Statistical Downscaling Prediction of Sea Surface Winds over the Global Ocean

Cangjie Sun School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

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Adam H. Monahan School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada

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Abstract

The statistical prediction of local sea surface winds from large-scale, free-tropospheric fields is investigated at a number of locations over the global ocean using a statistical downscaling model based on multiple linear regression. The predictands (the mean and standard deviation of both vector wind components and wind speed) calculated from ocean buoy observations on daily, weekly, and monthly scales are regressed on upper-level predictor fields from reanalysis products. It is found that in general the mean vector wind components are more predictable than mean wind speed in the North Pacific and Atlantic, while in the tropical Pacific and Atlantic the difference in predictive skill between mean vector wind components and wind speed is not substantial. The predictability of wind speed relative to vector wind components is interpreted by an idealized model of the wind speed probability density function, which indicates that in the midlatitudes the mean wind speed is more sensitive to the vector wind standard deviations (which generally are not well predicted) than to the mean vector winds. In the tropics, the mean wind speed is found to be more sensitive to the mean vector winds. While the idealized probability model does a good job of characterizing month-to-month variations in the mean wind speed in terms of the vector wind statistics, month-to-month variations in the standard deviation of speed are not well modeled. A series of Monte Carlo experiments demonstrates that the inconsistency in the characterization of wind speed standard deviation is the result of differences of sampling variability between the vector wind and wind speed statistics.

Corresponding author address: Adam Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065 STN CSC, Victoria BC V8W 3V6, Canada. E-mail: monahana@uvic.ca

Abstract

The statistical prediction of local sea surface winds from large-scale, free-tropospheric fields is investigated at a number of locations over the global ocean using a statistical downscaling model based on multiple linear regression. The predictands (the mean and standard deviation of both vector wind components and wind speed) calculated from ocean buoy observations on daily, weekly, and monthly scales are regressed on upper-level predictor fields from reanalysis products. It is found that in general the mean vector wind components are more predictable than mean wind speed in the North Pacific and Atlantic, while in the tropical Pacific and Atlantic the difference in predictive skill between mean vector wind components and wind speed is not substantial. The predictability of wind speed relative to vector wind components is interpreted by an idealized model of the wind speed probability density function, which indicates that in the midlatitudes the mean wind speed is more sensitive to the vector wind standard deviations (which generally are not well predicted) than to the mean vector winds. In the tropics, the mean wind speed is found to be more sensitive to the mean vector winds. While the idealized probability model does a good job of characterizing month-to-month variations in the mean wind speed in terms of the vector wind statistics, month-to-month variations in the standard deviation of speed are not well modeled. A series of Monte Carlo experiments demonstrates that the inconsistency in the characterization of wind speed standard deviation is the result of differences of sampling variability between the vector wind and wind speed statistics.

Corresponding author address: Adam Monahan, School of Earth and Ocean Sciences, University of Victoria, P.O. Box 3065 STN CSC, Victoria BC V8W 3V6, Canada. E-mail: monahana@uvic.ca

1. Introduction

Sea surface winds play a central role in influencing the exchange of heat, momentum, and mass between the ocean and the atmosphere (e.g., Garratt 1992; Bates et al. 2001; Jones and Toba 2001; Donelan et al. 2002). As well, sea surface winds represent a potentially significant energy resource (e.g., Liu et al. 2008; Capps and Zender 2009) and high sea surface winds represent hazards to shipping (e.g., Sampe and Xie 2007). While the output of global climate models represents the best tool available for studying large-scale climate variability, it is generally not directly relevant for inferences about local climate. The coarse resolution and approximate parameterizations of subgrid-scale processes both limit the accuracy of the representation of local variability, especially in the planetary boundary layer. In particular, local surface winds may be influenced by local small-scale processes that are not resolved well in climate models. The process of downscaling is designed to relate local, small-scale variability to variability on large scales. Dynamical downscaling approaches this problem by nesting finely resolved, local dynamical models within coarsely resolved, large-scale models. In contrast, statistical downscaling (SD) is a complementary strategy employed to empirically downscale large-scale variability through statistical methods. Although dynamical downscaling has the merits of being physically based and not assuming a stationary climate, potentially significant drawbacks such as possible errors associated with imperfect parameterizations of key processes (e.g., clouds and boundary layer processes), systematic biases, coarse spatial resolution, and extremely high computational demands constrain the use of pure dynamical downscaling. Although SD has the weakness that it assumes statistically stationary relationships between large-scale and local variables (as represented by historical observations), it is inexpensive and easy to implement.

This study considers the statistical relationships between local ocean winds and large-scale free-tropospheric circulation based on buoy observations and reanalysis products. Following earlier work by Monahan (2012a) and Culver and Monahan (2013), SD is used to investigate the predictability of sea surface winds (both wind speed and vector wind components) measured at buoys over the global oceans on daily, weekly (10 days), and monthly time scales. Along with the absolute predictive skills of surface-wind statistics, the predictability of the statistics of wind speed relative to those of the vector wind components is considered in this study.

A simple multiple linear regression will be used for the SD in place of a more sophisticated analysis (e.g., stepwise linear regression) in order to minimize the chances of overfitting the model. With appropriate cross-validation, multiple linear regression produces statistically robust prediction models. In the present study, the number of statistical degrees of freedom available for building the statistical model can be fairly small (e.g., 12 years of 3 months for a given season at a typical buoy). The number of model parameters increases with the complexity of a statistical model, requiring larger datasets for their robust estimation and to avoid overfitting the model. The use of a relatively simple statistical model reduces the potential risk of skill inflation due to model overfitting.

This study has four primary goals: 1) Characterize the predictive information that free-tropospheric large-scale predictors carry for the statistics of local sea surface winds across a range of wind climates. 2) Explore the predictive skills of different surface-wind statistics (mean and standard deviation) on different temporal scales (daily, weekly, and monthly). 3) Investigate the relationship between the predictability of mean wind speed and that of the vector wind components presented in Culver and Monahan (2013) over a wider range of wind climate using an idealized probability distribution model (IPM) of the wind speed probability density function introduced by Monahan (2012a). 4) Assess the modeling skill of the IPM for the submonthly standard deviation of wind speeds across the global ocean.

Monahan (2012a) investigated the predictability of surface winds in the subarctic northeast Pacific off of western Canada, while Culver and Monahan (2013) studied the statistical predictability of historical land surface winds over central Canada. Other studies (e.g., Salameh et al. 2009; van der Kamp et al. 2012) have investigated the predictability of various vector wind components in regions of complex topography. In contrast to land surface winds, sea surface winds are less influenced by stationary local features (e.g., topography or fixed surface inhomogeneities). Therefore, the connection between sea surface winds and upper-level large-scale atmospheric fields is expected to be simpler than that for surface winds over land. Furthermore, the range of wind climates is much greater over oceans than over land (because of the much weaker surface drag over water). Thus, consideration of sea surface winds allows for the analysis of surface wind SD in a relatively idealized setting over a relatively large parameter range. This study does not consider the temporal structure of winds [which is considered in detail in Monahan (2012b)]. The focus is on the “instantaneous” prediction of surface wind statistics on various averaging time scales from large-scale free tropospheric predictors on the same time scale.

