## 1. Introduction

Marine winds are an important output of modern operational numerical weather prediction (NWP) systems. They provide input to marine forecasts and warning systems, with their accuracy having direct implications for marine safety. The calculation of air–sea fluxes also relies directly on marine winds, in turn influencing estimates of heat and moisture fluxes within atmospheric models. The accuracy of systems that subsequently use these outputs such as wind energy forecast systems, wave models, and ocean models is also critically dependent on the quality of these forcing fields.

Increased computational power, improved modeling techniques, and increased availability of observations have facilitated the rapid development of NWP capabilities in recent years. Deficiencies remain, however, due to factors such as imperfect model physics and uncertainties in initial and boundary conditions. Identifying the nature and distribution of these errors provides valuable input to model developers to ascertain model shortcomings. For the purpose of obtaining the best winds possible, postprocessing of the direct model output can be used to mitigate these deficiencies.

Where systematic errors in the wind field are known a priori, these can be removed with statistical corrections (e.g., Tolman 1998b; Greenslade et al. 2005). However, these simple correction approaches cannot account for spatial variation in the wind error; winds corrected in this way can be expected to retain significant regional biases. In addition, changes in systematic wind biases with alterations and upgrades to the atmospheric model must be continually monitored, and corrections adjusted accordingly.

The Australian Bureau of Meteorology (Bureau) recently implemented the Australian Community Climate Earth System Simulator (ACCESS; NMOC 2010a), based on the Met Office Unified Model/Variational Assimilation (UM/VAR) system (Rawlins et al. 2007), for operational forecasts. Recent work has identified a systematic negative bias in the marine 10-m wind speed (*U*_{10}) from this model (Durrant and Greenslade 2012), in turn producing a negative bias in significant wave height (*H*_{s}) from the Bureau’s operational wave model (Durrant and Greenslade 2011).

In the present study, different means of statistically removing these wind biases are explored (for the remainder of this paper, “wind” will be used to specifically refer to surface winds at the 10-m height). These range in complexity from simple spatially homogeneous corrections, to corrections that vary both in space and time. In an effort to eliminate the need to manually monitor and update the applied corrections, an automatic, self-learning correction method is proposed, applicable to operational forecast winds. This work was performed with the intention of developing an operational system applicable to the Bureau’s forecasting environment. The work was carried out within the context of replacing the Bureau’s operational wave model (Durrant and Greenslade 2011) and is presented from this perspective. The concepts are, however, applicable to any downstream system requiring marine surface winds.

Previous work examining statistical corrections to surface winds is summarized in section 2. The methods for wind corrections are then outlined in section 3. Results are presented in section 4 and conclusions are given in section 5.

## 2. Background

The ability to forecast ocean waves relies to a large extent on the accuracy of NWP systems. Current third-generation wave models such as the Wave Action Model (WAM; WAMDI Group 1988) and WAVEWATCH III (WW3; Tolman 1991, 2009) have been found by many studies to produce highly accurate forecasts several days in advance. The skill of these models is such that the quality of the wave forecast in deep water is to a large extent determined by errors in the forcing wind field (e.g., Cardone et al. 1996; Rogers and Wittmann 2002). A 10% error in the estimate of surface wind speed can lead to a 10%–20% error in *H*_{s} and a 20%–50% error in wave energy (Cavaleri 1994, 283–284).

*European Remote Sensing Satellite-2*(

*ERS-2*) altimeter data. Based on his findings, a wind correction was introduced, prior to tuning the wave model. This correction consisted of a coastal error correction up to 150 km offshore, a deep-ocean correction beyond 300 km offshore, and a smooth blending in between. The original deep-water correction consisted of a substantial linear correction:

*U*

_{c}and

*U*

_{o}are the corrected and original wind speeds (m s

^{−1}), respectively. By early 1997 wind speed biases were greatly reduced, and the following error corrections were used in the initial operational implementation of WW3 (Tolman et al. 2002):

*H*

_{s}, when compared to buoy data around the Australian coast (Greenslade et al. 2005).

While such adjustments provide a useful means of removing known biases in the forcing winds, these simple corrections have a number of limitations. One is the inability to account for the expected spatial variation in the wind error. Durrant and Greenslade (2012) showed that the ACCESS *U*_{10} bias exhibits considerable variation over the global domain. This spatial variability has been similarly demonstrated by Chelton and Freilich (2005) in the case of both the European Centre for Medium-Range Weather Forecasts (ECMWF) and NCEP operational analyses. Applying a single, homogeneous correction to the entire field does not account for this spatial variation, and winds corrected in this way can be expected to retain significant regional biases. The verifications of the Bureau’s GASP marine winds (Kepert et al. 2005; Schulz et al. 2007) did not include a spatial assessment of error, and adopted a single correction proposed for the entire global domain. Subsequent work demonstrated that the GASP model showed considerable regional variation, with positive biases in the tropics, and negative biases in the extratropics (Durrant and Greenslade 2012). As such, it could be speculated that the corrections of Greenslade et al. (2005) noted above, while improving the extratropics bias, likely degraded the performance in the tropics, and subsequent improvements seen in wave model results at midlatitude buoy locations around Australia would be unlikely to occur everywhere.

