## 1. Motivation

Prediction of rainfall systems without positional and intensity errors has numerous applications in various fields in hydrometeorological risk management, including flood forecasting. In an attempt to achieve this goal, the Stormy Weather Group at McGill routinely generates short-term forecasts called nowcasts. The pioneering work by Germann and Zawadzki (2002) led to the development of a system called the McGill Algorithm for Precipitation Nowcasting by Lagrangian Extrapolation (MAPLE) (Turner et al. 2004). The nowcasts produced by this technique are simply the radar-observed rainfall fields extrapolated using a semi-Lagrangian scheme. A variational echo-tracking technique (VET) is used to estimate the motion field and in turn to retain semi-Lagrangian persistence. Thus, the nowcasts produced in this way are not accounting for the temporal changes in the precipitation field.

The changes in the rainfall field could be included in the nowcasts through adding a growth and decay term. The importance of the growth and decay terms in nowcasts is well documented (Bellon and Austin 1978; Tsonis and Austin 1981; Wolfson et al. 1999; Radhakrishna et al. 2012). Many attempts have been made to incorporate a growth and decay term in to the nowcasts but failed (Tsonis and Austin 1981; Wolfson et al. 1999) because of their short predictability (Radhakrishna et al. 2012). Thus, there remains a need to develop an algorithm for obtaining better nowcasts that allow for growth and decay while retaining the semi-Lagrangian persistence.

The problem of properly incorporating growth and decay could be approached through the use of model predictions. However, the accuracy of model-predicted rainfall systems depends on many factors like the microphysical schemes governing the state of the atmosphere, the quality of the data in the data assimilation, and so on. For example, the unevenly spaced observations results in errors in the numerical weather prediction (NWP) models’ output, as they require accurate information of atmospheric fields as initial conditions for better forecasts (Gustavsson 1981). Numerous attempts have been made to study the errors arising from imperfections inherent in models (Thiebaux et al. 1990; Mariano 1990; Brewster 1991). Early in the forecasts, model errors tend to be dominated by displacement errors. Thus, correcting model predictions for positional and intensity errors could solve this problem.

There are many ways to correct the positional and intensity errors in the model outputs and postprocessing technique is one such method. Postprocessing the model output fields can be done using statistical methods (Wilks 1995), nonlinear methods (Casaioli et al. 2003), and neural networks methods (Masters 1993; Bishop 1996). Each one has its own pros and cons.

Atmospheric flow is governed by large-scale waves, which can be superposed on each other, creating constructive or destructive interference, thus providing a possible mechanism for the growth and decay of precipitation. They can also travel at different velocities, as rain cells often do with respect to larger-scale rainfall. To test the validity of this model and in turn to postprocess the model predictions for correcting the positional and intensity errors, an algorithm is devised based on the interference of the waves. The phase information of the waves is obtained by decomposing the rainfall field into Fourier space. Adjusting the powers and phases of the model-predicted rainfall field to the radar-observed rainfall field powers and phases and inverting back to physical space would reproduce the radar-observed field. Since the power spectra of precipitation field is almost invariant on the nowcasting time scale (Zawadzki 1973), if one is able to predict the phases at different scales, then that phase information can be used to correct the positional and to some extent the intensity errors of the predicted precipitating systems (i.e., postprocessing the model outputs). Thus, the focus of the present study is to analyze the predictability of phase at different scales.

A comprehensive study is made on the scale dependence analysis of positional errors using the continental-scale Weather Surveillance Radar-1988 Doppler (WSR-88D) network [National Operational Weather Radar (NOWrad)] data and Weather Forecast and Research (WRF) model data. From this study, we identify the scales that are important for assessing the positional errors of the system but that are not properly predicted by the model. The information on the predictability of these scales can be used to correct for positional errors of the forecasts generated by the model as described in sections 3 and 4. An attempt has also been made to adjust the positional errors of the precipitating systems predicted by the model using the phase information from the observed radar composites in Fourier space. The predictability of phase at different scales in Fourier space is also studied in order to establish the lead time for which the phase correction algorithm can be applied to postprocess the model rainfall predictions. The positional and intensity errors of the model are represented in terms of phase and power errors and the characteristics of these errors as a function of scale are studied. The diurnal variation of phase and power errors of the model is also investigated using 240 h of 15-min time resolution rainfall data.