Section 2 describes the data used in this study, and section 3 presents both the methodology to be used and the prediction results of surface-wind statistics. Section 4 introduces the idealized model used to understand the relationship between the predictability of the statistics of the vector wind components and wind speed. A discussion and conclusions are presented in section 5.

2. Data

This study assesses the cross-validated statistical predictability of the statistics of historical sea surface wind observations from a total of 52 moored ocean buoys (the predictands) using free-tropospheric large-scale circulation data from global reanalysis products (the predictors).

a. Buoy data

The buoys considered in this study are situated in the tropical and North Pacific and Atlantic Oceans, with data durations of between 8 and 28 years (Table 1). The Southern Ocean and Indian Ocean are not considered in this study as we were not able to find buoy observations in those locations of sufficient duration to establish robust statistical relationships with the flow aloft. The wind and direction data from the 52 buoys were obtained from four sources:

  1. Prediction and Research Moored Array in the Atlantic (PIRATA) project 10-min averaged data (measured at 3–4 m above mean sea level) from five buoys in the tropical Atlantic (downloaded from http://www.pmel.noaa.gov/pirata/);

  2. Tropical Atmosphere Ocean (TAO)/Triangle Trans-Ocean Buoy Network (TRITON) project 10-min averaged wind data (measured at 3–4 m above mean sea level) from 12 buoys in the tropical Pacific (downloaded from http://www.pmel.noaa.gov/tao/disdel/disdel-pir.html);

  3. National Data Buoy Center (NDBC) hourly reports of 8-min averaged data (approximately 5 m above mean sea level) from 31 buoys off of the west and east coast of North America (downloaded from http://www.ndbc.noaa.gov/); and

  4. Japan Meteorological Agency (JMA) three-hourly reports of 10-min averaged data (approximately 5 m above mean sea level) from four buoys in the northwest Pacific; (downloaded from http://www.data.kishou.go.jp/kaiyou/db/vessel_obs/data-report/html/index_e.html).

Table 1.

The location, duration, anemometer height, and data archive of the buoys considered in this study. Buoy identification (ID) is given where applicable.

Table 1.

These data were then used to calculate means of vector wind components (projections along 36 directions around the compass) and wind speed on daily, weekly and monthly time scales. Standard deviations of these quantities on subaveraging time scales were also calculated. Other than removing missing data from the buoy time series, no other preprocessing was carried out on these datasets.

b. Global reanalysis products

Ten-meter and 850-hPa fields (zonal wind U, meridional wind V, and temperature T) were obtained from National Centers for Environmental Prediction (NCEP)/Department of Energy (DOE) Reanalysis 2 data (downloaded from http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis2.html). Wind speed fields W were computed from the U and V fields. These data are available 4 times daily from January 1979 to December 2011 at a resolution of 2.5° × 2.5°. The downscaling predictors were calculated from the 850-hPa data. An analysis using predictors at other pressure levels demonstrated that predictive information is largely independent of predictor pressure level throughout the free troposphere (Sun 2012).

c. Construction of surface-wind predictands

The statistics of both wind speed and vector wind components were predicted on three different time scales (daily, weekly, and monthly) in this study. For all predictions, the same averaging time scale was used for both the predictors and predictands. Predictions were carried out separately in each calendar season [December–February (DJF), March–May (MAM), June–August (JJA), and September–November (SON)] to minimize the influence of nonstationarities in the relationship between predictors and predictands resulting from the seasonal cycle. An inspection of the seasonal variations in surface wind data (not shown) demonstrated that these calendar seasons characterize the dominant nonstationarity in the data. A summary of the predictands is as follows:

  1. mean wind speed on the specified averaging time scale ,

  2. subaveraging time scale standard deviation of wind speed σw,

  3. mean vector wind components in the direction along the basis vector , (These components are considered at 10° increments around the compass. By construction, projections separated by 180° are the same up to a sign.), and

  4. subaveraging time scale standard deviation of the vector wind components along , .

Throughout this paper, an overbar will denote averaging on daily, weekly, or monthly time scale. If the time scale is not explicitly specified when discussing a given result, it will hold on any time scale. In particular, is the mean of the square of the wind speed.

Two further statistics, μ and σ, are also calculated: 1) the amplitude of the average vector wind and 2) the isotropic vector wind standard deviation , where and are means of two orthogonal vector wind components and σu and συ are standard deviations of two orthogonal vector wind components (u and υ generally denote arbitrary orthogonal components, unless otherwise explicitly specified). The quantities μ and σ arise in the context of the idealized model of the wind speed probability density function considered in section 4.

3. Results of downscaling predictions

a. Spatial correlation map

As described in the previous section, 850-hPa U, V, T, and W averaged on daily, weekly, and monthly time scales were used as the predictors in this study. The strength and spatial scale of the statistical relationship between surface wind statistics and upper-level large-scale predictors can be assessed through inspection of spatial correlation fields. Correlation fields of each of mean zonal wind and wind speed with , , , and during DJF for one buoy at Atlantic Ocean are displayed in Fig. 1. Both the monthly and daily time scales are displayed. It can be seen that is strongly correlated with on large scales: positive correlations are found locally while negative correlation fields are found to the north. The other predictor fields also show large-scale correlation structure with In contrast, the absolute correlation values for are much smaller than those for , particularly on the monthly time scale. The horizontal scales of strong correlations increase with the averaging time scale: on the daily time scale, the spatial scales of the correlation fields are smaller than those on monthly time scale (Sun 2012). This result is consistent with the fact that on the synoptic scale, the influence exerted on surface winds by large-scale circulation is more local while on longer time scales large-scale teleconnection patterns become more important.

Fig. 1.
Fig. 1.

(left) Correlation maps of mean zonal wind at buoy 41001 with large-scale predictors at 850 hPa on monthly time scales: (top) , (second row) , (third row) , and (bottom) . The position of the buoy is indicated by the white dot. The white boxes in the panels denote the domain used for the EOF decomposition of large-scale predictor fields. (left center) As in (left), but for monthly-mean wind speed. (right center) As in (left), but on a daily time scale. (right) As in (left center), but on a daily time scale.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

b. Combined EOF analysis

It is evident from Fig. 1 that predictive information for surface wind statistics is spatially distributed within individual predictor fields, and that some of this information is common across these different fields. To efficiently distribute the predictor variance among the smallest number of time series, a combined empirical orthogonal function (EOF) analysis following that in Monahan (2012a) is carried out. Predictions on daily, weekly, and monthly time scales are made using 26, 16, and 6 combined PCs, respectively, as predictors. The numbers of predictors chosen for each time scale were selected as the minimum number needed to explain over 85% of the total variance in the four large-scale predictor fields. The prediction results are not sensitive to reasonable changes in the number of predictors included in the model, or to reasonable changes in the EOF domain. To maintain the consistency across different time scales, the EOF domains are the same on daily, weekly, and monthly time scales at a particular site. As the spatial scales of correlation maps are smaller at shorter time scale, more PC predictors are included in the downscaling model on daily time scales than monthly time scales to reproduce the smaller structures of variability. As the number of statistical degrees of freedom is larger on smaller averaging time scales, the SD model can accommodate more predictors without overfitting.

