Statistical Downscaling Prediction of Sea Surface Winds

Statistical Downscaling Prediction of Sea Surface Winds over the Global Ocean

CANGJIE SUN AND ADAM H. MONAHAN

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

(Manuscript received 11 October 2012, in final form 7 March 2013)

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, andmonthly 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 andAtlantic, 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 de-

viation of speed are not well modeled. A series of Monte Carlo experiments demonstrates that the in-

consistency in the characterization of wind speed standard deviation is the result of differences of sampling

variability between the vector wind and wind speed statistics.

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 sig-

nificant 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 variabil-

ity, it is generally not directly relevant for inferences

about local climate. The coarse resolution and approx-

imate parameterizations of subgrid-scale processes both

limit the accuracy of the representation of local vari-

ability, 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 complemen-

tary strategy employed to empirically downscale large-

scale variability through statistical methods. Although

dynamical downscaling has the merits of being physi-

cally based and not assuming a stationary climate, po-

tentially 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 ex-

tremely high computational demands constrain the use

of pure dynamical downscaling. Although SD has the

weakness that it assumes statistically stationary re-

lationships between large-scale and local variables (as

represented by historical observations), it is inexpensive

and easy to implement.

This study considers the statistical relationships be-

tween local ocean winds and large-scale free-tropospheric

Corresponding author address:AdamMonahan, School of Earth

and Ocean Sciences, University of Victoria, P.O. Box 3065 STN

CSC, Victoria BC V8W 3V6, Canada.

E-mail: [email protected]

7938 JOURNAL OF CL IMATE VOLUME 26

DOI: 10.1175/JCLI-D-12-00722.1

� 2013 American Meteorological SocietyUnauthenticated | Downloaded 12/23/21 03:00 PM UTC

mailto:[email protected]

circulation based on buoy observations and reanalysis

products. Following earlier work by Monahan (2012a)

and Culver and Monahan (2013), SD is used to in-

vestigate 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 pre-

dictive 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 statisti-

cally 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 statisticalmodel, 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 sur-

face 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 con-

trast to land surface winds, sea surface winds are less

influenced by stationary local features (e.g., topography

or fixed surface inhomogeneities). Therefore, the con-

nection 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 dis-

cussion and conclusions are presented in section 5.

2. Data

This study assesses the cross-validated statistical pre-

dictability of the statistics of historical sea surface wind

observations from a total of 52 moored ocean buoys (the

predictands) using free-tropospheric large-scale circula-

tion 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

SouthernOcean and IndianOcean are not considered in

this study as we were not able to find buoy observations

in those locations of sufficient duration to establish ro-

bust statistical relationships with the flow aloft. The

wind and direction data from the 52 buoys were ob-

tained from four sources:

1) Prediction and Research Moored Array in the At-

lantic (PIRATA) project 10-min averaged data

(measured at 3–4m 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 3–4m 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 5m above

mean sea level) from 31 buoys off of the west and east

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http://www.pmel.noaa.gov/pirata/

http://www.pmel.noaa.gov/tao/disdel/disdel-pir.html

http://www.pmel.noaa.gov/tao/disdel/disdel-pir.html

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 5m

above mean sea level) from four buoys in the north-

west 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.