Tolman (1998b) did consider an aspect of the spatial variation of error, applying separate corrections around coastlines. However, in the open ocean the proposed correction simply prioritized bias minimization in the Northern Hemisphere, in accordance with NCEP’s forecasting priorities, which actually introduced a slight positive bias in the Southern Hemisphere (Tolman 1998a).

The second limitation of this correction method is that, once determined, any applied correction to an operational system will need to be continually monitored for changes in systematic wind biases. For a given atmospheric model configuration, bias could be expected to vary seasonally, or with changes due to long-lived modes of atmospheric variability such as El Niño–Southern Oscillation (ENSO). In an operational environment, model upgrades pose similar problems. Such changes to the forcing are an issue for any downstream model. Within the context of wave modeling, part of the motivation for performing wind corrections is an acknowledgment of this fact, with adjustments to statistical wind corrections, in general, being assessed as less burdensome than retuning the wave model itself. However, maintaining such wind corrections still requires considerable effort. Ideally, a correction method that can automatically adapt to these changes is desired.

One such technique is the operational consensus forecast (OCF) scheme of Woodcock and Engel (2005). In its complete form, OCF combines forecasts derived from a multimodel ensemble to produce an improved real-time forecast at locations where recent observations are available. Component model biases and weighting factors are derived from a training period of previous model forecasts and verifying observations. The next real-time OCF forecast is a weighted average of the set of latest-available, bias-corrected, component forecasts. Determining corrections based on a defined, moving, learning period in this way results in a correction that evolves in time with the changes in recent systematic model biases, without the need for manual monitoring.

Durrant et al. (2009) investigated the application of OCF to wave forecasts, using 10 models at 14 National Data Buoy Center (NDBC) buoy sites located around North America. The focus of their study was on the performance of weighted averages of several models, which is not the aim here. However, it included what the authors refer to as internal methods, those in which an individual model forecast is corrected according to a training set based only on that particular model. For *U*_{10}, an average 16% improvement in individual model RMSE was achieved by applying linear corrections, based on a 30-day learning window.

This technique is constrained by the need for observations with which to “train” the model. This has generally restricted its application to site-based locations, for which consistent historical observations are available. In the case of Durrant et al. (2009), for example, forecasts rely on buoy observations. Here, the aim is to correct the entire gridded wind field at each grid point.

In the absence of a dense observing network to represent “truth” for the purposes of training, an analysis field has been substituted in several studies as a reference to which to compare forecast products. Recent work at the Bureau has investigated the correction of gridded fields of surface temperature against the Mesoscale Surface Analysis System (MSAS) analysis (NMOC 2008, 2010b). Inherent systematic background error can be expected to grow with forecast period, and hence the potential to benefit from analysis-based corrections exists. This approach offers the advantage of a training dataset with regular output at every grid point being corrected. However, while assimilation techniques have undoubtedly provided significant gains in analysis accuracy, biases remain, as clearly shown for the specific case of surface winds by Chelton and Freilich (2005). The corrected product can do no better than replicate the systematic errors of the analysis.

The method developed here aims to produce an observation-based, spatially varying, learned correction scheme that can be applied to forecast winds, thus addressing the spatial variation in systematic *U*_{10} error, as well as removing the need to manually monitor and update the applied correction.

## 3. Method

The OCF methodology of Woodcock and Engel (2005) is a simple statistical scheme, which takes a weighted average of bias-corrected component model forecasts on a site-by-site and day-by-day basis. The scheme is based upon the premise that the forecast of model *i* (*f*_{i}) has three components: the true value (*o*), a systematic error component or bias (*b*_{i}) that can be approximated and removed, and a random error component (*e*_{i}) that can be reduced through compositing. Bias and weighting parameters are based on a moving window of historical data. Normalized weighting parameters (*w*_{i}) are calculated by using the inverse of the mean absolute error (MAE) from the bias-corrected error samples of the *n* contributing model forecasts over the training period.

The OCF scheme described above uses site-based observations to determine a learned correction. The aim here is to correct the entire wind field grid, applying an independent correction for each grid point. This is achieved with the use of remotely sensed wind data acquired by a scatterometer.

### a. Data

Wind observations from the SeaWinds scatterometer on board the Quick Scatterometer (QuikSCAT) are used here. This instrument produces winds over an 1800-km-wide swath at a horizontal spacing of 25 km, with an orbital period of 101 min, covering about 90% of the ice-free oceans per day. Scatterometer observations have become a valuable source of data for the study of marine surface winds due to their high quality and spatial coverage (e.g., Kelly 2004). These data have been extensively used for verification of NWP winds (e.g., Rogers and Wittmann 2002; Yuan 2004; Isaksen and Janssen 2004; Schulz et al. 2007).

The mission requirements of QuikSCAT specified a wind speed accuracy of 2 m s^{−1} for the range 3–20 m s^{−1} and 10% for the range 20–30 m s^{−1}, as well as a directional accuracy of 20**°** RMSE for wind speeds ranging from 3 to 30 m s^{−1} (Lungu and Callahan 2006). Several studies have shown accuracies exceeding these specifications (e.g., Stoffelen 1998; Freilich and Vanhoff 2006). In comparisons between NDBC open-ocean in situ observations and QuikSCAT over a 2-yr period, Chelton and Freilich (2005), for example, demonstrated RMSE values of around 1.7 m s^{−1} over all wind speeds (up to the maximum measured of 25 m s^{−1}). Even at hurricane strength winds, Sharma and D’Sa (2008) suggest that QuikSCAT winds maintain accuracy comparable to that of buoys. The wind direction accuracy is a sensitive function of wind speed at low wind speeds but improves rapidly with increasing wind speed. Chelton and Freilich (2005) report a directional error of about 14° at 6 m s^{−1}.