A “leave one year out” cross-validation strategy is employed in the multiple linear regression model to prevent model overfitting. For example, predictions of the first year were determined from a regression model built with data from all other years. The predictions of the second year were then obtained in a similar way (only the second year's data were withheld when estimating the model parameters). When the predictions for all the years were obtained, the r2 value (i.e., square of correlation between predictions and observations) was computed to measure the prediction skill.

c. Predictability of surface wind statistics: A case study of three representative buoys

Statistics of the vector wind components (both means and standard deviations) in 36 directions around the compass, along with the mean and standard deviation of wind speed, were predicted at all buoys on daily, weekly, and monthly time scales. Figure 2 shows the DJF predictive skills (r2) of each of the surface-wind statistics for the monthly time scale at three representative buoys. It is evident that the predictive skills of vector wind components are generally anisotropic, as had been previously noted for land surface winds by van der Kamp et al. (2012) and Culver and Monahan (2013). The speed prediction is isotropic by construction, as wind speed is a scalar quantity. Previous studies have suggested that the maximum prediction skill of vector wind components is aligned with topographic features in mountainous areas (van der Kamp et al. 2012), although vector prediction anisotropy is also observed in regions with little topographic variability (Culver and Monahan 2013). We were unable to determine any dominant factor determining the magnitude or orientation of this anisotropy. For example, for the buoys considered in this study, the maximum prediction skills were aligned both along and across shore. Note also that at buoy 41001 the predictive skill of the best predicted mean vector wind is much better than that of mean wind speed, while at buoy 51002 the mean wind speed is as well predicted as the best predicted mean vector wind component. Buoy 21001 represents an intermediate case.

Fig. 2.
Fig. 2.

Monthly-time-scale DJF r2 prediction skills at three representative buoys. Shown are vector wind means (solid red line) and standard deviations (red dashed line) in 36 directions, the mean wind speed (blue line), and the wind speed standard deviation (dashed blue line). The black circle denotes a reference prediction skill of r2 = 0.8.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

d. Wind statistics predictability distribution

Maps of the DJF prediction skills (correlation r2 values) of the best predicted mean vector wind components (i.e., the vector wind component that has the highest predictive skill among all the components in 36 directions), the mean wind speed , the best predicted standard deviations of vector wind component , and the standard deviation of wind speed σw on monthly time scales at all the 52 buoys are shown in Fig. 3. Several general results follow from these prediction maps.

  1. As was found in Monahan (2012a) and Culver and Monahan (2013), the prediction skills of the best predicted mean vector wind component are generally higher than those of the mean wind speed across all the 52 buoys. There is no general relationship between the predictability of the mean speed and the worst predicted mean vector wind component (not shown).

  2. The buoys which have relatively high prediction skills of mean wind speed are generally located in tropical regions. Through the midlatitudes, the prediction skills of mean wind speed are generally considerably lower. There is no general relationship between the predictability of speed and proximity to land.

  3. The subaveraging time scale standard deviations of both vector wind components and wind speed are generally poorly predicted at all geographic locations.

Corresponding maps for the other calendar seasons and averaging time scales produce results consistent with these general results (Sun 2012).
Fig. 3.
Fig. 3.

Cross-validated DJF r2 predictive skills on the monthly time scale. (top) Best predicted vector wind component; (second row) mean wind speed; (third row) best predicted standard deviation of vector wind component; and (bottom) standard deviation of wind speed.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

These results are also illustrated by scatterplots (across 52 buoys and 4 seasons) of the relative predictability of the vector wind component and wind speed statistics (Fig. 4). Each point in these plots represents the prediction skill of the specified surface-wind statistics in one season at one buoy on the specified time scale. In general, we see that mean quantities are generally better predicted than standard deviations, particularly on shorter averaging time scales. Furthermore, the best predicted vector wind component is almost always better predicted than the mean wind speed. To investigate the relative predictability of the statistics of vector wind components and wind speed, we now turn to an idealized model of the wind speed probability distribution.

Fig. 4.
Fig. 4.

(top) The prediction skills (cross-validated r2) of the standard deviations of wind speed relative to those of the mean wind speed, (middle) the best predicted standard deviations of vector wind components relative to the best predicted means of vector wind components, and (bottom) the mean wind speed relative to the best predicted means of vector wind components. (left) The daily time scale predictions, (center) the weekly time scale predictions, and (right) the monthly time scale predictions.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

4. Interpretation of the relative predictability of vector wind and wind speed statistics

a. A Gaussian model of the vector wind probability density function

The results of the previous section showed that the prediction skills of mean wind speed are generally smaller than those of the best predicted vector wind components. This result was also obtained for sea surface winds in the northeast subarctic Pacific (Monahan, 2012a) and for land surface winds across Canada (van der Kamp et al. 2012; Culver and Monahan 2013). Monahan (2012a) introduced an idealized probability model of the wind speed probability distribution to investigate the reason for these differences in predictability. Assuming that fluctuations in the vector winds are isotropic, uncorrelated, and Gaussian, Monahan (2012a) showed that the mean wind speed can be modeled as a function of the magnitude of the mean vector wind and the isotropic standard deviation :
e1
where and (σu, συ) are the mean and standard deviation of orthogonal vector wind components. The expression for F is as follows (Rice 1945):
e2
where Ij(x) is the associated Bessel function of the first kind of order j. It should be pointed out that no explicit assumptions are made about the temporal autocorrelation structure of the wind components. To assess the performance of this model, we compared the IPM modeled and the actual on monthly time scales using 10-m surface wind data from NCEP/DOE Reanalysis 2. We have also assessed the IPM's performance with buoy data; the results are consistent with those obtained with the reanalysis data (Sun 2012). For each of the four calendar seasons, the following calculations were carried out:
  1. at each grid point and for each month in the record, we computed μ and σ from monthly means and standard deviations of the 10-m zonal wind and meridional wind;

  2. these values of μ and σ are used to compute monthly using the IPM; and

  3. we calculated the correlation between the modeled monthly from the IPM and the monthly computed directly from NCEP/DOE Reanalysis 2 data. The square of the correlation (r2), which describes the fraction of variance held in common between the two time series, provides a linear measure of the model performance in modeling mean wind speed.

It was found (not shown) that the modeled mean wind speed from the IPM has a high correlation with the mean wind speed from the reanalysis data on a global scale (r2 above 0.9 at all grid points). The mean wind speed model derived from the IPM is demonstrated to work well across the global ocean. In later section, we will demonstrate that this model is less successful in modeling month-to-month variations of the submonthly wind speed standard deviation.

b. Sensitivity of to μ and σ

Having provided evidence that the IPM is able to characterize the variability of in terms of the variability of μ and σ, we can use this model to investigate the sensitivity of to changes in these vector wind statistics. While is a function of μ and σ, the sensitivities of the mean wind speed to μ and σ are functions of the ratio alone:
e3
e4
For convenience we can define the bounded scalar quantity θ (Monahan 2012a; Culver and Monahan 2013):
e5
The sensitivities of to μ and σ as functions of θ can be computed numerically and are shown in Fig. 5a.
Fig. 5.
Fig. 5.