The present paper is organized as follows. Section 2 describes the data used in this study. Phase correction methodology is illustrated in section 3. Scales responsible for the structure of the rain envelope are explained using Fourier space in section 4. Important scales that are responsible for the positional errors but that are not well predicted by the model are presented in section 5. The WRF model errors are characterized in terms of phase and power and are shown in section 6. The predictability of phase at different scales as a function of time (i.e., the lifetime of phases at different scales) is discussed in section 7. The phase-extrapolation method is also explained in section 7. The results are summarized in the last section.

## 2. Data

The data used in the present study are the rainfall maps derived from U.S. composite level II radar reflectivity mosaics generated by Weather Decision Technologies (WDT) and the rainfall prediction from WRF model, version 3.2. Mesoscale systems observed on 4, 5, 15, 16, 19, 20, 25, 26, and 27 April and 2 May 2011 over North America are considered for this analysis.

The WSR-88D network volumetric data is remapped by WDT, in partnership with the National Severe Storms Laboratory (NSSL), using cutting-edge techniques to quality control and merge data at the lowest possible elevation from multiple radars with high temporal (5 min) and spatial (1 km × 1 km) resolution (Zhang et al. 2005). The spatial coverage of the radar network extends from 20° to 52°N and from 130° to 60°W. The present study utilizes data from WDT radar composites at the lowest elevations, with a minimum reflectivity of 15 dB*Z* (to ignore echoes from nonmeteorological targets). The WDT radar composite data are remapped to a spatial resolution of 3 km (east–west and north–south). Following the method 1 described in Fabry et al. (1994), the 5-min time resolution radar reflectivity data is downscaled to a 15-min time resolution in order to compare with the precipitation fields obtained with WRF model.

The WRF model is a next-generation mesoscale NWP system designed to serve both operation forecasting and atmospheric research needs. Figure 1 shows the domain considered in the present study. Because of the poor radar data coverage over the Rocky Mountain region, we considered the radar and model data between 20° and 45°N and 110° and 75°W. For model runs, we used a single domain with a 3-km spatial resolution, 15-min temporal resolution, and 52 vertical levels. WRF model runs are started with a cold start using the North American Mesoscale (NAM) initial conditions, which has a spatial resolution of 12 km and a temporal resolution of 3 h. To avoid comparisons over the 6-h spinup time, model runs are started 6 h prior to the starting time of radar data (0000 UTC): that is, 1800 UTC of the previous day and continued for 30 h.

The Thompson et al. (2004) scheme with ice, snow, and graupel processes suitable for high-resolution simulations was used to govern the microphysical processes. In the present simulations, the cumulus parameterization scheme was turned off. Radiation is treated utilizing the Rapid Radiative Transfer Model (RRTM) for longwave radiation and the Goddard shortwave two-stream multiband scheme with ozone from climatology and cloud effects for shortwave radiation. Boundary layer physics is accounted with the help of Mellor–Yamada–Janjic scheme.

In this study, we have used reflectivity (dB*Z*) rather than rainfall rate (mm h^{−1}) fields because reflectivity fields are smoother and follow a Gaussian distribution. WRF model–predicted surface rain accumulations are converted to reflectivity values using the reflectivity–rain rate (*Z*–*R*) relation *Z* = 300*R*^{1.5} following Carbone et al. (2002). Fourier processing is based on a periodicity assumption (Bloomfield 1976) and to get rid of the Gibbs effect we have applied a Hanning window (Blackman and Tukey 1959) to both radar and model fields to dampen them at the edge of the domain.