QuikSCAT data were obtained from the Center for Ocean–Atmospheric Prediction Studies (COAPS; http://coaps.fsu.edu/). The product used here is the so-called compact level 2B swath product produced by the Jet Propulsion Laboratory (JPL) utilizing the QSCAT1 geophysical model function (Dunbar et al. 2006).

The model winds to be corrected are those of the Bureau’s recently implemented ACCESS system. The winds are from a test configuration of the global ACCESS model. Evaluations are carried out for the 4-month period from July to October 2008. Unfortunately, a data-archiving issue restricted the available ACCESS data to this period. Due to the short period of available ACCESS winds, some additional analysis is also performed using the Bureau’s previous model, GASP, for the entire year of 2008. In the case of both models, QuikSCAT data have been assimilated, and for verification purposes, these data cannot be considered independent at the analysis time. Hence, short-term forecast winds from 0–12-h lead time are used. This ensures high quality winds are considered with minimal phase error, while still maintaining the independence of the verification data.

### b. Correction methods

static homogenous—a single statistical correction applied over the whole grid;

spatially varying—independent corrections calculated for each model grid point, retrospectively for the entire period; and

spatially and temporally varying—spatially varying corrections as above, extended to also vary in time.

*U*and

*U*

_{c}are the wind speed and corrected wind speed, respectively, and the constants

*b*,

*s*, and

*l*

_{1}and

*l*

_{2}are to be determined via least squares regression applied to the entire set of collocations. Following Woodcock and Engel (2005), observations are treated here as truth. The implicit assumption here is that the observations have more desirable (and, in particular, lower) error characteristics than the model.

Static homogenous corrections, similar to those of Tolman (1998b) and Greenslade et al. (2005), consider the entire set of collocations together. These serve primarily to provide a benchmark against which to assess the relative gains achieved by more complex variations of the linear regression method. Spatially varying corrections consider an independent correction calculated for each model grid point. Utilizing the comprehensive spatial coverage of the QuikSCAT observations, data are placed into 1° spatial bins and corrections calculated for each bin retrospectively for the entire period. This spatially varying correction is then extended to vary in time. Based on the learned correction concept of the OCF technique, appropriate adjustments are determined from comparisons between scatterometer observations and preceding forecasts.

## 4. Results

Results are presented in three stages, according to the complexity of the linear regression approach used. Section 4a presents results from simple corrections to the whole field (i.e., spatially static), the applications of spatially varying corrections are then explored in section 4b, and, finally, results for spatially and temporally varying learned corrections are presented in section 4c. Global verification statistics refer to statistics calculated from the full set of scatterometer and (corrected) model collocations. Note that in the case of the static homogeneous (section 4a) and spatially varying (section 4b) results, the data used to determine the corrections are the same data used to verify the corrected winds, and cannot be considered independent. In the case of the learned correction, the use of past observation–model comparisons to correct the winds at the current forecast time has the advantage of maintaining the independence of the verifying data. This is because data valid at a given time step have not been used in the construction of the correction applied to that time step. In all cases, additional independent verification is conducted using surface wind observations from the homogenized GlobWave altimeter dataset (Queffeulou and Croizé-Fillon 2012).

### a. Static and homogeneous

*U*is the model wind speed and

*U*

_{c}is the corrected wind speed. Table 1 shows global verification statistics for each of these corrections. RMSE has been reduced by between 5% and 6%, and the bias has been almost removed in all cases. The scatter index (SI), defined as the standard deviation of the difference between model and observations, normalized by the observed mean, does not show much sensitivity. This is not unexpected from these simple systematic corrections for the whole period. The linear correction performs best, but only slightly.

Global verification statistics for ACCESS *U*_{10} (m s^{−1}) against QuikSCAT scatterometer data and independent GlobWave altimeter data before and after the application of homogeneous corrections calculated retrospectively over the entire 4-month period.

Examining the spatial bias, however, highlights the danger of considering global statistics alone. The case of a slope correction is used for illustration here, which amounts to a 5% increase in *U*_{10}. The spatial *U*_{10} biases for the whole period both before and after applying this correction are shown in Figs. 1a and 1b, respectively. Despite improvement, and an average bias of close to zero, the corrected winds still show significant regional biases in the open ocean and strong negative biases are still evident in coastal regions.

### b. Spatially varying corrections

The homogeneous corrections applied in section 4a considered only the scalar wind speed. Wind is a vector quality, and also contains directional error. For global homogeneous corrections, ignoring directional error is justified by the fact that the mean directional bias over the globe is small (see Durrant and Greenslade 2012). When considering regional corrections, directional effects may play a larger role. In the case of a coastal bias, for example, the nature of the bias may be expected to differ for offshore, onshore, or alongshore wind conditions.