(a) Sensitivity of to μ and σ as functions of θ and (b) sensitivity of σw to μ and σ as functions of θ.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

From the sensitivity plot, it is clear that in the low θ regime, is more sensitive to the variability of σ than of μ. In contrast, in the high θ regime is more sensitive to the variability of μ. For intermediate values of θ, has similar sensitivity to both σ and μ. This result indicates that in a high θ regime, variability of is determined by variability of μ, irrespective of the variability of σ. In contrast, in a low θ regime, variations in are determined by those of σ and are insensitive to changes of μ. These different regimes can be illustrated by considering the skill of modeled by the IPM allowing μ to vary from month to month at each point while holding σ constant at its climatological value. The results of this calculation for DJF are displayed in Fig. 6a. Consistent results are obtained for other seasons. For comparison, we also calculated monthly time scale θ values at each grid point using the monthly time scale μ and σ, and then averaged these across all months to produce one climatological θ field (Fig. 6b). These maps demonstrate the following:

  1. The θ regimes are geographically organized. In general, high θ values are found in the tropical Pacific and Atlantic as well as the Indian Ocean. The low θ regime is in the subtropical and subpolar latitudes. Intermediate θ values are found predominantly in the midlatitudes.

  2. The model with fixed σ represents month-to-month variations in well in some regions (e.g., the tropics) and poorly elsewhere (e.g., subpolar and subtropical regions).

  3. The two maps in Figs. 6a and 6b match closely. They clearly indicate that where high θ dominates, the model with fixed σ can successfully represent monthly-time-scale variability of , while where low θ prevails, the model with fixed σ cannot accurately characterize month-to-month variations in . This result is consistent with the sensitivity plots in Fig. 5a: for low θ regimes, computed from the IPM with fixed climatological σ is not very accurate because variations of are more sensitive to those of σ in this regime. In contrast, in the high θ regimes, the performance of the model with fixed σ remains good as is primarily dependent on μ. Note that the maps in Figs. 6a and 6b are similar but not identical, because there is not expected to be an exactly linear relationship between θ and the modeling skill r2.

The results of this analysis suggest that the scalar quantity θ is a good measure of the dependence of on μ and σ for observed sea surface winds.
Fig. 6.
Fig. 6.

(a) Modeling skill of DJF mean wind speed by the IPM [Eq. (1)] with month-to-month variations in μ but σ held constant (at its long-term average value). (b) Climatological DJF θ distributions on monthly time scale, with positions of all 52 buoys. (c) As in (b), but on a weekly time scale. (d) As in (b), but on a daily time scale. Note that the color bar for (a) is between 0 and 1, while for (b)–(d) it is between 0 and 1.5.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

Maps of the DJF θ distribution on each of the three averaging time scales (daily, weekly, and monthly) with the locations of all the 52 buoys superimposed are presented in Figs. 6b–d. It can be seen that θ generally decreases as averaging time scales increase: on daily time scales, the mid-to-high θ regime (above 0.9) dominates all the buoys. On weekly time scale, an intermediate θ regime (0.5–0.9) appears on the flanks of the surface westerlies. On the monthly time scale, the low θ regime appears in subtropical and subpolar latitudes. As the variability of extratropical sea surface winds is strongest on the synoptic time scale of several days, subdaily variability is smaller than the subweekly or submonthly variability. In consequence, σ is much smaller than μ on the daily time scales, resulting in a broadly distributed high θ regime. It is noteworthy that in the tropical Pacific and Atlantic, a high θ regime generally dominates most buoys on daily, weekly, and monthly time scales. In the tropics the major forms of variability, such as the Madden–Julian oscillation (MJO), have time scales much longer than those of midlatitude synoptic eddies. The relatively steady tropical trade winds result in μ values that are much larger than σ, resulting in a high θ regime. The θ field also displays seasonal variations (Sun 2012).

The distribution of the buoys considered provides good coverage of the full θ range on weekly and monthly time scales. In consequence, we have a representative sample of buoys within each of the three θ regimes with which to establish statistical relationships.

c. Predictability of relative to μ and σ

The results of the previous section suggest that the predictability of relative to that of μ and σ is a function of θ. Scatterplots of SD predictive skills (correlation r2 values) of relative to those of μ and σ are shown in Fig. 7 for daily, weekly, and monthly averaging time scales. Corresponding values of θ are indicated by color. The SD predictions of each wind statistic on each of three averaging time scales were done separately for each of the four calendar seasons and each of the 52 buoys (resulting in a total of 208 points in each plot). The following are observed: 1) On the daily time scale, for which θ is consistently large, the predictive skill of is strongly correlated with the predictive skill of μ across all stations and seasons. In contrast, the predictive skill of has no strong relationship with that of σ (except for the smallest values of θ). 2) On weekly and monthly time scales, the points in the scatterplot of r2(μ) with gather around the 1:1 line for high θ values, while the points are more broadly scattered for low θ values. In contrast, in the scatterplot of r2(σ) with , the data points gather around the 1:1 line for low θ values and scatter away from the 1:1 line for high θ values.

Fig. 7.
Fig. 7.

The correlation-based predictive skill of relative to that of (left) μ and (right) σ on (top) daily, (middle) weekly, and (bottom) monthly averaging time scales. The color of the data points denotes the value of θ. One-to-one lines are given in solid blue.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

The above results indicate that θ is a good metric for characterizing the statistical predictability of relative to that of μ and σ. In the high θ regime, good predictions of require accurate predictions of μ. On the other hand, in low θ regimes, how well can be predicted depends on the predictability of σ. In a medium θ regime, the situation is complicated as has comparably strong and nonlinear dependence on μ and σ. While these results have been demonstrated using a linear statistical downscaling model, the sensitivity results of the IPM suggest that they should hold irrespective of how the predictions are made.

d. Predictability of relative to that of the best predicted vector wind component

Having related the predictability of to that of μ and σ, we are on the way toward understanding the predictability of relative to the vector wind components . In the low θ regime, the predictive skill of is determined by that of σ. Predictions of the isotropic standard deviation σ are not better than ; it follows from Fig. 4 that σ will generally not be as well predicted as the . It follows that in the low θ regime, in which variations are dominated by those of σ, the predictability of should be less than that of the best predicted mean vector wind component. On the other hand, in the high θ regime, the predictive skill of is determined by that of μ. To complete the connection between the predictability of speeds and of vector wind components, we need to relate the predictability of the amplitude of the mean vector wind μ to the predictability of the vector wind components themselves. Culver and Monahan (2013) provided the following theoretical expression for the linear predictability of μ relative to that of the vector wind component aligned along the long-term mean wind:
e6
where and r2(μ) are the (correlation-based) predictabilities of the wind component along the long-term mean and of μ by a single predictor x (assumed to have a Gaussian distribution), respectively. The result in Eq. (6) is based on the assumption that variations of (e.g., from month to month) are isotropic, uncorrelated, and Gaussian. The quantities and are the mean and standard deviation of over the entire record, respectively. For convenience, we introduce the quantity , which gives us
e7

From Eq. (7) it is clear that the predictability of μ is bounded above by that of the along-mean wind component. Furthermore, will itself be bounded above by the predictability of the best predicted component (by definition). Figure 8 displays the predictive skill of μ relative to that of in high θ regimes (θ ≥ 1) in relation to γ on daily, weekly, and monthly time scales. It can be seen that in general, when γ ≫ 1, the predictability of μ approaches that of . On the other hand, when γ decreases, the predictive skill of μ becomes smaller than that of . As discussed in the previous section, a majority of the buoys are in a high θ regime on daily time scales across all seasons and locations, while on weekly and monthly time scales, many buoys are in a medium or low θ regime. As a result, fewer data points with θ > 1 are displayed for the plots on weekly and monthly time scales.

