step then uses the identified predictors in a non-parametric setting to estimate the precipitation response.Gauge precipitation is used as a ground reference or true precipitation in this study.
The conditional estimation of precipitation using the two covariates can be described as follows: Here, (R est (t)) is the estimated ground precipitation from a given pair of radar rain rate (R(t)) and incident air temperature (T (t)) values at a given geographical location in the two-dimensional space (x, y) and time, t.
The conditional estimation in Eq. ( 1) uses two covariates, in contrast to Hasan et al. (2016a, b) where a nonparametric kernel regression estimator using a single covariate (R(t)) was adopted.Readers are referred to (Mehrotra and Sharma, 2006;Sharma Hydrol. Earth and Mehrotra, 2014;Sharma et al., 2016) for further details on the nonparametric modelling framework used in this work.
This study uses the k-nearest neighbour (k-nn) regression estimator as the nonparametric predictive model.This model can be expressed as: Where k denotes the number of observed pairs of radar precipitation rate and temperature considered "similar" to the current conditioning vector [R, T ].Similarity here is defined on the basis of a weighted Euclidean distance that is further explained below.E(.) denotes the expectation operator, in the absence of which the uncertainty about the expected value can be computed.
The term g k represents the observed gauge precipitation corresponding to k th neighbour of the conditioning vector.K is a maximum number of neighbours permissible and it is an important parameter in the k-nearest neighbour method.In the present study, K is taken as equal to the square root of the sample size as suggested by Lall and Sharma (1996).
The order of each neighbour is ascertained based on a weighted Euclidean distance metric, written as: Here, ξ i is the distance of the conditioning vector [R, T ] to the i th data point (r i , t i ) in a two-dimensional space.s R and s T are sample standard deviations of the radar precipitation rate and temperature, and β R and β T are partial weights denoting the relative importance each conditioning variable has on the ensuing response respectively (Sharma and Mehrotra, 2014).The sample standard deviations are used to standardise the predictor variables to make them independent of their measurement scale, while the partial weights allow elimination of a predictor variable if not relevant to the prediction being made.Readers are referred to Sharma and Mehrotra (2014) for the informational theory rationale that allows for the estimation of these partial weights, and the NPRED, R package ( (Sharma et al., 2016), downloadable from http://www.hydrology.unsw.edu.au/download/software/npred) that enables their estimation for any sample data set.

Model evaluation criteria
A number of metrics have been used in literature to evaluate and compare the performance of models (Hasan et al., 2016b;Villarini et al., 2008).The root mean square error (RMSE) is commonly used as a performance measure and it provides the overall skill measure of a predictive model (Hasan et al., 2016b).We used primarily RMSE as a quality metric to evaluate the performance of the proposed model.Mean absolute error (MAE) and mean error (ME) were used as additional quality metrics.
Definition of RMSE, MAE and ME can be found in the literature (e.g., Hasan et al., 2016b;Villarini et al., 2008).