Buoy ID Latitude (8N) Longitude (8W) Duration

Anemometer

height (m) Source

41001 32.31 75.48 1973–2001 5 NDBC

41004 32.50 79.10 1978–98 5 NDBC

41006 29.30 77.40 1982–96 5 NDBC

41009 28.52 80.17 1988–98 5 NDBC

41010 28.91 78.47 1988–99 5 NDBC

42001 25.89 89.66 1975–2001 5 NDBC

42002 25.79 93.67 1973–2001 5 NDBC

42003 26.04 85.61 1976–2001 5 NDBC

42019 27.91 95.35 1990–98 5 NDBC

42020 26.97 96.70 1990–2001 5 NDBC

44004 38.48 70.43 1977–98 5 NDBC

44005 43.19 69.14 1978–98 5 NDBC

42007 30.09 88.77 1981–99 5 NDBC

51001 23.45 162.28 1981–2000 5 NDBC

51002 17.09 157.81 1984–98 5 NDBC

51003 19.35 160.62 1984–98 5 NDBC

51004 17.53 152.38 1984–98 5 NDBC

46002 42.59 130.47 1975–98 5 NDBC

46035 57.07 177.75 1985–97 5 NDBC

46005 46.10 131.00 1976–2000 5 NDBC

46041 47.35 124.73 1987–98 5 NDBC

46030 40.42 124.53 1984–98 5 NDBC

46028 35.74 121.88 1983–96 5 NDBC

46027 41.85 124.38 1983–2001 5 NDBC

46025 33.75 119.05 1982–98 5 NDBC

46026 37.76 122.83 1982–99 5 NDBC

46023 34.71 120.97 1982–99 5 NDBC

46042 36.79 122.40 1987–98 5 NDBC

46012 37.36 122.88 1980–98 5 NDBC

46013 38.24 123.30 1981–98 5 NDBC

— 12.00 38.00 2000–09 4 PIRATA

— 8.00 38.00 2001–09 4 PIRATA

— 4.00 38.00 2001–09 4 PIRATA

— 0.00 35.00 2001–08 4 PIRATA

— 15.00 38.00 2001–09 4 PIRATA

— 8.00 155.00 2000–08 4 TAO/TRITON

— 2.00 155.00 1998–2005 4 TAO/TRITON

— 22.00 155.00 1998–2005 4 TAO/TRITON

— 9.00 140.00 1998–2009 4 TAO/TRITON

— 2.00 140.00 2000–08 4 TAO/TRITON

— 2.00 125.00 2001–09 4 TAO/TRITON

— 22.00 125.00 1999–2008 4 TAO/TRITON

— 2.00 195.00 1997–2006 4 TAO/TRITON

— 5.00 195.00 1999–2006 4 TAO/TRITON

— 28.00 125.00 2001–09 4 TAO/TRITON

— 5.00 170.00 2001–09 4 TAO/TRITON

— 2.00 180.00 1999–2004 4 TAO/TRITON

22001 28.30 233.85 1978–2000 5 JMA

21001 39.50 214.50 1978–91 5 JMA

21003 25.60 224.00 1978–87 5 JMA

21004 29.00 225.00 1982–2000 5 JMA



http://www.ndbc.noaa.gov/

http://www.ndbc.noaa.gov/

http://www.data.kishou.go.jp/kaiyou/db/vessel_obs/data-report/html/index_e.html



These data were then used to calculate means of

vector wind components (projections along 36 direc-

tions 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, me-

ridional 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 2.58 3 2.58. The down-

scaling predictors were calculated from the 850-hPa data.

An analysis using predictors at other pressure levels

demonstrated that predictive information is largely in-

dependent 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 pre-

dictions, 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 be-

tween predictors and predictands resulting from the sea-

sonal cycle. An inspection of the seasonal variations

in surface wind data (not shown) demonstrated that

these calendar seasons characterize the dominant non-

stationarity in the data. A summary of the predictands is

as follows:

1) mean wind speed on the specified averaging time

scale w,

2) subaveraging time scale standard deviation of wind

speed sw,

3) mean vector wind components in the direction along

the basis vector ~e, ~u � ~e (These components are

considered at 108 increments around the compass.

By construction, projections separated by 1808 arethe same up to a sign.), and

4) subaveraging time scale standard deviation of the

vector wind components along ~e, s~u�~e.

Throughout this paper, an overbar will denote aver-

aging 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, w2 is

the mean of the square of the wind speed.

Two further statistics, m and s, are also calculated: 1)

the amplitude of the average vector wind m5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 1 y2

p

and 2) the isotropic vector wind standard deviation

s5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1/2)(s2

u 1s2y)

p, where u and y are means of two

orthogonal vector wind components and su and sy are

standard deviations of two orthogonal vector wind

components (u and y generally denote arbitrary or-

thogonal components, unless otherwise explicitly spec-

ified). The quantities m and s 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-hPaU, 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 u and wind speed w with U, V,W, and

T during DJF for one buoy at Atlantic Ocean are dis-

played in Fig. 1. Both the monthly and daily time scales

are displayed. It can be seen that u is strongly correlated

with U 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 u In contrast, the absolute

correlation values for w are much smaller than those for

u, 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.