Fig. 8.
Fig. 8.

The predictive skill of μ relative to that of the along-mean vector wind component in high θ regimes (θ ≥ 1) in relationship to γ [Eq. (7); as indicated by the color of the data points].

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

It should be emphasized that γ increases with averaging time scales. In Fig. 8 it can be observed that the upper limit of γ increases from 6 to 10 as the averaging time scale increases from daily to monthly. On longer averaging time scales, a smaller fraction of the variance of the wind is retained in variations of , and a larger fraction is contained in subaveraging time scale variability (i.e., σ). For an increasingly large fraction of stations and seasons, as the averaging time scale is increased the variability in the averaged vector wind becomes much smaller than the climatological mean wind [i.e., ].

In general, shorter averaging time scales are associated with larger values of θ and smaller values of γ, while longer averaging time scales with smaller θ and larger γ. Predictability of is generally smaller than that of the vector wind components on long averaging time scales because of lower θ values (and the fact that σ has weak predictability). In contrast, predictability of can be limited on short averaging time scales when μ is more poorly predicted than because of the small value of γ. It is possible that there may be an optimal averaging time scale on which the relationship among , μ, and is balanced in such a way as to yield optimal predictability of relative to .

e. Prediction of subaveraging time scale standard deviation of wind speed with the IPM

The IPM introduced above has been shown to be able to successfully model in terms of μ and σ. An analytic expression for the standard deviation of wind speed σw can also be derived from the IPM. By definition,
e8
from which it follows that
e9
From Eq. (9), we have an exact expression for the wind speed standard deviation σw in terms of μ and σ.

As with the modeled , the sensitivities ∂μσw and ∂σσw are functions of the scalar variable θ, as illustrated in Fig. 5b. In contrast to modeled , the modeled σw is most sensitive to variations in σ over the entire θ range. In the same way that we assessed the ability of the IPM to represent observed variability of in terms of variability in μ and σ, we now consider a similar calculation to test the performance of Eq. (9) in capturing observed the month-to-month variability in the submonthly standard deviation of wind speed:

  1. at each grid point, we calculated monthly μ and σ from monthly means and standard deviations of 10-m NCEP/DOE Reanalysis 2 zonal wind and meridional wind;

  2. from these, we computed monthly σw using Eq. (9); and

  3. we then calculated the correlation between the modeled month-to-month variations in σw from Eq. (9), and those directly computed from NCEP/DOE Reanalysis 2 data [the squared value of the correlation (r2) is used to assess the model performance in modeling the month-to-month changes in the submonthly standard deviation of wind speed].

The results of this calculation for DJF are presented in Fig. 9a. In contrast to the highly accurate representation of by the IPM on a global scale, the model fails to represent month-to-month changes in σw over most of the ocean. It can be seen that in the midlatitude and high-latitude regions (of both the Northern and Southern Hemispheres), the r2 prediction skill of the model is generally below 0.5. In the tropical regions, the IPM generally performs better although its performance is still poor in a number of places. The failure in reproducing month-to-month variations of the submonthly standard deviation of wind speed from Eq. (9) is perplexing: we should be able to obtain knowledge of these statistics, as long as we have the correct wind speed probability density function pw(w). The fact that we can simulate but not σw with our model leads us to reexamine the three assumptions on which the IPM is based: that the vector wind fluctuations are Gaussian, uncorrelated, and isotropic. While these approximations are reasonable for modeling first-order statistics (mean wind speed), they may not be good approximations for modeling the second-order statistics. Monahan (2006) demonstrated that the along- and across-wind components u and υ are close to being uncorrelated and have nearly isotropic fluctuations on a global scale (with some exceptions in monsoon and ITCZ regions). However, the skewness and kurtosis of the along-mean vector wind components can differ substantially from zero (Monahan 2006). Therefore, we will investigate the influence of the non-Gaussianity of vector wind components on modeling the standard deviation of wind speed.

Fig. 9.
Fig. 9.

(a) The modeling skill of DJF submonthly time scale σw from the IPM [Eq. (9)]. (b) As in (a), but with the non-Gaussian vector wind model obtained from Eq. (11).

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

1) Wind speed PDF from non-Gaussian vector winds

We first decompose the vector winds into components along and across the time-mean vector wind. In this section, these will be denoted by u and υ respectively. Following Monahan (2006), non-Gaussian surface wind components are included in the model through a Gram–Charlier expansion (Johnson et al. 1994) of the probability density function (PDF) of the along-mean wind component as follows:
e10
where He3(x) = x3 − 3x, He4(x) = x4 − 6x2 + 3, (the monthly along-mean wind skewness), and (the monthly along-mean wind kurtosis). The cross-mean wind component is modeled as Gaussian (as is broadly consistent with observations; Monahan 2006, 2007). Note that pu(u) defined in this way is not strictly nonnegative, and therefore is not necessarily a proper PDF. Nevertheless, the resulting function has the correct moments and is a useful model of the PDF of u so long as realizations of this random variable are not required (e.g., Johnson et al. 1994). A more complicated expression for the wind speed PDF can then be obtained:
e11
where Ij is the associated Bessel function of the first kind of order j (Monahan 2006). As with previous analyses, we use this model to simulate month-to-month variations in σw given observed month-to-month variations in , σ, , and . The performance in capturing σw is slightly improved by including month-to-month changes in the skewness and kurtosis of u, but model performance remains much poorer than for at most locations. (Fig. 9b). Evidently, non-Gaussianity of the vector wind components is not the primary cause degrading the IPM's performance in characterizing variability in σw. Why should it be the case that the model performs well for modeling monthly-mean wind speed but largely fails when modeling submonthly wind speed standard deviation? As we will now show, a contributing factor is related to differences in sampling variability of these statistics.

2) Sampling variability in month-to-month fluctuations of μ, σ, , and σw

Having demonstrated that the model assumption of Gaussian-distributed vector winds is not the primary cause of the difficulty in modeling month-to-month variations in σw, we now ask the question: might the poor simulation of σw result from different sampling variability of μ, σ, and σw? At any location, within each month, the wind fluctuations have on the order of 15–30 statistical degrees of freedom as the surface winds generally have an autocorrelation time scale on the order of one to two days (Monahan 2012b). As a result, there will be nonnegligible sampling variability in all surface wind statistics, which may be different from one statistic to another. To assess the influence of sampling variability, a series of Monte Carlo experiments were conducted to examine how the potential sampling errors in μ and σ influence the Gaussian model's performance in modeling and σw.