## 3. Phase correction methodology

*P*(

*x*,

*y*,

*t*) be the precipitation field at any given time

*t*with positions (

*x*,

*y*). Applying a 2D Fourier transform

*F*to this field yieldswhere

*A*(

*k*,

_{x}*k*,

_{y}*t*) and Φ(

*k*,

_{x}*k*,

_{y}*t*) represent the amplitude and phase of the precipitation field at wavenumber (

*k*,

_{x}*k*), respectively, and are given bywhere

_{y}*I*(

*k*,

_{x}*k*,

_{y}*t*) and

*R*(

*k*,

_{x}*k*,

_{y}*t*) are, respectively, the imaginary and real parts of the Fourier elements at wavenumber (

*k*,

_{x}*k*). The application of 2D Fourier transform to radar [

_{y}*P*(

_{R}*x*,

*y*,

*t*)] and model [

*P*(

_{M}*x*,

*y*,

*t*)] precipitation fields yields two sets of Fourier elements; their amplitudes and phases are given by [

*A*(

_{R}*k*,

_{x}*k*,

_{y}*t*), Φ

_{R}(

*k*,

_{x}*k*,

_{y}*t*)] and [

*A*(

_{M}*k*,

_{x}*k*,

_{y}*t*), Φ

_{M}(

*k*,

_{x}*k*,

_{y}*t*)]. To retrieve a model-predicted rainfall field that is closer to reality, Φ

_{M}values are replaced with the Φ

_{R}values in Fourier space, keeping

*A*(amplitude/power spectra) constant, and then transforming the new set of Fourier elements back into physical space using an inverse Fourier transform. This process can be mathematically represented as follows:where

_{M}*P*(

_{RM}*x*,

*y*,

*t*) is the phase-corrected precipitation field of the model output.

*a*—radar [yes], model [yes]), misses (

*b*—radar [yes], model [no]), false alarms (

*c*—radar [no], model [yes]), and correct negatives (

*d*—radar [no], model [no]), the verification scores are estimated using the following equations:where

*w*= (

*a*+

*b*)(

*a*+

*c*)/(

*a*+

*b*+

*c*+

*d*). The state “yes” at a position represents the existence of rainfall field (

*Z*≥ 15 dB

*Z*in our case) at that position and the state “no” stands for the absence of rainfall. Note that for rainfall field, we set a threshold value of 15 dB

*Z*, which corresponds to a rain rate of 0.2 mm h

^{−1}.

The outcome of the phase correction is illustrated in Fig. 2. The 15-min rainfall accumulations observed with the radar network and predicted by the WRF model at one instance of four cases are shown on the top and middle panels of Fig. 2, respectively. The bottom panels of Fig. 2 depict the phase-corrected WRF model at zero lag time. The overlaid contours represent the 15-dB*Z* reflectivity thresholds obtained from radar data before applying the Hanning window. When compared with radar observations, the spatial errors of the model predictions vary from case to case. Model-predicted precipitation fields at 1600 UTC 4 April 2011 (first column of Fig. 2) and at 1600 UTC 15 April 2011 (third column) deviate slightly from radar observations; in the other two cases, model predictions are quite different from radar observations both in position and intensity.

Table 1 presents the verification scores of the WRF model predictions before and after phase correction at zero lag time. The verification scores are computed between radar and model and between radar and phase corrected model rainfall fields using Eqs. (5)–(8). The low verification scores of the model before phase correction and high verification scores after phase correction illustrate that incorporating phases from the radar composites brings model forecasts much closer to the reality. After phase correction, one can clearly see that the model image is not only displaced spatially but has also a modified intensity (Fig. 2). Surprisingly, this process can generate new cells where the model had not predicted any precipitation (in the same manner as the interference of gravity waves could lead to growth and decay of the rainfall). When applying the phase correction to model-predicted fields to improve forecasts, three questions come to mind: 1) What scales determine the envelope structure (in this study it is the 15-dB*Z* contour)? 2) What scales are not properly predicted by the model? and 3) For how long are the phases of these scales are predictable?

Verification scores of rainfall fields predicted by WRF model before and after phase correction.

## 4. Scales responsible for envelope structure

In this section we characterize the various scales in Fourier space; that is, we want to know which scales are responsible for the envelope structure. Two separate but complementary tests are performed to answer this question.