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 in-

formation is common across these different fields. To

efficiently distribute the predictor variance among the



http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis2.html

http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis2.html

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 theminimum number needed to explain over

85% of the total variance in the four large-scale pre-

dictor 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 do-

main. 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 vari-

ability. 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 esti-

mating themodel parameters).When the predictions for

all the years were obtained, the r2 value (i.e., square of

FIG. 1. (left) Correlationmaps of mean zonal wind at buoy 41001 with large-scale predictors at 850 hPa onmonthly time scales: (top)U,

(second row) V, (third row) W, and (bottom) T. 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.



correlation between predictions and observations) was

computed to measure the prediction skill.

c. Predictability of surface wind statistics: A casestudy 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 pre-

dictive 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 compo-

nents 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 pre-

diction 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 topo-

graphic variability (Culver andMonahan 2013). Wewere

unable to determine any dominant factor determining

the magnitude or orientation of this anisotropy. For ex-

ample, for the buoys considered in this study, the maxi-

mum 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 predictedmean

vector wind component. Buoy 21001 represents an in-

termediate case.

d. Wind statistics predictability distribution

Maps of the DJF prediction skills (correlation r2 values)

of the best predicted mean vector wind components

max[r2(~u � ~e)] (i.e., the vector wind component that has

the highest predictive skill among all the components in

36 directions), the mean wind speed w, the best pre-

dicted standard deviations of vector wind component

max[r2(s~u�~e)], and the standard deviation of wind speed

sw on monthly time scales at all the 52 buoys are shown

in Fig. 3. Several general results follow from these pre-

diction 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 gener-

ally 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 pre-

diction 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).

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 pre-

diction 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

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 5 0.8.



predicted than standard deviations, particularly on

shorter averaging time scales. Furthermore, the best

predicted vector wind component is almost always bet-

ter 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 ideal-

ized model of the wind speed probability distribution.

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

a. A Gaussian model of the vector wind probabilitydensity 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 sur-

face 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 in-

vestigate the reason for these differences in predictability.

Assuming that fluctuations in the vector winds are iso-

tropic, 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

m5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 1 y2

pand the isotropic standard deviation

s5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1/2)(s2

u 1s2y)

p:

w(m,s)5sF�ms

�, (1)

where (u, y) and (su, sy) are the mean and standard

deviation of orthogonal vector wind components. The

expression for F is as follows (Rice 1945):

F(x)5

ffiffiffiffip

2

rexp

�x2

4

��11

x2

2

�I0

�x2

4

�1

x2

2I1

�x2

4

��,

(2)

where Ij(x) is the associated Bessel function of the first

kind of order j. It should be pointed out that no explicit

assumptions aremade about the temporal autocorrelation

structure of the wind components. To assess the perfor-

mance of this model, we compared the IPM modeled w

and the actual w on monthly time scales using 10-m sur-

face 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 re-

analysis data (Sun 2012). For each of the four calendar

seasons, the following calculations were carried out:

1) at each grid point and for eachmonth in the record, we

computed m and s from monthly means and standard

deviations of the 10-m zonal wind and meridional

wind;

2) these values of m and s are used to compute monthly

w using the IPM; and

3) we calculated the correlation between the modeled

monthly w from the IPM and the monthly w com-

puted 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 IPMhas a high correlation with themean

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 de-

viation of vector wind component; and (bottom) standard de-

viation 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 com-

ponents, 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.



wind speed from the reanalysis data on a global scale (r2

above 0.9 at all grid points). Themeanwind speedmodel

derived from the IPM is demonstrated to work well

across the global ocean. In later section, we will dem-

onstrate that this model is less successful in modeling

month-to-month variations of the submonthly wind speed

standard deviation.

b. Sensitivity of w to m and s

Having provided evidence that the IPM is able to char-

acterize the variability of w in terms of the variability of m

ands, we can use thismodel to investigate the sensitivity of

w to changes in these vector wind statistics. While w is

a function of m and s, the sensitivities of the mean wind

speed to m and s are functions of the ratio m/s alone:

›mw5›w

›m5

›hsF

�ms

�i›m

5s

sF 0�ms

�5F 0

�ms

�and (3)

›sw5›w

›s5

›hsF

�ms

�i›s

5F�ms

�2

m

sF 0�ms

�. (4)

For convenience we can define the bounded scalar

quantity u (Monahan 2012a; Culver andMonahan 2013):

u5 tan21�ms

�. (5)

The sensitivities of w to m and s as functions of u can be

computed numerically and are shown in Fig. 5a.