By construction in these idealized calculations the vector winds were Gaussian, uncorrelated, and isotropic so the wind speed population statistics are exactly related to those of the vector winds by Eqs. (1) and (9). The strength of this analysis is that it is a “perfect model” calculation—we know exactly what is true about the underlying relationship between the statistics of vector winds and wind speed, and within this can investigate the role of sampling variability.

In our first experiment, we let
e12
e13
where δ1 and δ2 are random numbers with a uniform distribution on (−½, ½). We will interpret μ0 and σ0 as the climatological mean and standard deviation of the vector winds, while the random numbers δ1 and δ2 describe month-to-month fluctuations in these statistics with strengths scaled by rm and rs, respectively. Note that fluctuations of and s represent true month-to-month variability in the vector wind statistics. That is, these are the signal that we are interested in capturing with our model.

We then generated N = 120 realizations of and s each representing a separate month. Within each month, we randomly sampled M days (the number of independent wind realizations within each month) of vector wind components (u, υ) from the Gaussian model with the mean and isotropic standard deviation s for that month. For each month, we computed the sample mean and standard deviation of wind speed from the sample u and υ, as well as the sample μ and σ. The sample μ and σ were then used to compute the and σw from the Gaussian model using Eqs. (1) and (9). We then correlated the observed and modeled wind speed moments, to characterize the modeling skills of and σw. This procedure was repeated for different values of rm and rs, with μ ranging from 1 to 10 m s−1 and σ kept fixed at 3 m s−1 to assess sampling variability under different θ regimes. An ensemble of 300 estimates of the modeling skill was computed.

We consider both M = 500 and M = 20. The second of these is closer to the real number of statistical degrees of freedom within any month, while the first is considered to illustrate how sampling variability changes as sample size increases. The modeling skills of and σw are plotted as functions of θ in Fig. 10, from which the following can be observed:

  1. The modeling skill of is generally high with little sensitivity to the values of rm and rs. Consistent with the results presented earlier, the model is able to reproduce month-to-month variability in for different sizes of the true signal strength.

  2. The modeling skill of σw can be quite poor for small values of rs (the true month-to-month variability of σ). When rs = 0, the modeling skill of σw is substantially poorer than that with rs = 0.45. For instance, for M = 500, when rm = 0.45, rs = 0 (Fig. 10a), and the r2 modeling skill of σw is about 0.6. In contrast, when rm = 0, rs = 0.45 (Fig. 10c), and the modeling skill of σw is close to 0.95. The value of rm (the true month-to-month variability in μ) does not substantially influence the modeling skill of σw (not shown).

  3. The modeling skill of σw increases with M. Under the same set of rm and rs values, the modeling skill of σw is better with M = 500 than with M = 20. For M = 20, the modeling skill of σw is low even when rs is relatively large.

Fig. 10.
Fig. 10.

Monte Carlo experiment derived modeling skills r2 of and σw by the Gaussian models [Eqs. (1) and (9)] for different values of rs and rm, and for (left) M = 500 and (right) M = 20.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

When rs = 0, the population vector wind standard deviation remains the same from month to month so all fluctuations of s are produced by sampling fluctuations alone. In this case fluctuations in σw modeled by Eq. (9) differ significantly from the true fluctuations in the wind speed submonthly standard deviation. As rs increases, the real month-to-month fluctuations of s (the signal) increase in size relative to those of the sampling fluctuations (the noise). Thus, the signal-to-noise ratio (SNR) increases, and the model does a better job of simulating month-to-month changes in σw.

Similarly it is observed that the modeling skill of σw increases as M increases for specified rs and rm values. By increasing M, while the signal stays the same, the noise is reduced so the SNR increases and model performance is better. For small values of the SNR, the IPM has difficulties modeling the month-to-month variability in the wind speed standard deviation even when the observed vector wind components are Gaussian, uncorrelated, and isotropic without approximation.

This analysis demonstrates that the skill of the IPM in simulating month-to-month variations of σw is determined by a SNR that is related to the size of the true month-to-month fluctuations of σ (characterized by rs) and the number of independent wind realizations M within the month. Sampling fluctuations in σw are distinct from those of μ and σ, so variability in σw is only well predicted when the signal of true month-to-month variability is sufficiently large relative to the sampling noise. We will now develop a quantitative measure of the SNR and use this to interpret the modeling results of Fig. 9a.

This first two steps of this analysis are similar to those of the previous Monte Carlo experiment, sampling a broader range of values of rs and M (rs ranges from 0 to 1.5; rm is set to 0.3; and M ranges from 1 to 1000). We define the signal-to-noise ratio as
e14
where , is the ensemble mean value of the sampled vector wind standard deviation over all 120 months, and is the corresponding standard deviation. As discussed above, it is expected that the size of the variability in will depend on the true signal strength rs and the number of degrees of freedom M. The SNR defined by Eq. (14) characterizes the month-to-month fluctuation of σ (the signal) in the dataset relative to the sampling variability (noise) given by . The SNR can be computed from Monte Carlo simulations and compared with the modeling skill of σw.

Consistent with the qualitative analysis described earlier, the modeling skill of σw is determined by the SNR as shown in Fig. 11a: as the SNR increases, σw is better modeled. To obtain a model r2 skill better than 0.9, the SNR has to achieve a value above 3. The relationship among M, rs, and SNR is illustrated in Fig. 11b. Consistent with the previous analysis, the signal-to-noise ratio increases with both M and rs. When rs = 0.2, the number of independent realizations M has to exceed 1000 to get a signal-to-noise ratio of 3. When rs = 0.9, a SNR of 3 can be obtained with M below 50.

Fig. 11.
Fig. 11.

(a) The r2 modeling skill of σw as a function of the signal-to-noise ratio [Eq. (14)] and (b) the SNR as a function of rs and M.

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

These results indicate that for real sea surface wind data with a typical value of M smaller than 30, the month-to-month fluctuation of σ has to be relatively large to result in a sufficiently high SNR, to obtain a good modeling skill of σw with the IPM. We will now estimate the SNR from the NCEP/DOE Reanalysis 2 surface wind dataset and compare it with the distribution of modeling skill of σw previously shown in Fig. 9a. To estimate SNR, we need to have the estimates of submonthly and the value of M at each grid point.

The number of independent wind realizations within a month can be estimated as follows:
e15
where N is the duration of a month and Te is the autocorrelation time scale. In computing Te, the autocorrelation function of the vector wind components was modeled as a decaying exponential:
e16
where t is the time lag. The value of Te was then obtained from the observed autocorrelation function by linear regression. In fact, the autocorrelation structures are not generally exponential and in many locations over the ocean the vector wind autocorrelation structure is anisotropic (Monahan 2012b). For this calculation, Te was estimated from the zonal wind component, for which the autocorrelation time scale is generally the largest.