The scales that govern the envelope structure are evaluated through considering random noise within the envelope of the rainfall field. Similar to a reflectivity field (positive field values with nonzero mean), we have generated a nonzero-mean random field with values between 0 and 1. Random fields and structured random fields are shown in Fig. 3. Structured random fields are simply the radar and model fields in which the reflectivity values greater than 15 dB*Z* are replaced with random field values. The power spectra of the radar, model, random, and structured random fields are shown in Fig. 4. The power spectrum of the random field is flat, indicating the absence of any correlation in the field. On the other hand, the power spectrum of the structured random field has slightly more power at scales greater than 100 km and shows some correlation within the field.

*D*(

*k*,

_{x}*k*,

_{y}*t*) and

*λ*(

*k*,

_{x}*k*) respectively denote the phase distance and wavelength corresponding to the wavenumber (

_{y}*k*,

_{x}*k*). In two-dimensional space, waves can travel in any direction and their direction of propagation can be represented by phases ranging from 0 to 2

_{y}*π*but in this study we choose a −

*π*to

*π*range. To represent the magnitude of the phase distance, instead of the actual phase difference (which can be either positive or negative), the absolute phase difference is considered.

The phase distance between the radar and model, random fields, and structured random fields are estimated using Eq. (9) and shown in Fig. 5. The black solid line denotes the *λ*/4 line. When phases are random, the average phase difference at a subset of scales becomes about *π*/2 (after dealiasing for first-order aliasing, the phase differences lie between 0 and *π*) and the estimated spectral distance oscillates around the *λ*/4 line. From Fig. 5, it is apparent that the phase distance of the random field is oscillating around the *λ*/4 line at all scales. On the other hand, the phase distance of radar model and structured random fields oscillates around the *λ*/4 line up to approximately 200 km but deviates from that line after 200 km and possesses a similar structure. In all the three cases, the spectral distance of meso-*β* and meso-*γ* scales oscillates around the *λ*/4 line, indicating that the phase differences between radar and model are random at these scales. By replacing rainfall fields with random fields, the phase distance at meso-*α* scales were not significantly changed. This indicates that meso-*α* scales are mainly responsible for shaping the pattern of the rainfall field while meso-*β* and meso-*γ* scales are responsible for redistributing the rainfall field within the envelope of the pattern.

## 5. Important scales in phase correction

To obtain a forecast close to reality by incorporating radar-observed precipitation pattern phases in model outputs, the scales that are not well predicted by the model but that could cause errors in model predictions should be identified. For this purpose, we have used 240 h of radar rainfall data and the corresponding WRF model forecasts, where the latter are displaced with respect to the radar observations.

Cross correlation, RMS error, and verification scores are used to identify the scales that are responsible for positional and intensity errors and are not properly predicted by the WRF model. Figure 6 represents the cumulative correlation and RMS error values that are computed between the radar observed fields and the phase-corrected WRF model fields (left panels) as well as between the radar and the phase- and amplitude-corrected model fields with intensities greater than or equal to 15 dB*Z* (right panels) at zero lag time. Also shown in Fig. 6 are the respective mean values of correlation and RMS error values of 240 h of data. Before correcting the phases, the correlation between radar and model images is the one depicted in the figure at the largest wavelength (i.e., at 4000 km). From Fig. 6, one can observe that the correlation and RMS error values, for scales greater than around 500 km, on average change little despite the replacement of the model phases (Fig. 6, left) and of the model phases and amplitudes (Fig. 6, right) by their radar counterparts. Below 500 km, the correlation increases with decreasing scale and most of the increase observed between the 500- and 50-km scales. The increase in correlation for scales below 50 km is minor. Similarly, the RMS error starts decreasing below 500 km up to 50 km. Application of the phase correction, the mean correlation, and RMS error between radar and model improve from 0.42 and 22.8 dB*Z* to 0.94 and 9.6 dB*Z*, respectively. Higher correlation and lower RMS error values after phase correction in all cases clearly suggest that the phase correction is successful in morphing radar and model images. Scales larger than 500 km and smaller than 50 km contribute little to the adjustment of correlation and RMS errors between model and radar observations, although Fig. 5 shows appreciable phase distances at the larger scales.