This dependence on directionality is explored here for the eight distinct regions shown in Fig. 2. Figure 3 shows wind roses for both the modeled wind (left) and the observations (middle), based on all collocations within each specified area. The right-hand-side panels in Fig. 3 show the directionally dependent slope correction, constructed by binning collocations into 20° directional bins, according to observation direction, and calculating the slope correction for each bin. Where a given 20° bin contains less than 1% of the total observations, no slope is plotted. The blue line shows, for reference, the slope calculated from all collocations in the region, irrespective of direction (note that wind roses show the direction the wind is traveling to, consistent with oceanographic conventions).

Regions for which wind roses and the directional dependence of slope are calculated. See Fig. 3.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Regions for which wind roses and the directional dependence of slope are calculated. See Fig. 3.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Regions for which wind roses and the directional dependence of slope are calculated. See Fig. 3.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

For each region shown in Fig. 2, (left) the model wind roses, (middle) observation wind roses, and (right) the directionally dependent slope correction based on the entire period: (a) north Pacific, (b) eastern tropical Pacific, (c) western tropical Pacific, (d) south Pacific, (e) Indian Ocean, (f) south Indian Ocean, (g) north Atlantic, and (f) east Australia coastal.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

For each region shown in Fig. 2, (left) the model wind roses, (middle) observation wind roses, and (right) the directionally dependent slope correction based on the entire period: (a) north Pacific, (b) eastern tropical Pacific, (c) western tropical Pacific, (d) south Pacific, (e) Indian Ocean, (f) south Indian Ocean, (g) north Atlantic, and (f) east Australia coastal.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

For each region shown in Fig. 2, (left) the model wind roses, (middle) observation wind roses, and (right) the directionally dependent slope correction based on the entire period: (a) north Pacific, (b) eastern tropical Pacific, (c) western tropical Pacific, (d) south Pacific, (e) Indian Ocean, (f) south Indian Ocean, (g) north Atlantic, and (f) east Australia coastal.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

A correction based only on the scalar wind speed will work well where (a) the directional distributions in the wind roses for modeled and observed values do not significantly deviate from one another and (b) the slope correction does not vary significantly with direction. For the plots in Fig. 3, where the directionally dependent slope correction (red line) matches the overall slope correction (blue line), correcting the scalar wind speed is likely to produce a good result in all conditions.

On the whole, the directional distribution appears to be well captured by the model in all regions. The directionally dependent slope correction for the tropical Pacific, the Indian Ocean as a whole, and the south Indian Ocean region show little variation in the range of directions for which the wind is blowing for the majority of the time. Some variation in slope with direction is seen in the midlatitude corrections, however. For the North Pacific and North Atlantic a minimum in the slope correction occurs for winds blowing in northeasterly directions. In the South Pacific this minimum is to the southeast. In these midlatitude regions, the maximum slope correction is located in approximately the opposite direction to the minimum.

One means of accounting for a directionally dependent bias is by applying different corrections to the *u* and *υ* components, as was done by Greenslade et al. (2005). Such separate *u* and *υ* corrections can account for symmetric east–west or north–south variations; errors of this nature would show up as an ellipse rather than a circle in the plots in Fig. 3. Such corrections cannot, however, account for a maximum in one direction and a minimum in the opposite direction, as in the off-center circles seen in these midlatitude cases. As such, considering the *u* and *υ* components separately is unlikely to provide improvements over simpler, scalar wind speed corrections. In any event, these midlatitude directional slope variations are minor. Even at the coast, where the directional dependence might be expected to be most prevalent, the variation is slight, as shown here for the Australian east coast (Fig. 3h).

Proceeding then with scalar wind speed corrections, collocation data for the whole period are distributed into 1° latitude–longitude bins, and appropriate corrections are determined for each bin, producing an independent correction to be applied at each model grid point. The resulting correction coefficients [i.e., those corresponding to Eqs. (5)–(7)] for bias (Fig. 4a), slope (Fig. 4b), and linear (Figs. 4c,d) corrections are shown.

Spatially varying correction coefficients, calculated independently for each 1° × 1° model grid point: (a) bias, (b) slope, (c) slope of linear, and (d) intercept of linear.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Spatially varying correction coefficients, calculated independently for each 1° × 1° model grid point: (a) bias, (b) slope, (c) slope of linear, and (d) intercept of linear.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Spatially varying correction coefficients, calculated independently for each 1° × 1° model grid point: (a) bias, (b) slope, (c) slope of linear, and (d) intercept of linear.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Global verification statistics for *U*_{10} after the application of these spatially varying corrections are presented in Table 2. Overall, significant improvement is seen over the static homogeneous corrections. Slope corrections show the best overall results, though again the difference between these methods is small. The simplicity of having to only calculate a single correction field favors the use of slope corrections over linear corrections.

Global verification statistics for ACCESS *U*_{10} (m s^{−1}) against QuikSCAT scatterometer data and independent GlobWave altimeter data before and after the application of spatially varying corrections calculated retrospectively over the entire 4-month period.

More importantly, applying spatially varying corrections effectively removes the bias over the whole global domain. Figure 5 shows the *U*_{10} bias after applying the slope corrections of Fig. 4b. The broad-scale regionally varying biases have been almost eliminated, and the finer-scale coastal biases have been greatly reduced. At any given grid point, the removal of the *U*_{10} biases is not surprising, given that verifications are based here on the same scatterometer data used to determine the correction. However, for the purposes of reducing the bias over the entire domain, it is clear that spatially varying corrections offer significant gains over homogeneous corrections.