Determination of phase
In order to assess the usefulness of the proposed approach, it was compared against an alternate approach where the precipitation phase for first ascertained, followed by the application of different Z-R relationships for snow and rain.For the classification of precipitation phase at gauge level, we adopted the method from Finnish Meteorological Institute which is used operationally in Finland for phase classification (Koistinen et al., 2004;Saltikoff et al., 2015): Here, P lp represents the probability of liquid precipitation, T ( • C) the air temperature, and H (%) the relative humidity at a height of 2 m.If P lp < 0.2, precipitation is considered as solid and if P lp > 0.8, precipitation is considered as liquid.For the case of 0.2 ≤ P lp ≤ 0.8, precipitation is considered as mixed (Koistinen et al., 2004;Saltikoff et al., 2015).
4 Study area and data to detect snowfall and clear air echoes (Koistinen et al., 2004).The wave length of the Hurum radar is 5.319 cm.The Norwegian radar network scans the atmosphere with a 7.5 minute temporal resolution.The met.no processes the raw radar volume scan from the radar stations.The data goes through extensive quality control and data transformations before the radar products are distributed to end users (Elo, 2012).The met.no performs a routine that removes clutter and other noise (non-meteorological echo) from the radar scan first.Then it reconstructs the gap in the data caused by clutter.The processing algorithm segments the volumetric radar reflectivity data as convective or stratiform precipitation type and it computes the Vertical Profile of Reflectivity (VPR) depending on precipitation types.VPRs of convective and stratiform precipitation types are distinctly different (Abdella, 2016;Chumchean et al., 2008).Bright band effect and non-uniform vertical profile of reflectivity are major sources of uncertainties in radar precipitation estimation in high latitude regions (Abdella, 2016;Joss et al., 1990;Koistinen et al., 2004;Koistinen and Pohjola, 2014).The radar data are corrected for bright band effects that appear in the VPR.
After the processing, the met.no generates and distributes various radar products.One of the radar precipitation rate products available for the public to use in hydrological applications is the Surface Rainfall Intensity (SRI).The SRI product uses the lowest Plan Position Indicator (PPI) and projects the aloft reflectivity data down to a reference height (1 km) near to the ground.The projection method is known as VPR correction that takes the vertical variability of reflectivity and bright band effect into account (Elo, 2012).The VPR corrected reflectivity is transformed from polar to Cartesian coordinate system with 1 km × 1 km spatial resolution and the mosaic of nine weather radar imageries is merged to single SRI product covering the entire Norway.Finally, the reflectivity is converted to precipitation rate by using parametric Z -R relationship (Z = 200R 1.6 ) derived by Marshall and Palmer (1948) and the precipitation rate is accumulated to the temporal resolution desired (hourly in this case).It can be noted that the Norwegian meteorological institute uses the single Z -R relationship (Marshall-Palmer for rain) for all seasons throughout the year.
Data for the period from January 2011 to May 2015 were used for this study.A spatial subset of accumulated hourly radar precipitation rate with 1 km × 1 km spatial resolution for the study area was downloaded from the met.no's "thredds" server .More details on the procedure adopted for catch correction are provided in the next sub-section.
The gridded temperature dataset for Norway is spatially interpolated based on the historical air temperature observations from Norwegian meteorological stations.The interpolation is based on Optimal Interpolation in a Bayesian setting (Lussana et al., 2016).In this three-dimensional spatial interpolation, the elevation of each grid point is obtained from a high-resolution digital elevation model and the real elevation of stations stored as metadata used.The resulted interpolated air temperature is on the regular grid which is 2 m above the ground terrain elevation.For further details of the interpolation method, readers are referred to the Norwegian meteorological institute's report by Lussana et al. (2016).This gridded temperature data with an hourly temporal resolution was used to derive temperature time series for the precipitation gauge locations.
The gridded hourly wind speed datasets are derived from a statistical downscaling of a 10 km numerical model dataset onto a 1 km grid (same grid as the hourly gridded air temperature).Gridded wind speed data was used to correct wind induced under-catch of precipitation gauges.Hourly measured relative humidity data is available at 25 gauge locations within the study area.Relative humidity data together with air temperature were used to compute the phase of the precipitation at gauge level in this study.Spatial variation of relative humidity is relatively small within 50 -100 km distances and hence simple interpolation techniques can be used (Beek, 1991).It can be noted that nearest gauge with relative humidity measurement is less than 50 km for most gauges in this study.In this study, for gauge locations with missing relative humidity data, relative humidity data available from nearest gauge were used.
As precipitation gauge locations and radar precipitation rate grids are in the same UTM33 coordinate system, they were simply overlaid and the radar pixel of 1 km 2 overlapping each precipitation gauge was located.One location near Oslo has three precipitation gauges within a 1 km × 1 km pixel.Except for that, all pixels consist of a single gauge.The pixel value (precipitation rate) for each hour was extracted and continuous hourly time series of radar precipitation rates for all gauges were generated.The precipitation intensities in the study area (high latitudes) is relatively low.An analysis of statistical properties of precipitation rates in mid Norway showed that intensities less than 1.76 mm h −1 contributes to 50 % of the total precipitation volume while less than 6 mm h −1 contributes to 88 % (Engeland et al., 2014).Further, the same study found that precipitation intensities below 0.1 mm h −1 contributes little to the total precipitation and might be treated as zero precipitation.Timesteps with gauge precipitation or radar precipitation rate less than 0.1 mm h −1 were therefore removed in this study.Finally, an observed dataset of hourly gauge precipitation and corresponding radar precipitation rate and air temperature for those hourly timesteps were prepared for all precipitation gauge locations.The length of the dataset at each gauge location used in this study is shown with the size of the circles in Fig. 1.