From the sensitivity plot, it is clear that in the low u

regime, w is more sensitive to the variability of s than of

m. In contrast, in the high u regime w is more sensitive

to the variability of m. For intermediate values of u, w

has similar sensitivity to both s and m. This result in-

dicates that in a high u regime, variability of w is de-

termined by variability of m, irrespective of the variability

of s. In contrast, in a low u regime, variations in w are

determined by those of s and are insensitive to changes

of m. These different regimes can be illustrated by con-

sidering the skill of modeled w by the IPM allowing m to

vary frommonth to month at each point while holding s

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 u values at each

grid point using themonthly time scalem ands, and then

averaged these across all months to produce one cli-

matological u field (Fig. 6b). These maps demonstrate

the following:

1) The u regimes are geographically organized. In

general, high u values are found in the tropical Pacific

and Atlantic as well as the Indian Ocean. The low u

regime is in the subtropical and subpolar latitudes.

Intermediate u values are found predominantly in

the midlatitudes.

2) The model with fixed s represents month-to-month

variations inw 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 u dominates, themodel

with fixed s can successfully represent monthly-time-

scale variability of w, while where low u prevails, the

FIG. 5. (a) Sensitivity of w to m and s as functions of u and (b) sensitivity of sw to m and s as

functions of u.



model with fixed s cannot accurately characterize

month-to-month variations in w. This result is consis-

tent with the sensitivity plots in Fig. 5a: for low u

regimes, w computed from the IPM with fixed clima-

tological s is not very accurate because variations of w

are more sensitive to those of s in this regime. In

contrast, in the high u regimes, the performance of the

model with fixed s remains good as w is primarily

dependent on m.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

u and the modeling skill r2.

The results of this analysis suggest that the scalar

quantity u is a good measure of the dependence of w on

m and s for observed sea surface winds.

Maps of the DJF u distribution on each of the three

averaging time scales (daily, weekly, and monthly) with

the locations of all the 52 buoys superimposed are pre-

sented in Figs. 6b–d. It can be seen that u generally de-

creases as averaging time scales increase: on daily time

scales, the mid-to-high u regime (above 0.9) dominates

all the buoys. On weekly time scale, an intermediate u

regime (0.5–0.9) appears on the flanks of the surface

westerlies. On the monthly time scale, the low u 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, s is much smaller thanm on

the daily time scales, resulting in a broadly distributed

high u regime. It is noteworthy that in the tropical Pa-

cific and Atlantic, a high u 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 m

values that are much larger than s, resulting in a high u

regime. The u field also displays seasonal variations (Sun

2012).

The distribution of the buoys considered provides

good coverage of the full u range on weekly andmonthly

time scales. In consequence, we have a representative

sample of buoys within each of the three u regimes with

which to establish statistical relationships.

c. Predictability of w relative to m and s

The results of the previous section suggest that the

predictability of w relative to that of m and s is a func-

tion of u. Scatterplots of SD predictive skills (correlation

r2 values) of w relative to those of m and s are shown

in Fig. 7 for daily, weekly, and monthly averaging time

scales. Corresponding values of u 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 u is consistently large, the predictive skill of

w is strongly correlated with the predictive skill of m

across all stations and seasons. In contrast, the pre-

dictive skill of w has no strong relationship with that of

s (except for the smallest values of u). 2) On weekly

and monthly time scales, the points in the scatterplot

of r2(m) with r2(w) gather around the 1:1 line for high u

values, while the points are more broadly scattered for

low u values. In contrast, in the scatterplot of r2(s) with

r2(w), the data points gather around the 1:1 line for low

FIG. 6. (a) Modeling skill of DJF mean wind speed by the IPM

[Eq. (1)] with month-to-month variations in m but s held constant

(at its long-term average value). (b) Climatological DJF u distri-

butions onmonthly 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.



u values and scatter away from the 1:1 line for high u

values.