The *U*_{10} bias after applying the spatially variant slope correction shown in Fig. 4b.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The *U*_{10} bias after applying the spatially variant slope correction shown in Fig. 4b.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The *U*_{10} bias after applying the spatially variant slope correction shown in Fig. 4b.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

In summary, the corrections based on scalar wind speed only appear reasonable, with more complex, directionally dependent corrections unlikely to offer significant gains. Accounting for the regional variation in model bias is effectively addressed by applying spatially varying corrections. Corrections that are a function of *U*_{10} (i.e., slope and linear corrections) were found to perform better than bias corrections, with the small difference between the former two favoring the simpler, slope correction.

### c. Spatially and temporally varying learned corrections

As described in section 3, the method used here is based on the calculation of corrections based on a moving window of historical data. Spatially varying corrections are calculated as above at each model output time, with the period used to calculate the correction grid restricted to a given temporal window prior to this time. For example, if a 30-day learning window is used, the correction to be applied to the 0000 UTC forecast for 1 July would be calculated from collocations over the period 2100 UTC 31 May to 2100 UTC 30 June. The correction of 0300 UTC would use data from 0000 UTC 1 June to 0000 UTC 1 July, etc. This is schematically represented in Fig. 6. Updating the correction at every model output time in this way produces a correction that automatically evolves to match the recent historical bias of the model.

Schematic of the learned correction method, where *w* represents an arbitrary learning window. A time step of 3 h is used in the present study with the calculated correction varying geographically as well as with time. See text for details.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Schematic of the learned correction method, where *w* represents an arbitrary learning window. A time step of 3 h is used in the present study with the calculated correction varying geographically as well as with time. See text for details.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Schematic of the learned correction method, where *w* represents an arbitrary learning window. A time step of 3 h is used in the present study with the calculated correction varying geographically as well as with time. See text for details.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Based on the results of the previous section, spatially varying slope corrections of the scalar wind speed are used. The time step from one correction calculation to the next is 3 h, the time step at which the model output is available.

When developing correction methodologies such as the learned linear-regression corrections proposed here, some consideration must be given to the scales being targeted. As described in section 3 regarding the OCF methodology, the error in a model forecast can be somewhat simply described as having both a systematic bias component, and a random error component. These errors, though, occur at a range of temporal and spatial scales.

Consider the case of a low pressure system passing over a given location. If this system is predicted with the correct intensity, but with the timing delayed by several hours, a point location may experience a negative bias as it approaches and a positive bias after it passes. Within the context of the corrections applied here, this would be attributed as random error. This technique is not able to remove this type of error; rather, systematic bias on the scale of weeks to months is the target.

The most obvious means of targeting these larger-scale systematic errors is to increase the length of the training window. Figure 7, for example, shows slope corrections for 1 August 2008, calculated from learning windows of 2, 5, 10, and 30 days. The spatial coherency of the corrections increases with the length of the learning window as the influence of the phase error decreases. The total number of observations contributing to the correction calculation at each grid point (Fig. 8) also increases with window size, with implications for statistical robustness.

Correction grids calculated for 1 Jul 2008 for a given learning window in days (a) 2, (b) 5, (c) 10, and (d) 30.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Correction grids calculated for 1 Jul 2008 for a given learning window in days (a) 2, (b) 5, (c) 10, and (d) 30.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Correction grids calculated for 1 Jul 2008 for a given learning window in days (a) 2, (b) 5, (c) 10, and (d) 30.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The total number of observations used to construct the spatial corrections in Fig. 7 for (a) 2, (b) 5, (c) 10, and (d) 30 days.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The total number of observations used to construct the spatial corrections in Fig. 7 for (a) 2, (b) 5, (c) 10, and (d) 30 days.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The total number of observations used to construct the spatial corrections in Fig. 7 for (a) 2, (b) 5, (c) 10, and (d) 30 days.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

An alternate method of targeting synoptic-scale systematic error is to increase the area over which a correction is calculated. Returning to the case of our mistimed low pressure system, calculating the error over the area of the whole system targets the larger-scale systematic biases in the same way as increasing the temporal window. This is achieved by spatially smoothing the calculated corrections. Gaussian spatial filtering is used here, applying decreasing weight to surrounding grid points with increasing distance. Figure 9 shows a correction grid calculated with a 10-day learning period with smoothing applied over 2°, 5°, and 10° boxes. Though increasing the spatial smoothing achieves a qualitatively similar effect as increasing the temporal length of the window, this has the drawback that it reduces the ability to resolve small spatial scale features, such as those around coastlines and in the lee of islands.