Using the estimated value of M, we calculated values of at each grid point from the Monte Carlo simulation. The value of was estimated from the observed month-to-month values in the standard deviation of the vector winds. From these, the field of SNR was computed (Fig. 12). Comparison of the SNR map with that of the IPM modeling skill of σw (Fig. 9a) demonstrates that the agreement between these two fields is generally good. In the tropical regions, SNR is generally high. Correspondingly, in Fig. 9a, the modeling skill of σw in these regions is relatively good. In contrast, extratropical regions have smaller SNR values, which correspond with the poor modeling skill of σw found in these regions. Although the two maps do not match perfectly, their high degree of correspondence indicates that the Gaussian model's performance in modeling σw is strongly related to sampling variability as measured by the SNR given by Eq. (14). It follows that while the IPM provides a useful tool for describing variability of in terms of that of μ and σ, it will not generally be useful for doing so with σw, and presumably for other wind speed statistics comparably sensitive to sampling fluctuations.

Fig. 12.
Fig. 12.

Spatial distribution of the monthly-time scale DJF SNR as defined by Eq. (14).

Citation: Journal of Climate 26, 20; 10.1175/JCLI-D-12-00722.1

5. Summary of results

This study has investigated the predictability of local sea surface wind statistics from those of large-scale free-tropospheric flow fields. A statistical downscaling (SD) model based on multiple linear regression was used to predict the means and standard deviations of observed vector wind components and wind speed at 52 ocean buoys on daily, weekly, and monthly time scales. A summary of our general results is as follows:

  1. The predictive skill of the best predicted mean vector wind component is generally higher than that of the mean wind speed. Furthermore, the mean quantities are generally better predicted than the subaveraging time scale standard deviations for both vector wind components and wind speed.

  2. An idealized model of the wind speed probability density function introduced in Monahan (2012a) was used to investigate the relationship between and the statistics of vector wind components. This model indicates that the predictability of relative to the magnitude of the mean vector wind μ and the vector wind standard deviation σ can be characterized by the scalar quantity , which is dependent on season, geographic location, and averaging time scale. The quantity θ characterizes the local wind climate: specifically, whether the local vector winds are sustained or highly variable. Consistent with the results of the IPM, the predictability of the observed was found to be determined by that of σ for low values of θ and by μ for high values of θ.

  3. The subaveraging time scale variability σ was generally found to be poorly predicted by large-scale predictors. Therefore, in the low θ regimes, the predictive skill of (which is determined by that of σ) is generally lower than that of the best predicted vector wind component. On the other hand, in the high θ regimes, the predictive skill of is determined by that of μ. The predictive skill of μ relative to that of the best predicted vector wind is determined by the quantity . When γ ≫ 1, the predictive skill of μ relative to that of the vector winds is greatest. For smaller values of γ, the predictive skill of μ is much lower than that of the vector wind components. Correspondingly, the predictive skill of is bounded above by that of the best predicted vector wind component, and can be much lower, even in high θ regimes.

  4. The IPM generally fails to capture month-to-month variations of the subaveraging time scale standard deviation of wind speed, σw, in terms of variations in μ and σ. With a series of Monte Carlo experiments, we demonstrated that this can be understood to be a result of differences in sampling variability between the vector wind statistics and the wind speed statistics. The amplitude of the real month-to-month fluctuation in the vector wind standard deviation relative to that associated with sampling variability (characterized by a signal-to-noise ratio) accounts for the mismatch between modeled and real σw.

The first three of these conclusions are consistent with earlier results by Monahan (2012a) and Culver and Monahan (2013). However, these earlier studies considered the SD predictive skill of surface winds in a much more limited range of θ values. The θ range considered in the present study allows the conclusions of these earlier studies to be generalized to a much broader range of wind climates. The focus of this study has been the features of wind speed and vector wind predictability that are common across many locations and seasons. In consequence, no detailed analysis of predictability at any individual buoy has been carried out. Such detailed analyses—such as of the buoy in the Gulf of Mexico for which the standard deviations are considerably better predicted than the means (Fig. 3)—represent an interesting direction of future study.

In the idealized setting of the Monte Carlo experiments, in which the population statistics of the vector wind are specified, the IPM fails to model variations of the monthly standard deviation of wind speed despite performing well in modeling monthly means of wind speed. It can be expected that month-to-month variations of other higher-order statistics (e.g., 95th percentile, skewness, and kurtosis) of wind speed would not be well modeled by the IPM either. The failure of the more general wind speed PDF model derived from non-Gaussian vector winds to represent variations in σw indicates that this difficulty will persist irrespective of how the PDF of vector winds is modeled. As well, this difficulty cannot be circumvented by considering longer averaging time scales. On longer time scales, while the number of statistical degrees of freedom M will increase, the true variability of the subaveraging time scale standard deviation (the signal) will decrease. In general, we cannot expect that the IPM will do a better job modeling changes in the higher-order statistics of w on seasonal or annual averaging time scales. It follows that the general approach of using variations in the vector wind statistics to model variations in these higher-order wind speed statistics will be compromised by this strong sensitivity to differences in sampling variability.

The results of this study demonstrate that the direct SD predictive skill of mean sea surface wind speeds is generally low outside of the tropics. The potential exists that the SD prediction skills for the quantities that have been considered in this study could be improved by considering other sets of predictors or other SD techniques. A more detailed investigation of alternative SD approaches is an interesting direction of future study. Furthermore, previous studies have shown that the anisotropy in the predictability of land surface winds can be related to topographic features (van der Kamp et al. 2012; Salameh et al. 2009) although this is not always the case (Culver and Monahan 2013). Unlike the land surface, the sea surface is more homogeneous, and those heterogeneities that are present (such as sea surface temperature fronts) tend not to be fixed in place. The control on the strength of this anisotropy, and the orientation of the best predicted vector wind, are not well understood. A detailed examination of the anisotropy in the predictability of sea surface winds is another interesting direction of future study.

Acknowledgments

The authors would like to thank Aaron Culver, Andrew J. Weaver, Bill Merryfield, and Julie Zhou for their comments, as well as those of three anonymous reviewers. This work was funded by the Natural Sciences and Research Council of Canada's Collaborative Research and Training Experience Program in Interdisciplinary Climate Science.

REFERENCES

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  • Capps, S. B., and C. S. Zender, 2009: Global ocean wind power sensitivity to surface layer stability. Geophys. Res. Lett., 36, L09801, doi:10.1029/2008GL037063.

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  • Culver, A. M. R., and A. H. Monahan, 2013: The statistical predictability of surface winds over western and central Canada. J. Climate,in press.

  • Donelan, M., W. Drennan, E. Saltzman, and R. Wanninkhof, Eds., 2002: Gas Transfer at Water Surfaces. Geophys. Mongr., Vol. 127, Amer. Geophys. Union, 383 pp.

  • Garratt, J., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Johnson, N., S. Kotz, and N. Balakrishnan, 1994: Continuous Univariate Distributions. Vol. 1. Wiley, 756 pp.

  • Jones, I. S., and Y. Toba, Eds., 2001: Wind Stress over the Ocean. Cambridge University Press, 307 pp.

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  • Monahan, A. H., 2006: The probability distribution of sea surface wind speeds. Part I: Theory and SeaWinds observations. J. Climate, 19, 497520.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind speeds. J. Climate, 20, 57985814.

  • Monahan, A. H., 2012a: Can we see the wind? Statistical downscaling of historical sea surface winds in the subarctic northeast Pacific. J. Climate, 25, 15111528.

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  • Monahan, A. H., 2012b: The temporal autocorrelation structure of sea surface winds. J. Climate,25, 6684–6700.