Figure 7 illustrates the variation of cumulative verification scores (POD, FAR, CSI, and ETS) as a function of scale after incorporating the phases (top two panels) as well as both phases and amplitudes (bottom two panels) for 10 case studies at zero lag time. The verification scores are estimated using Eqs. (5)–(8) at different scales. The initial low verification scores (represented at 4000-km scale) of all of the cases with high variance clearly indicate that the model-predicted precipitation fields differ considerably from the observations. The average verification scores of the WRF model predictions are POD = 0.45, FAR = 0.63, CSI = 0.26, and ETS = 0.24. Incorporating Φ_{R} instead of Φ_{M} on a scale-by-scale basis results in a significant increase of the verification scores and after replacing Φ_{M} with Φ_{R} at all scales the verification score values are POD = 0.86, FAR = 0.03, CSI = 0.83, and ETS = 0.83. Higher verification scores after phase correction indicates that the false alarms are eliminated through adjusting the field closer to observations. From Figs. 6 and 7, a predominant change seen in all scores below 500 km implies that the WRF model is not properly predicting the scales smaller than 500 km. Also, one decade of wavelengths, from 50 to 500 km, is playing a crucial role in predicting the continental-scale precipitating systems, which are not properly captured by the model. At larger scales (>500 km), though the phase differs from model to radar, correlation, RMS error, and verification scores are not changing after incorporating Φ_{R} in Φ_{M}, implying that the these scales are contributing very less to the signal.

A visual representation of effect of the phase correction algorithm at zero lag time at different scales in Fourier space is shown in Fig. 8. This figure depicts the rainfall field predicted by WRF model at 0000 UTC 5 April 2011 after replacing the phases from the corresponding radar composites for different subset of scales. For example, the field at *λ* ≥ 100 km represents the phase corrected WRF model field with reflectivity greater than 15 dB*Z* after replacing Φ_{M} of scales greater than or equal to 100 km with Φ_{R}. The intensity values smaller than 15 dB*Z* are replaced with no rain (in our case, 0 dB*Z*). Replacing phases larger than 500 km is not resulting in any changes in the pattern of the precipitation field predicted by the WRF model. The pattern of the continental-scale rainfall field predicted by the model converges close to reality after replacing the radar phases of scales between 500 and 50 km. Scales below 50 km are only responsible for small fluctuations in the field; that is, these scales are mainly responsible for the redistribution of intensities of the field as discussed in section 4.

## 6. Quantification of model error

_{error}and

*P*

_{error}denote cumulative phase and power errors up to a scale of

*λ*, respectively. PCM and PACM indicate phase corrected model and phase- and amplitude-corrected model, respectively. The estimated phase and power contributions (errors) in terms of the usual scores are shown in Figs. 9 and 10.

As shown in section 5, both phases and amplitudes appreciably contribute to morphing below 500 km and most of the contribution is seen between the scales 50 and 500 km. The mean contribution of phase in morphing the radar and model images in terms of correlation, RMS error, and CSI are 0.52, 13.24, and 0.57, whereas amplitude has a contribution of 0.06, 9.61, and 0.17, respectively. Based on a comparison of the power spectra, phases have a greater impact on the morphing process than amplitude. The large contribution of the phase correction to the correlation and CSI suggests that incorporating radar phases in the model fields results in a good pattern match with the radar observations, but possessing errors in intensity values. Comparing the contributions of phase and amplitude in terms of the RMS error, we find that while phase correction eliminates major positional and intensity errors, amplitude correction also plays a considerable role in removing the intensity errors. Correcting amplitudes at meso-*γ* (<20 km) scales shows a steep decrease in the RMS error, confirming that meso-*γ* scales are responsible for the eliminating the intensity errors from the model predictions. This gives support to the statement that scales smaller than 50 km are mainly responsible for the redistribution of intensities.