The above results indicate that u is a good metric for

characterizing the statistical predictability of w relative

to that of m and s. In the high u regime, good predictions

of w require accurate predictions of m. On the other

hand, in low u regimes, how well w can be predicted

depends on the predictability of s. In a medium u re-

gime, the situation is complicated as w has comparably

strong and nonlinear dependence on m and s. 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 w relative to that of the bestpredicted vector wind component

Having related the predictability of w to that of m and

s, we are on the way toward understanding the pre-

dictability of w relative to the vector wind components~u � ~e. In the low u regime, the predictive skill of w is

determined by that of s. Predictions of the isotropic

standard deviation s are not better than max[r2(s~u�~e)]; it

follows from Fig. 4 that s will generally not be as well

predicted as the max[r2(~u � ~e)]. It follows that in the low

u regime, in which w variations are dominated by those

of s, the predictability of w should be less than that of

the best predicted mean vector wind component. On the

FIG. 7. The correlation-based predictive skill of w relative to that of (left) m and (right) s on

(top) daily, (middle) weekly, and (bottom)monthly averaging time scales. The color of the data

points denotes the value of u. One-to-one lines are given in solid blue.



other hand, in the high u regime, the predictive skill of

w is determined by that of m. To complete the connec-

tion between the predictability of speeds and of vector

wind components, we need to relate the predictability of

the amplitude of the mean vector wind m to the pre-

dictability of the vector wind components themselves.

Culver and Monahan (2013) provided the following

theoretical expression for the linear predictability of m

relative to that of the vector wind component aligned

along the long-term mean wind:

r2(m)5mean(u)2

std(u)21mean(u)2r2(u) , (6)

where r2(u) and r2(m) are the (correlation-based) pre-

dictabilities of the wind component u along the long-

term mean and of m by a single predictor x (assumed

to have aGaussian distribution), respectively. The result

in Eq. (6) is based on the assumption that variations

of u (e.g., from month to month) are isotropic, un-

correlated, and Gaussian. The quantities mean(u) and

std(u) are the mean and standard deviation of u over the

entire record, respectively. For convenience, we intro-

duce the quantity g5mean(u)/std(u), which gives us

r2(m)5

�1

g21 1

�21

r2(u) . (7)

From Eq. (7) it is clear that the predictability of m is

bounded above by that of the along-mean wind compo-

nent. Furthermore, r2(u) will itself be bounded above by

the predictability of the best predicted component (by

definition). Figure 8 displays the predictive skill of m

relative to that of u in high u regimes (u $ 1) in relation

to g on daily, weekly, and monthly time scales. It can be

seen that in general, when g � 1, the predictability of m

approaches that of u. On the other hand, when g de-

creases, the predictive skill of m becomes smaller than

that of u. As discussed in the previous section, a majority

of the buoys are in a high u 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 u regime. As a result, fewer data points with u. 1

are displayed for the plots on weekly and monthly

time scales.

It should be emphasized that g increases with aver-

aging time scales. In Fig. 8 it can be observed that the

upper limit of g 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 u, and a larger

fraction is contained in subaveraging time scale vari-

ability (i.e., s). For an increasingly large fraction of

stations and seasons, as the averaging time scale is in-

creased the variability in the averaged vector wind u

becomes much smaller than the climatological mean

wind [i.e., std(u) � mean(u)].

In general, shorter averaging time scales are associ-

ated with larger values of u and smaller values of g, while

longer averaging time scales with smaller u and larger g.

Predictability of w is generally smaller than that of the

vector wind components on long averaging time scales

because of lower u values (and the fact that s has weak

FIG. 8. The predictive skill ofm relative to that of the along-mean

vector wind component u in high u regimes (u $ 1) in relationship

to g [Eq. (7); as indicated by the color of the data points].



predictability). In contrast, predictability of w can be

limited on short averaging time scales when m is more

poorly predicted than u because of the small value of g.