  • Rice, S.O., 1945: Mathematical analysis of random noise (part 2). Bell Syst. Tech. J., 24, 46156.

  • Salameh, T., P. Drobinski, M. Vrac, and P. Naveau, 2009: Statistical downscaling of near-surface wind over complex terrain in southern France. Meteor. Atmos. Phys., 103, 253265, doi:10.1007/s00703-008-0330-7.

    • Search Google Scholar
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  • Sampe, T., and S.-P. Xie, 2007: Mapping high sea winds from space: A global climatology. Bull. Amer. Meteor. Soc., 88, 1965–1978.

  • Sun, C., 2012: Statistical downscaling of sea surface winds over the global oceans. M.S. thesis, School of Earth and Ocean Sciences, University of Victoria, 111 pp.

  • van der Kamp, D., C. Curry, and A. Monahan, 2012: Statistical downscaling of historical monthly mean winds over a coastal region of complex terrain. II: Predicting wind components. Climate Dyn., 38, 13011311.

    • Search Google Scholar
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Save
  • Bates, N. R., and L. Merlivat, 2001: The influence of short-term wind variability on air–sea CO2 exchange. Geophys. Res. Lett., 28, 32813284.

    • Search Google Scholar
    • Export Citation
  • Capps, S. B., and C. S. Zender, 2009: Global ocean wind power sensitivity to surface layer stability. Geophys. Res. Lett., 36, L09801, doi:10.1029/2008GL037063.

    • Search Google Scholar
    • Export Citation
  • Culver, A. M. R., and A. H. Monahan, 2013: The statistical predictability of surface winds over western and central Canada. J. Climate,in press.

  • Donelan, M., W. Drennan, E. Saltzman, and R. Wanninkhof, Eds., 2002: Gas Transfer at Water Surfaces. Geophys. Mongr., Vol. 127, Amer. Geophys. Union, 383 pp.

  • Garratt, J., 1992: The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Johnson, N., S. Kotz, and N. Balakrishnan, 1994: Continuous Univariate Distributions. Vol. 1. Wiley, 756 pp.

  • Jones, I. S., and Y. Toba, Eds., 2001: Wind Stress over the Ocean. Cambridge University Press, 307 pp.

  • Liu, W. T., W. Tang, and X. Xie, 2008: Wind power distribution over the ocean. Geophys. Res. Lett., 35, L13808, doi:10.1029/2008GL034172.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2006: The probability distribution of sea surface wind speeds. Part I: Theory and SeaWinds observations. J. Climate, 19, 497520.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2007: Empirical models of the probability distribution of sea surface wind speeds. J. Climate, 20, 57985814.

  • Monahan, A. H., 2012a: Can we see the wind? Statistical downscaling of historical sea surface winds in the subarctic northeast Pacific. J. Climate, 25, 15111528.

    • Search Google Scholar
    • Export Citation
  • Monahan, A. H., 2012b: The temporal autocorrelation structure of sea surface winds. J. Climate,25, 6684–6700.

  • Rice, S.O., 1945: Mathematical analysis of random noise (part 2). Bell Syst. Tech. J., 24, 46156.

  • Salameh, T., P. Drobinski, M. Vrac, and P. Naveau, 2009: Statistical downscaling of near-surface wind over complex terrain in southern France. Meteor. Atmos. Phys., 103, 253265, doi:10.1007/s00703-008-0330-7.

    • Search Google Scholar
    • Export Citation
  • Sampe, T., and S.-P. Xie, 2007: Mapping high sea winds from space: A global climatology. Bull. Amer. Meteor. Soc., 88, 1965–1978.

  • Sun, C., 2012: Statistical downscaling of sea surface winds over the global oceans. M.S. thesis, School of Earth and Ocean Sciences, University of Victoria, 111 pp.

  • van der Kamp, D., C. Curry, and A. Monahan, 2012: Statistical downscaling of historical monthly mean winds over a coastal region of complex terrain. II: Predicting wind components. Climate Dyn., 38, 13011311.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    (left) Correlation maps of mean zonal wind at buoy 41001 with large-scale predictors at 850 hPa on monthly time scales: (top) , (second row) , (third row) , and (bottom) . The position of the buoy is indicated by the white dot. The white boxes in the panels denote the domain used for the EOF decomposition of large-scale predictor fields. (left center) As in (left), but for monthly-mean wind speed. (right center) As in (left), but on a daily time scale. (right) As in (left center), but on a daily time scale.

  • Fig. 2.

    Monthly-time-scale DJF r2 prediction skills at three representative buoys. Shown are vector wind means (solid red line) and standard deviations (red dashed line) in 36 directions, the mean wind speed (blue line), and the wind speed standard deviation (dashed blue line). The black circle denotes a reference prediction skill of r2 = 0.8.

  • Fig. 3.

    Cross-validated DJF r2 predictive skills on the monthly time scale. (top) Best predicted vector wind component; (second row) mean wind speed; (third row) best predicted standard deviation of vector wind component; and (bottom) standard deviation of wind speed.

  • Fig. 4.

    (top) The prediction skills (cross-validated r2) of the standard deviations of wind speed relative to those of the mean wind speed, (middle) the best predicted standard deviations of vector wind components relative to the best predicted means of vector wind components, and (bottom) the mean wind speed relative to the best predicted means of vector wind components. (left) The daily time scale predictions, (center) the weekly time scale predictions, and (right) the monthly time scale predictions.

  • Fig. 5.

    (a) Sensitivity of to μ and σ as functions of θ and (b) sensitivity of σw to μ and σ as functions of θ.

  • Fig. 6.

    (a) Modeling skill of DJF mean wind speed by the IPM [Eq. (1)] with month-to-month variations in μ but σ held constant (at its long-term average value). (b) Climatological DJF θ distributions on monthly time scale, with positions of all 52 buoys. (c) As in (b), but on a weekly time scale. (d) As in (b), but on a daily time scale. Note that the color bar for (a) is between 0 and 1, while for (b)–(d) it is between 0 and 1.5.

  • Fig. 7.

    The correlation-based predictive skill of relative to that of (left) μ and (right) σ on (top) daily, (middle) weekly, and (bottom) monthly averaging time scales. The color of the data points denotes the value of θ. One-to-one lines are given in solid blue.

  • Fig. 8.

    The predictive skill of μ relative to that of the along-mean vector wind component in high θ regimes (θ ≥ 1) in relationship to γ [Eq. (7); as indicated by the color of the data points].

  • Fig. 9.

    (a) The modeling skill of DJF submonthly time scale σw from the IPM [Eq. (9)]. (b) As in (a), but with the non-Gaussian vector wind model obtained from Eq. (11).

  • Fig. 10.

    Monte Carlo experiment derived modeling skills r2 of and σw by the Gaussian models [Eqs. (1) and (9)] for different values of rs and rm, and for (left) M = 500 and (right) M = 20.

  • Fig. 11.

    (a) The r2 modeling skill of σw as a function of the signal-to-noise ratio [Eq. (14)] and (b) the SNR as a function of rs and M.

  • Fig. 12.

    Spatial distribution of the monthly-time scale DJF SNR as defined by Eq. (14).

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