The variations in the power spectra of radar-observed and model-predicted rainfall fields are illustrated in Fig. 11 at zero lag time. The thin gray lines shown in Fig. 11 are the power spectra corresponding to 15-min accumulations and the continuous thick line is the average power spectrum. The variance of powers at the meso-*α* and meso-*β* scales of the radar is greater than those of the model prediction. The largest bias is observed during the decay process (spectra having less powers at large scales) of the rainfall field, which suggests that models are not able to capture it. On average though, the power spectra of the radar and model seem to be similar at meso-*α* scales and show more differences at the meso-*β* and meso-*γ* scales. The ratio of the radar to model powers illustrated in Fig. 11 is greater than 1 at meso-*γ* scales and smaller than 1 at meso-*β* scales. In other words, the model is underestimating powers at meso-*γ* scales and overestimating them at meso-*β* scales. Thus, correcting for amplitudes in addition to correcting for phases at meso-*γ* scales considerably reduces the errors in precipitation intensity. The steep decrease in RMS error (Fig. 9) observed in the amplitude correction at meso-*γ* scales is partly due to the removal of the artifact caused by thresholding the rainfall field at 15 dB*Z*. While adjusting the phases may lead to constructive interference, replacing the *A _{M}* values with the

*A*values that are larger when compared to

_{R}*A*values at the meso-

_{M}*γ*scales often results in increasing the intensity values. At the boundaries of the 15-dB

*Z*threshold the intensity values less than 15 dB

*Z*that are removed by thresholding before amplitude correction will come into picture after incorporating radar powers, and thus the intensities of the field values at the boundaries exceed 15 dB

*Z*and in turn decrease the RMS error.

As seen from Fig. 9, correcting amplitudes at meso-*β* scales has little effect on the correlation value computed between radar model images corrected for phase and amplitude, but at the same time we can see the decrease in the RMS error obtained by reducing the intensity errors. This is due to the high powers at meso-*β* scales of the model predictions compared with the radar observations. Thus, incorporating radar amplitude information in the model amplitude spectra does not lead to useful improvements in terms of pattern matching but reduces the intensity errors.

The variance seen in all scores of the model presented at the largest (4000 km) scale in Figs. 7 and 8 is mainly due to the diurnal variation of the precipitating systems that is not properly captured by the model. Utilizing the U.S. radar mosaics, Surcel et al. (2010) showed a varying diurnal cycle of precipitation patterns over different regions of the North America. They also showed that the Canadian Global Environmental Multiscale (GEM) model is not capturing that diurnal cycle. The inability of the models to capture the diurnal variation that increases (growth phase) or decreases (decay phase) the rainfall area will clearly lead to forecast errors in position as well as in intensity and subsequently to phase and amplitude errors. The effect of the diurnal variation on model prediction quality in terms of phases and amplitudes in the spectral space is illustrated in Fig. 12. The diurnal variation shown here is different from the diurnal variation shown by Surcel et al. (2010) in the sense that we considered the whole observation domain instead of subgridding the domain.

The diurnal variation of the CSI score as a function of scale at zero lag time shown in the top panels of Fig. 12 is the averaged CSI value of 15-min rainfall data in each hour for 10 days. The CSI scores computed between the radar and phase-corrected model as well as between the radar and phase- and amplitude- corrected model indicate model predictions are better in the early hours during the night than later in the day. Model predictions have maximum CSI values around 0900 UTC and minimum values at 2300 UTC. The effect of diurnal variation on positional and intensity errors is explained with the help of phase and power contributions to the CSI scores in correcting the model errors with zero lag time. The phase and power contributions are estimated using Eqs. (10) and (11) and are shown in the bottom panels of Fig. 12. The phase contribution is smaller during the early hours compared with that from the later hours of the day. The high CSI values and low phase contribution in the early hours suggest that the model predictions have greater positional errors at the later hours than at the earlier hours. On the other hand, the power contribution is smaller between 1200 and 2000 UTC and higher at other hours, indicating that the intensity errors are greater early and late in the day.