(a) Correction grids for 1 Jul using a 10-day learning period with no smoothing applied and (b)–(d) Gaussian smoothing filters applied of 5, 10, and 30 days, respectively. Note that the setup in (a) is the same as in Fig. 7c.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

(a) Correction grids for 1 Jul using a 10-day learning period with no smoothing applied and (b)–(d) Gaussian smoothing filters applied of 5, 10, and 30 days, respectively. Note that the setup in (a) is the same as in Fig. 7c.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

(a) Correction grids for 1 Jul using a 10-day learning period with no smoothing applied and (b)–(d) Gaussian smoothing filters applied of 5, 10, and 30 days, respectively. Note that the setup in (a) is the same as in Fig. 7c.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Central to these considerations is the fact that learned corrections are applied to a subsequent forecast. Corrections calculated over small temporal and spatial scales can be expected to vary on similar scales, making them inappropriate for this type of application. Conversely, the strength of a learned correction scheme is its ability to automatically adapt to the evolution of systematic error on monthly and seasonal time scales, or indeed with changes due to alterations to the atmospheric model. Increasing the length of the learning window decreases the speed with which the correction can adapt to these changes. Thus, a balance must be struck when choosing the scales of the learning windows.

Learning windows of 2, 5, 10, 20, and 30 days with both no smoothing (i.e., calculating corrections for each model grid point), and smoothing of 2°, 3°, 5°, 10°, and 20° are explored here, resulting in 30 different combinations of learning window characteristics. Corrected winds for the whole 4-month period were calculated and verified for each case. Figure 10 shows the percentage reduction in *U*_{10} RMSE associated with each combination. Full statistics for each combination are given in Table 3.

Percentage improvement in ACCESS RMSE relative to scatterometer data for corrected wind fields over the uncorrected wind field for different-length learning windows (days) and different levels of smoothing (°) applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in ACCESS RMSE relative to scatterometer data for corrected wind fields over the uncorrected wind field for different-length learning windows (days) and different levels of smoothing (°) applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in ACCESS RMSE relative to scatterometer data for corrected wind fields over the uncorrected wind field for different-length learning windows (days) and different levels of smoothing (°) applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Full statistics for all combinations of learning windows and smoothing applied to the ACCESS model.

Consistent with the considerations discussed above, applying corrections corresponding to small time and space scales, results in very little improvement, with SI in fact being degraded. For short time windows, increasing the spatial scales results in significant improvement; however, this trend is not apparent for learning windows of 5 and 10 days, and reverses for longer learning windows. This is consistent with the increased smoothing reducing the ability to resolve small-scale features such as those along coastlines and in the lee of islands, as seen in Fig. 9. Overall, the best results here are achieved by using the 30-day learning window to target the desired error scales, with no spatial smoothing applied, maintaining the ability to correct systematic errors on small spatial scales. Based on this assessment, these spatial and temporal windows are used for the remainder of this work. Note that these optimal values apply to the specific case of the ACCESS winds examined here, and may differ between models. Additionally, due to the short time frame examined here, learning periods longer than 30 days are not explored; we return to this point in the next section.

The evolution of these corrections is shown in Fig. 11, at monthly intervals, over the period examined. This evolution is slow as a result of our using learning windows of 30 days. This is not unexpected, given the small variation in spatial bias from month to month over the period of this study, as found by Durrant and Greenslade (2012).

Calculated spatially and temporally varying learned slope corrections for ACCESS at monthly intervals over the period of this study: (a)–(e) from 1 Jul to 1 Nov 2008. Corrections shown here represent those applied at the model dates indicated. In each case, corrections are calculated from the 30-day learning window.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Calculated spatially and temporally varying learned slope corrections for ACCESS at monthly intervals over the period of this study: (a)–(e) from 1 Jul to 1 Nov 2008. Corrections shown here represent those applied at the model dates indicated. In each case, corrections are calculated from the 30-day learning window.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Calculated spatially and temporally varying learned slope corrections for ACCESS at monthly intervals over the period of this study: (a)–(e) from 1 Jul to 1 Nov 2008. Corrections shown here represent those applied at the model dates indicated. In each case, corrections are calculated from the 30-day learning window.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Global *U*_{10} verification statistics following the application of these corrections are given in Table 4. Overall, the results are comparable to those of the spatially varying corrections calculated retrospectively over the entire period (section 4b). Spatial *U*_{10} bias, shown in Fig. 12, is similarly comparable, showing negligible remaining regional biases. This is an encouraging result, noting that verification and training data are independent here while the temporally fixed corrections are verified with the same data that are used to determine the corrections. Furthermore, the temporal variation of the corrections applied over the 4 months examined here is rather small. Over longer periods, where the *U*_{10} bias could be expected to show greater variation, the gains from temporally evolving corrections would be expected to be greater still.

Global verification statistics for ACCESS *U*_{10} (m s^{−1}) against QuikSCAT scatterometer data and independent GlobWave altimeter data before and after the application of spatially and temporally varying learned corrections.

Bias of ACCESS winds corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Bias of ACCESS winds corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Bias of ACCESS winds corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

An overall improvement of 7% in RMSE has been achieved. Examining the spatial distribution of these gains, Fig. 13 shows the percentage improvement in RMSE at every grid point, indicating a large amount of regional variation. In general, large improvement is seen in areas where significant biases were present in the uncorrected winds. Bias is almost eliminated in each region. Figure 14 shows quantile–quantile plots for both uncorrected and corrected winds against scatterometer observations for the whole globe, as well as separately for the tropics (20°S–20°N), the northern extratropics (north of 20°N), and the southern extratropics (south of 20°S). Clear improvements are evident across the wind range, though a slight underestimation of high winds remains apparent in the corrected data.