Catch correction for precipitation gauges
Accuracy of precipitation gauge measurement is essential to achieve better results from water balance calculations, hydrological modelling and calibration of remote sensing algorithms.Solid precipitation exhibits significant under-catch in windy conditions.Consideration of catch errors is more important in high latitude Nordic and mountainous regions due to large catch errors for snow.Field study in Norway showed that precipitation gauges, even with wind shield, catch 80 % of true precipitation at wind speeds of 2 m s −1 , 40 % at 5 m s −1 , and only 20 % at 7 m s −1 for solid precipitation at temperatures equal or below et al., 2015).As this study uses gauge observation as a ground observed truth, corrected gauge observation is required for a reliable outcome from the investigation.
We corrected gauge precipitation for wind induced under catch by using the Nordic precipitation correction model (Førland et al., 1996).The Nordic model classifies the precipitation phase using air temperature and uses different equations for solid and liquid precipitation and average value of the two equations was used for mixed precipitation.The correction equations use wind speed and air temperature at each gauge location.As mentioned above, gridded hourly wind speed data was used for aerodynamic correction.It was found that correlation between the corrected precipitation by using measured wind speed data (15-gauge locations) and gridded data are over 0.97 for all those 15 gauge locations.Based on the catch error computations in this study, the mean correction factor of hourly precipitation (corrected precipitation/observed precipitation) is 1.61 for solid and 1.14 for liquid precipitation while median are 1.53 and 1.11 for solid and liquid precipitation respectively.Corrected gauge precipitation was used as true observed precipitation in this study.

Results and Discussion
The performance of nonparametric radar precipitation estimation using air temperature as an additional covariate is presented in this section.The bivariate model is compared with the bench mark of the univariate nonparametric model where radar precipitation is used as the sole predictor.We tested the proposed method for a number of criteria and the results are presented below.

Partial weight of predictors
For each precipitation gauge location, we estimated the partial weights associated with radar precipitation rate and incident air temperature using the observed hourly radar precipitation rate and air temperature and the corresponding gauge precipitation data.
Figure 2 shows the histogram of partial weight of radar precipitation rate (β R ) computed for 68 precipitation gauge locations in the study area of 50 km radius from the Radar station as shown in Fig. 1.It is noted that the summation of partial weights of radar precipitation rate (β R ) and air temperature (β T ) is scaled to 1. Hence, the partial weight associated with air temperature   that partial weight of radar precipitation rate (β R ) is equal to 1 for nearly 13 % of the gauge locations and the partial weight associated with temperature (β T ) is therefore zero.There, the bivariate problem collapsed into a univariate problem with radar precipitation rate as a single predictor.
Table 1 shows the summary statistics of computed partial weights among the precipitation gauge locations in the study area.
It can be seen that the partial weight associated with air temperature is in the range of mean +/ − 0.1 for more than 70 %