## 7. Predictability of phase at different scales

As shown in section 4, applying a phase correction algorithm to model-predicted rainfall fields using the existing radar observations reduces the positional and intensity errors of the model predictions. To apply the phase correction to forecasted fields (i.e., for postprocessing the model output for a certain time period in the near future with existing radar images), Φ_{R} needs to be extrapolated for that particular time period. This requires that phase be predictable at different scales. Knowing the predictability of radar phases at scales that are not well predicted by the model, the extrapolated phase information could then be used to postprocess the model output. Phase predictability at various scales is studied with the help of 15-min rainfall accumulations from radar mosaics. The Phase Stochastic model by Metta et al. (2009) (PhaSt) depends on the extrapolated phase information at different scales; thus, the phase predictability at various scales is also useful to test the PhaSt model.

*G*(

*x*) be a time series of data and

*G*(

_{A}*x*) the autocorrelation function of

*G*(

*x*), then

*G*(

_{A}*x*) is estimated as follows:where

*F** represents the conjugate values of the fast Fourier transform. Fifteen-minute-accumulation radar mosaics data are used to estimate the correlation and are interpolated to 1-s time interval to fit the one-dimensional exponential decay function to obtain decorrelation time.

Figure 13 displays the decorrelation time of phases at various scales in both zonal and meridional directions. The thin gray lines indicate the decorrelation time of phase at each scale during each day and the thick black line indicates its mean. On average, the decorrelation time increases with increasing scale. The observed spikes at a few scales are due to long time persistence of those scales for particular events. From Fig. 13 one can observe that the differences in the lifetime of phase of zonal and meridional waves at individual scales are minor.

Advection of rainfall systems is associated with meso-*α* scales and correcting the phases associated with these scales in the model predictions will correct most of the positional errors. The meso-*α* scales have a lifetime of more than 2 h in both the zonal and meridional directions. Hence, using the phase information from the time of the previous radar data, the positional errors of the model-predicted precipitation systems could be corrected up to a lead time of more than 2 h. This can be achieved if one could extrapolate phases at various scales in time. The measured decorrelation time at small scales is almost constant, which is an artifact that comes from the limited data resolution and the smoothing of small-scale patterns by the 15-min accumulation process. Using the PhaSt model, Metta et al. (2009) have made forecasts up to a lead time of 2 h. This is because large scales are predictable up to a lead time of about 2 h.

*α*scales have a persistency exceeding 2 h, in principle phases could then be extrapolated and used to postprocess the model outputs. Let Φ

_{R}(

*k*,

_{x}*k*,

_{y}*t*) and Φ

_{R}(

*k*,

_{x}*k*,

_{y}*t*− 1) be the phases of 15-min accumulations of radar-observed rainfall fields at time steps

*t*and

*t*− 1, respectively. The phases are extrapolated linearly with a time step of 15-min using the following equation:where Φ

_{RE}(

*x*,

*y*,

*t*+

*n*) is the extrapolated radar phase for the

*n*th time step. Though small scales are unpredictable for longer times, instead of filtering the smaller scale’s phase, we have applied the same extrapolation technique to obtain a new set of Fourier elements. Here,

*n*represents the time step of forecast with a time period of 15 min. Assuming models can predict the evolution of the rainfall systems in time, the time evolution of the phases of the predicted model rainfall images can also be used in a phase extrapolation process that is similar to the extrapolation of radar phase. The phases extrapolated from model predictions are given by the following equation:After extrapolating the phases at all scales, the postprocessed model rainfall field for

*n*th time step is obtained as follows:where

*P*(

_{RM}*x*,

*y*,

*t*+

*n*) is the phase-corrected model rainfall field and

*A*(

_{M}*k*,

_{x}*k*,

_{y}*t*+

*n*) is the amplitude spectra of the

*n*th time-step model-predicted rainfall field. Since we have at our disposal two possible set of amplitudes (the ones from radar data at time

*t*and the ones from the model prediction at time

*t*+

*n*) and two ways of extrapolating phases using Eqs. (13) and (14), we have three other possible ways of generating nowcasts:

The relative performance of each method is shown in Fig. 14. Also shown in the figure are the CSI values of the model predictions. The mean CSI values clearly demonstrate that nowcasts generated using the extrapolated phase exhibit higher CSI values than the model predictions up to a lead time of about 4 h. Though wavelengths from 50 to 500 km are crucial in the phase correction in removing the positional errors, meso-*α* scales ranging from 200 to 500 km are mainly responsible for correcting the positional errors as shown in section 4. Nowcasts generated from radar amplitudes and extrapolated radar phases show high CSI values compared with the extrapolated phase from model as well as radar extrapolated phase with model amplitudes. Comparing to the use of phases and amplitudes of model predictions, radar phases and amplitudes provide better nowcasts. Thus, model predictions are not providing any additional information in adding the growth and decay of rainfall to the nowcasts.