Percentage improvement in ACCESS RMSE achieved by applying a learned slope correction with a window length of 30 days and no spatial smoothing.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in ACCESS RMSE achieved by applying a learned slope correction with a window length of 30 days and no spatial smoothing.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in ACCESS RMSE achieved by applying a learned slope correction with a window length of 30 days and no spatial smoothing.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The Q–Q plots for corrected and uncorrected ACCESS winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The Q–Q plots for corrected and uncorrected ACCESS winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The Q–Q plots for corrected and uncorrected ACCESS winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Most importantly, this method addresses the need for a spatially varying correction, while simultaneously removing the need to manually monitor and update that correction.

### d. Application to a different wind field

The above analysis has shown that, in the case of the ACCESS model for the 4-month period examined, spatially and temporally varying learned corrections based on scatterometer data provide a robust and effective means of correcting marine surface wind forecast data in an operational setting. However, a number of questions remain from the above analysis. The techniques described in this paper appear to work well for this specific model and season, and while there is no apparent reason why the technique should not work with other datasets, this assertion has not been demonstrated. It has been claimed that one of the primary strengths of this self-learning technique is that it can automatically account for seasonal variation in error; however, the 4-month period examined here prevents this from being explicitly examined. Similarly, the examination of learning periods longer than 30 days was not possible for these short time frames, leaving us to speculate whether longer learning periods lead to better results. To address these questions, the same techniques were applied to the Bureau’s previous operational model (GASP) for the full year of 2008, overlapping the 4-month period examined for ACCESS. The focus of this work remains with the current system, ACCESS; the intention here is not to repeat the entire analysis, but simply to address some of the specific points raised above.

Before this is explored, we first consider the optimal length of the learning period. Figure 15 shows the improvement in RMSE relative to scatterometer data for different combinations of learning window length and spatial smoothing. Figure 15 is comparable to Fig. 10 for ACCESS; however, longer learning windows of 45, 60, 75, and 90 days have additionally been assessed. The full statistics for each combination are given in Table 5. The conclusions drawn here are similar to those for the ACCESS case, with smoothing adding value for short learning windows, but not for longer windows. Windows longer than 30 days appear not to produce any further gains in the case of no smoothing, while slight gains are apparent when spatial smoothing is applied. Considered within the context of the extra computational expense of longer windows, and the increased delay for the correction to adjust to changes, as well as the ability to correct smaller-scale features when no spatial smoothing is applied, the choice of a 30-day learning window appears to be the best practical choice here.

Percentage improvement in RMSE relative to scatterometer data for corrected GASP wind fields over the uncorrected wind field for different length learning windows and different levels of smoothing applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in RMSE relative to scatterometer data for corrected GASP wind fields over the uncorrected wind field for different length learning windows and different levels of smoothing applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Percentage improvement in RMSE relative to scatterometer data for corrected GASP wind fields over the uncorrected wind field for different length learning windows and different levels of smoothing applied to the calculated correction field.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Full statistics for all combinations of learning windows and smoothing applied to the GASP model.

Results from the entire year of 2008 for the various correction techniques discussed in this paper are given in Table 6. While some improvement is evident, gains are less than those for the ACCESS winds, primarily due to a less marked systematic error in GASP in need of correction. Despite this, there remain a number of arguments for applying corrections.

Global verification statistics for GASP *U*_{10} (m s^{−1}) against QuikSCAT scatterometer data and independent GlobWave altimeter data before and after the application of simple bulk, spatially varying, and spatially and temporally varying learned corrections.

Focusing again on learned spatially and temporally varying corrections using a 30-day learning window, Fig. 16 shows quantile–quantile plots for both uncorrected and corrected winds for the globe, as separated into the tropics and the northern and southern extratropics. Though improvements are evident, it is clear that systematic error in the GASP winds are less than those of ACCESS (Fig. 14), resulting in smaller gains in terms of RMSE. However, it is also apparent that unlike ACCESS, which showed negative bias in each region, GASP shows positive bias in the tropics, and a slight negative bias elsewhere. This is more clearly visible in Fig. 17, showing global bias before and after the application of the corrections. The effective removal of this spatial variation in bias shows that even in the case of small overall bias, this correction technique offers benefits at regional scales that cannot be achieved with simple overall corrections.

The Q–Q plots for corrected and uncorrected GASP winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The Q–Q plots for corrected and uncorrected GASP winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The Q–Q plots for corrected and uncorrected GASP winds for (a) the entire global domain, (b) the tropics, (c) the southern extratropics, and (d) the northern extratropics.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

(a) Bias of GASP winds against scatterometer data for the uncorrected winds and (b) those corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

(a) Bias of GASP winds against scatterometer data for the uncorrected winds and (b) those corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

(a) Bias of GASP winds against scatterometer data for the uncorrected winds and (b) those corrected with a 30-day learning window applied every 3 h.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