Performance of k-nn prediction model
The k-nearest neighbour regression based estimator was used to predict precipitation at each gauge location.The observed dataset and the computed partial weights of predictors were used with the NPRED k-nn regression tool to specify the model.
For comparison, a reference model using the k-nn regression estimator but a single predictor variable (hourly radar precipitation rate) was also developed.
We calculated the k-nn regression estimate of expected response by using the leave-one-out cross-validation (LOOCV) procedure, whereby leaving out one observed response value (gauge precipitation) from the regression and estimating the expected response value for that observed response.This ensures the modelled outcomes represent the results that will be obtained using a new or independent data set.The improvement in radar precipitation estimation with the use of air temperature as an additional covariate is measured as a percentage reduction in RMSE compared to the reference model.
All the gauge locations with an associated partial weight of air temperature (β T > 0) show an improvement in radar precipitation estimation.The mean improvement in RMSE is 9 % while the median is 7.5 % and it is more than 5 % for 80 % of the gauge locations where air temperature was identified as an additional covariate.It can be noted that partial weight for each gauge location was calculated independently using the data from that specific location and then the RMSE was estimated by LOOCV estimated using the entire data at that gauge location.However, a split sample test was done to verify the results, where two-thirds of the data were used to estimate partial weight and one-third of the data were used to estimate RMSE for each gauge location.The split sample test gave similar results as before.
We also examined the spatial cross-validation of computed partial weights.First, a single average partial weight was calculated by taking the arithmetic mean of partial weights of all gauge locations which were computed independently at each gauge location and presented in Fig. 2 and Table 1.This single average value of partial weight (0.68, 0.32) was used with the predictive models to estimate radar precipitation and the improvement in RMSE estimated.Then, for each gauge location, an average partial weight was calculated by leaving that gauge out and adopting the mean partial weight from five nearest gauges.The k-nn prediction model was again re-specified for each gauge location using the computed average partial weight of 5 nearest gauges.The results of percentage improvement in RMSE showed a strong resemblance to the results with a single mean value of partial weight for the study area.It is possible, therefore, that a regional or nearest neighbour average value of partial weight can be used for ungauged locations.As with the partial weight, improvement in RMSE at gauge locations does not clearly show any systematic pattern of spatial variation.
Based on above examinations, spatial variation of station specific partial weight can be discarded and a single average value adopted.Hence, in the results that follow, we use a single average partial weight computed for the study area.As shown in Table 1, the mean value of partial weight for radar precipitation rate is 0.68 and air temperature is 0.32.The k-nn regression prediction model with radar rain rate and air temperature as two predictors at each gauge location was specified with this single average partial weight.
Figure 3 shows the percentage improvement in RMSE for the proposed model with radar precipitation and air temperature as two predictors with the single average partial weight of (0.68, 0.32) compared to the reference model with radar precipitation rate as a single predictor.The precipitation gauges' locations are shown by circles and their sizes are proportional to the length of the data used in the nonparametric predictive models at each gauge location.A filled discrete colour scale is used to show percentage improvement in RMSE.All the gauge locations show improvement in RMSE with the use of temperature as an additional covariate comparing with the reference model of radar precipitation as a single predictor.Looking at Fig. 3, majority of gauge locations have a green colour and the improvement is 5 -10 % on those locations.Mean value of improvement is 8.5 % while the median is 7 %.Over 80 % of the gauge locations in the study area show more than 5 % improvement in RMSE while nearly 15 % show more than 15.0 % improvement.As discussed earlier and as seen in Fig. 3, this study did not find any pattern of spatial variation in the results.However, this spatial plot clearly shows that the improvement in RMSE with the use of temperature as an additional predictor is spread throughout the study area.
In addition to RMSE, we computed MAE and ME for the proposed model and the reference model with radar precipitation as a single predictor at gauge locations.The above quality metrics were also computed for the original data of radar precipitation rates for comparison.
Figure 4 shows the computed quality metrics for the two predictive models (knn-R and knn-RT) and the original data of radar precipitation rates.Looking at Fig. 4, the mean error of the original data (denoted as MP) was negative for almost all gauge locations.This shows the under estimation of radar precipitation compared to precipitation measured by gauges.Both nonparametric predictive models reduce the mean error considerably and bring it to near zero while they reduce the RMSE and 13 Although the main focus of this paper is to investigate the benefit of using temperature as an additional covariate in radar The above results demonstrate the usefulness of air temperature as an additional predictor variable in deriving radar precipitation in cold climates.Some further investigations of when this improvement can be expected to be most are presented next.

Performance for different threshold intensities
This study used the precipitation intensities of radar precipitation and gauge precipitation equal or above 0.1 mmh −1 .As described in Sect.4, precipitation intensities are relatively low in this region, consistent with intensities in cold climates.A data analysis showed that intensities are lower than 0.5 mm h −1 for around 60 % of the observations and only 5 % of the data have either gauge or radar precipitation rates above 2.0 mm h −1 .
To investigate whether very low intensities dominate the results presented earlier, we tested our proposed model for a range of intensities for both gauge and radar precipitation.Figure 5 shows the box plot of RMSE values estimated at gauge locations for threshold intensities 0.1 mm h −1 , 0.5 mm h −1 and 2.0 mm h −1 .Looking at Fig. 5, the improvement with the use of air temperature as an additional covariate is still significant for more severe intensities as well.

Variation with Temperature Classes
For each gauge location, we also estimated partial weights for different temperature classes.Partial informational correlation and hence the partial weight was found to vary with temperature class.For temperature above 10 • C, more than 85 % of the gauge locations were estimated as having zero partial weight for air temperature.It is therefore likely that radar precipitation estimation depends on air temperature for colder climates dominantly.The presence of hail may be the reason for a few precipitation gauge locations still exhibiting non-zero partial weight for air temperature above 10 • C. Further, we estimated RMSE for the dataset above 10 • C for each gauge location using the average partial weight (0.68, 0.32) and estimated the improvement compared to the reference model with radar precipitation rate as a single predictor.Nearly 70 % of gauge locations still showed improvement in RMSE; however, the improvement is insignificant when the air temperature is above 10 • C.