In the PhaSt model, Metta et al. (2009) extrapolated phases by adding a random Gaussian noise to generate ensemble members and finally to produce the forecast fields. The PhaSt model is able to generate forecast fields up to a lead time of 2 h. But the linear extrapolation of phases is giving high CSI values up to a lead time of 4 h. Thus, linear extrapolation of phases should provide better forecast fields in the PhaSt model instead of using random values.

## 8. Summary

To morph the radar and model rainfall maps and in turn to minimize the intensity and positional errors of the rainfall systems predicted by the WRF model, an algorithm is developed utilizing the WDT radar composites in Fourier space. Morphing of radar and model images is done with help of a phase-correction algorithm that utilizes phase information from radar composites and power information from model predictions in Fourier space. The analysis of the performance of the phase-correction algorithm applied on continental-scale precipitating systems shows the following:

- Waves of wavelength greater than 500 km are contributing much less, and the one decade of scales between 50 and 500 km contains the important positional and intensity errors.
- Compared to power, phase correction eliminates most of the positional and intensity errors. Correcting powers at meso-
*γ*scales plays a significant role in eliminating the intensity errors due to the underestimation of powers at these scales by the model. - Though phase correction eliminates most of the errors in the model predictions, there is a need to correct the power spectra also for better morphing the radar and model images.
- The effect of the diurnal variation on the rainfall systems changes the contribution of phase and power in correcting the model predictions. The phase contribution is minimum near the model run starting time (0000 UTC) and starts increasing with advancing time, whereas the power contribution is maximum early and late in the day.
- Phases of meso-
*β*and meso-*γ*scales are random and could be predictable for a short time. The lifetime of meso-*α*scales, which are mainly responsible for positional errors in model predictions, is more than 2 h. - Compared to WRF model predictions, the radar-extrapolated and phase-corrected model rainfall fields show high CSI values up to a lead time of 4 h.
- Linear extrapolation of phases provides forecasts up to a lead time of 4 h; instead of adding random Gaussian noise to generate the forecast fields, linearly extrapolated phases could be used to generate forecast fields in the PhaSt model.
- Model predictions are not providing any useful information in adding growth and decay of rainfall to the nowcasts through a phase correction algorithm.

We compared the extrapolated radar fields from phase correction algorithm with McGill Algorithm for Precipitation Nowcasting by Lagrangian Extrapolation (MAPLE). In terms of verification scores, the phase-correction algorithm gave results close but not exceeding those of MAPLE (the results are not shown here because this is not the main goal of this paper). We also tested various phase extrapolating techniques. Instead of linearly extrapolating radar phases with the help of first derivatives, we also utilized combination of first and second derivatives in the extrapolation process. The rainfall fields extrapolated with these processes are compared with the radar images. The results, not shown here, illustrate that the extrapolated rainfall fields utilizing the first derivatives of radar phases possess higher verification scores when compared to the other extrapolation techniques. Extrapolating phases with high accuracy is the promising work for short-term forecasting and can be carried out in the near future.

Another relevance of this work is the possibility of using the phase and amplitude distance as an alternative method for model evaluation.

The authors express their gratitude to the National Centers for Environmental Prediction for providing NAM analysis to run the WRF model and also Weather Decision Technologies (WDT) for providing the radar composites at lowest possible elevation. This project was undertaken with the financial support of the Government of Canada provided through the Department of the Environment. Special thanks go to all of the McGill group members for their fruitful discussions during the group meetings. We thank the two anonymous reviewers for their valuable suggestions in improving the quality of the work.

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