Once again, the spatially varying aspect of the corrections appears valuable; we now consider further the advantages gained by the addition of temporal variation. Figure 18 shows the temporal evolution of the bias for uncorrected GASP winds and those corrected with a simple bulk correction, a fixed spatially varying correction, and finally a learned spatially and temporally varying correction. In each case, average values across the tropics, the northern extratropics, and the southern extratropics are shown in addition to the global average, with a 30-day running Hanning window applied over the daily values. For the case of the uncorrected winds, although the global bias is near zero, the tropics show a positive bias, while the northern extratropics show a negative bias, with a mixed bias for the southern extratropics. Temporally, the global bias also remains relatively steady; however, there is a notable opposing annual cycle in the northern and southern extratropics, with the bias becoming increasingly negative in their winters. The tropics show an annual cycle that tends to follow the northern extratropics. As expected the bulk slope correction does little to reduce any of these features, simply moving all curves down a little. As discussed above, the spatially varying corrections do a good job of reducing the hemispheric differences, but the annual cycle remains. The spatially and temporally varying corrections retain the positive features of the spatially varying case while, additionally, effectively removing the annual cycle in all regions. This clearly demonstrates the advantages of the self-learning aspect of these final corrections. While it is not explicitly displayed here, the ability to remove variations in systematic error due to interannual variation as well as physical changes to the model is clearly apparent.

The 30-day running average of daily bias against scatterometer data for (top) uncorrected GASP winds, and those corrected using (second row) bulk, (third row) spatially varying, and (bottom) spatially and temporally varying corrections. Results are shown for the entire globe, as well as the tropics, the Southern Hemisphere extratropics, and the Northern Hemisphere extratropics separately.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The 30-day running average of daily bias against scatterometer data for (top) uncorrected GASP winds, and those corrected using (second row) bulk, (third row) spatially varying, and (bottom) spatially and temporally varying corrections. Results are shown for the entire globe, as well as the tropics, the Southern Hemisphere extratropics, and the Northern Hemisphere extratropics separately.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The 30-day running average of daily bias against scatterometer data for (top) uncorrected GASP winds, and those corrected using (second row) bulk, (third row) spatially varying, and (bottom) spatially and temporally varying corrections. Results are shown for the entire globe, as well as the tropics, the Southern Hemisphere extratropics, and the Northern Hemisphere extratropics separately.

Citation: Weather and Forecasting 29, 2; 10.1175/WAF-D-12-00101.1

The analysis of GASP winds for the full year has demonstrated that this technique can be generally applied to any model. The fact that GASP has less systematic error than ACCESS results in smaller gains in terms of the overall RMSE however, learned spatially and temporally varying corrections have been shown to effectively reduce the regional variation in the systematic error, as well as reducing the seasonal cycle in that error.

## 5. Concluding remarks

The application of three variations of linear-regression corrections to surface wind speeds from the Australian Bureau of Meteorology’s ACCESS simulator have been explored. These corrections, based on comparisons against QuikSCAT observations, were designed to produce improved forcing for a wave model. The simplest of these examined the application of a fixed correction over the entire global domain. While such corrections produced overall improvements, they were unable to adequately account for regional variation in *U*_{10} bias over the globe. These regional biases were effectively removed with a spatially varying correction, calculated independently for each model grid point, retrospectively, for the entire period.

These spatially varying corrections were then extended to vary in time. A variation of the linear-regression method was proposed whereby automatically evolving corrections are calculated in real time from a moving window of historical comparisons between observations and preceding forecasts. A number of spatial and temporal learning windows were explored, with a 30-day learning window, calculated individually for each model grid point found to give the best improvement in terms of model RMSE. This relatively long learning period effectively targeted the removal of synoptic-scale systematic biases, while applying independent corrections at each model grid point enabled the removal of persistent biases on fine spatial scales, such as those present along coastlines and in the lee of islands. This technique addresses the need for geographically varying corrections, as well as eliminating the need to monitor and manually adjust these corrections with time. Correcting the winds in this way has resulted in a reduction in *U*_{10} RMSE of about 7% and regional biases in the winds have been effectively removed. The impact of these wind corrections on wave forecasts is presented in Durrant et al. (2013).

An analysis of winds from the Bureau’s previous global model, GASP, for a full year demonstrated that this technique can be generally applied to any model, and that learned spatially and temporally varying corrections can effectively reduce regional variation in systematic error, as well as reducing the seasonal cycle of that error.

It should also be noted that before this correction method could be applied in an operational system, further work is needed. Issues such as real-time quality control and the ability to handle data delays and dropouts are essential for practical applications. It is also worth noting that QuikSCAT is no longer operational; the Advanced Scatterometer (ASCAT; EUMETSAT 2010) instrument, succeeding *ERS-2*, would provide a suitable replacement for an operational system.

This work has only examined a single, short forecast period, focusing mainly on the removal of systematic bias. The strength of this approach is that it can be independently applied across the range of available forecast periods. In fact, as inherent model errors can be expected to become more prevalent with increasing forecast periods, such bias correction techniques have been shown to give greater relative gains at these longer forecast periods (e.g., Durrant et al. 2009). Given that operational weather centers usually receive a number of surface wind products from other centers in real time, the possibility also exists to extend this work to include the weighted compositing aspects of a true OCF scheme. If the gains seen by Durrant et al. (2009) at site locations could be replicated over the entire domain, significant improvements in forecast skill could be expected in addition to the reduction in bias shown here.

## Acknowledgments

The authors would like to thank Eric Schulz, as well as two anonymous reviewers, for their helpful comments on the manuscript.

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