Separate parametric equations for rain and snow as a benchmark
As we discussed in Sect.2, the switch between a snow and rain Z -R relation is fast becoming a standard for weather radar operations in cold climates.We compared the proposed nonparametric radar precipitation estimation models with radar precipitation estimation by using two different parametric Z -R relationships, one for snow and other for rain.In this study, we used the radar snow equation of Finish Meteorological Institute (Z e = 100S 2 ) while keeping the Marshall and Palmer equation  For this investigation, we converted the original radar precipitation rates back to reflectivity using inversion of the Marshall and Palmer equation (R = (Z/200) 1/1.6 ).We estimated also the probability of liquid precipitation (P lp ) using Eq. ( 4) in order to classify and hence apply different Z -R relationships according to the precipitation phase.Hourly air temperature and relative humidity at each gauge location were used in this study for the estimation of probability of liquid precipitation (P lp ).Data were classified as solid or liquid or mixed precipitation using the computed value of probability of liquid precipitation (P lp ).

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The back calculated reflectivity was converted to precipitation rates using snow equation (Z e = 100S 2 ) for solid phase and rain equation (Z = 200R 1.6 ) for liquid phase.A weighted combination of solid and liquid was used for mixed precipitation by using the value of P lp as recommended by Koistinen et al. (2004); Saltikoff et al. (2015).Precipitation rates estimated by the two equations as described above is denoted by FMIMP.
For each gauge location, RMSE was calculated for the estimated radar precipitation rates by two equations (FMIMP).Here 10 wind corrected gauge precipitation was used as a true observed value.RMSE of FMIMP is compared with the RMSE of original radar precipitation rates (MP) and the two nonparametric predictive models (knn-R and knnRT).q q q q q q q q q q q q q q q q q q 0.4  Looking at Fig. 6, the use of two equation (FMIMP) with the snow equation for solid and partially for mixed phase reduces the RMSE for solid and mixed precipitation phase classes and hence the RMSE of entire dataset compared to the original precipitation rates estimated by Marshall and Palmer (MP).The application of a different equation for snow reduces the phase dependent bias in the Norwegian radar precipitation estimation.The average reduction in RMSE at gauge locations is 6 % of RMSE value of the original precipitation rates.However, it can be seen in Fig. 6 that the use of different equations for snow 5 and rain does not reduces the RMSE to the level of the nonparametric approach (knn-RT).Comparing FMMP and knn-RT, there is a further reduction of nearly 10 % in RMSE.
It should be noted that the phase classification used in this study is a model-based classification even though it is used operationally.The estimated phase can differ from actual observed phase at gauge level.Observations from disdrometers can provide a more accurate phase information at gauge level.However, disdrometers are not available everywhere.Even if a few 10 disdrometers were located within the study region, their representativeness in space and time would be limited (Saltikoff et al., 2015).Further, our phase classification is at gauge level, and represents near surface conditions.The phase of the precipitation can be different at the elevation where the radar measures the reflectivity.The measurement of phase information with the use of dual polarized radars can be a useful data source for further investigation.

Figure 1 .
Figure 1.Precipitation gauge locations (blue circles) and length of the observations at each precipitation gauge location (size of the circles) and radar station (purple star mark) overlaid on topography of the study area, Oslo region of Norway.Hypsometric (elevation) distribution of the gauges is on the top left corner

(
http://thredds.met.no/).The data are in netCDF file format and the gridded array is in Universal Transverse Mercator (UTM) 33 projected coordinate system.The hourly precipitation measurements from precipitation gauges are downloaded from the met.no's web portal for accessing meteorological data for Norway, "eKlima" (http://eklima.met.no).Within the study area, 88 precipitation gauges are in operation with hourly observation, however only 68 gauges are available during the period from 2011 to 2015.The precipitation gauges' locations (68 gauges) used in the study are shown in Fig.1overlaid on the topography of the study area.As shown in Fig.1, precipitation gauges are not evenly distributed.The urban areas (Oslo, Drammen, Lillestrom and Tonsberg) are densely gauged (Nearly 0.25 gauges/km 2 near Oslo and approximately 0.1 gauges/km 2 near other major cities) and rest of the area is sparsely gauged with hourly observation.Further, the precipitation data from precipitation gauges come with varying length because some gauges are in operation since 2013 or later and some gauges have a number of missing values during their operation.However, we used data from all available precipitation gauges for this study.Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.Some of the gauging stations are equipped with hourly temperature and other meteorological measurements (including wind speed and relative humidity).For this study, we used gridded hourly temperature and wind speed dataset with 1 km × 1 km grid resolution.The data are available from the Norwegian meteorological institute.The gridded wind speed data is available until May 2015.Even though, radar precipitation rates and air temperature data are available from January 2011 to date, due to the unavailability of wind speed data for catch correction of gauge precipitation, the study period is limited to four years (January 2011 -May 2015)

Figure 2 .
Figure 2. Percentage of precipitation gauge locations against estimated partial weight of radar precipitation rate (βR) at those gauge locations and the mean partial weight (red dash line) for gauge locations (68 gauges) in the study area.Partial weights provide a measure of relative importance of predictor variables on the response (refer Eq. (3)) and the summation of partial weights (βR + βT ) is equal to 1.

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of gauge locations.The gauge locations which resulted in associated partial weight for air temperature (β T > 0) are spread throughout the study area.However, we have not found a clear pattern of spatial variation in the estimated partial weights at gauge locations within the study area.11 Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.

Figure 3 .
Figure 3. Percentage of improvement in RMSE at each gauge locations (colour scale) for predictive model with radar precipitation rate and air temperature as two predictors with the singe average partial weight (βR = 0.68 and βT = 0.32) compared to radar precipitation rate as a single predictor and length of the data (circle size), which are used in the predictive model, overlaid on the coastline of the study area.

Figure 4 .
Figure4.Bar plot representing three quality metrics (RMSE, MAE and ME) at gauge locations for the original data (MP) and for the two nonparametric models (knn-R and knn-RT).Here, knn-R denotes the nonparametric model with radar precipitation rate as a single predictor, while knn-RT denotes the nonparametric model with radar precipitation rate and air temperature as two predictors with fixed partial weight of (0.68, 0.32).

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precipitation estimation, the results of the nonparametric radar precipitation estimation in this study are comparable with the results ofHasan et al. (2016b), although in a different setting.They tested their nonparametric method of radar rainfall estimation (radar reflectivity as a single predictor) in Sydney, Australia and they have reported 10 % improvement in RMSE compared to the traditional parametric Z -R relationship.In our study, k-nearest neighbour nonparametric method with radar 14 Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.precipitation rate as a single predictor resulted in a mean 6 % reduction in RMSE.The bivariate model with air temperature as an additional predictor resulted in a mean 14 % reduction in RMSE compared to the original radar precipitation rate data derived using a parametric equation (Z = 200R 1.6 ).

DatasetFigure 5 .
Figure5.Box plot of RM SE (mm h −1 ) values estimated at gauge locations for the original data (MP) and the two nonparametric models (knn-R and knn-RT) using data with intensities of radar precipitation rate and gauge precipitation greater than or equal 0.1 mm h −1 , 0.5 mm h −1 and 2.0 mm h −1 .Mean value of RMSE for each model by red diamond point.Here, knn-R -nonparametric model with radar precipitation rate as single predictor and knn-RT -nonparametric model with radar precipitation rate and air temperature as two predictors with the partial weight of (0.68, 0.32) Figure 6 shows the box plot comparison of RMSE values in mm h −1 estimated at gauge locations for entire data and phase classes separately.16 Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.

Figure 6 .
Figure 6.Box plot of comparison of RM SE (mm h −1 ) estimated at gauge locations for the original precipitation rates by Marshall and Palmer equation (MP) and precipitation rates estimated by different equation for snow and rain (FMIMP) and for the nonparametric model (knn-RT).RMSE values shown for entire data and separately for solid, mixed and liquid phase classes.Mean value of RMSE for each model by red diamond point.Here knn-RT -nonparametric model with radar precipitation rate and air temperature as two predictors with the partial weight of (0.68, 0.32).

Table 1 .
Summary statistics of computed partial weights for radar precipitation rate and air temperature in the study area.
tool, which is used for computation in the study, is available as R package and it can be downloadable from the following link as follows: http://www.hydrology.unsw.edu.au/download/software/npredCompetinginterests.The authors declare that there are no competing interests.Acknowledgements.The authors gratefully acknowledge Norwegian meteorological institute (met.no) for providing radar rain rate, gauge precipitation and air temperature data for this study.The authors would particularly like to thank Christoffer Artturi Elo at met.no for assisting 10 to get the radar precipitation rate data.A great appreciation goes to Water Research Centre, University of New South Wales (UNSW), Sydney, Australia for hosting the first author for research practicum.The authors acknowledge the Norwegian Research Council and Norconsult for funding this research work under the Industrial Ph.D. scheme (Project No.: 255852/O30).19 Hydrol.Earth Syst.Sci.Discuss., https://doi.org/10.5194/hess-2018-351Manuscript under review for journal Hydrol.Earth Syst.Sci. Discussion started: 28 June 2018 c Author(s) 2018.CC BY 4.0 License.