It is possible that there may be an optimal averaging

time scale on which the relationship among u,m, andw is

balanced in such a way as to yield optimal predictability

of w relative to u.

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

The IPM introduced above has been shown to be able

to successfully model w in terms of m and s. An analytic

expression for the standard deviation of wind speed sw

can also be derived from the IPM. By definition,

s2w 5w2 2w2 , (8)

from which it follows that

sw5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2s2 1m22

hsF

�ms

�i2r5sG

�ms

�. (9)

From Eq. (9), we have an exact expression for the wind

speed standard deviation sw in terms of m and s.

As with the modeled w, the sensitivities ›msw and

›ssw are functions of the scalar variable u, as illustrated

in Fig. 5b. In contrast to modeled w, the modeled sw is

most sensitive to variations in s over the entire u range.

In the same way that we assessed the ability of the IPM

to represent observed variability of w in terms of vari-

ability in m and s, 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 m and s

frommonthlymeans and standard deviations of 10-m

NCEP/DOEReanalysis 2 zonal wind andmeridional

wind;

2) from these, we computedmonthlysw usingEq. (9); and

3) we then calculated the correlation between the

modeled month-to-month variations in sw from Eq.

(9), and those directly computed from NCEP/DOE

Reanalysis 2 data [the squared value of the correla-

tion (r2) is used to assess the model performance in

modeling the month-to-month changes in the sub-

monthly standard deviation of wind speed].

The results of this calculation for DJF are presented

in Fig. 9a. In contrast to the highly accurate represen-

tation of w by the IPM on a global scale, the model fails

to represent month-to-month changes in sw 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 perfor-

mance is still poor in a number of places. The failure

in reproducing month-to-month variations of the sub-

monthly 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 w but not sw with our model leads us to re-

examine 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 compo-

nents u and y 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 onmodeling

the standard deviation of wind speed.

FIG. 9. (a) The modeling skill of DJF submonthly time scale sw

from the IPM [Eq. (9)]. (b) As in (a), but with the non-Gaussian

vector wind model obtained from Eq. (11).



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 y respectively. Following

Monahan (2006), non-Gaussian surface wind components

are included in the model through a Gram–Charlier expan-

sion (Johnson et al. 1994) of the probability density func-

tion (PDF) of the along-mean wind component as follows:

pu(u)51ffiffiffiffiffiffi2p

psexp

"2(u2 u)2

2s2

#

3h11

n

6He3

�u2 u

s

�1

k

24He4

�u2 u

s

�i, (10)

where He3(x) 5 x3 2 3x, He4(x) 5 x4 2 6x2 1 3,

n5 skew(u)5mean[(u2 u)3]/std(u)3 (the monthly

along-mean wind skewness), and k5 kurt(u)5fmean[(u2 u)4]/std(u)4g2 3 (the monthly along-

mean wind kurtosis). The cross-mean wind compo-

nent 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. Nev-

ertheless, the resulting function has the correct mo-

ments and is a useful model of the PDF of u so long

as realizations of this random variable are not re-

quired (e.g., Johnson et al. 1994). A more compli-

cated expression for the wind speed PDF can then be

obtained:

pw(w)5

h12

n

6He3

�us

�1

k

24He4

�us

�iI0

�wus2

�1

1

2

�ws

�hnHe2

�us

�2

k

3He3

�us

�iI1

�wus2

�

11

8

�ws

�2h22n

�us

�1 kHe2

�us

�ihI0

�wus2

�1 I2

�wus2

�i1

1

24

�ws

�3hn2 k

�us

�ih3I1

�wus2

�

1 I3

�wus2

�i1

k

192

�ws

�4h3I0

�wus2

�1 4I2

�wus2

�1 I4

�wus2

�io�w

s2

�exp

�2w21 u2

2s2

�, (11)

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 varia-

tions in sw given observedmonth-to-month variations in

u,s, n, and k. The performance in capturingsw is slightly

improved by including month-to-month changes in

the skewness and kurtosis of u, but model performance

remains much poorer than for w 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 sw.

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 de-

viation? 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 m, s, w, AND sw

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 sw, we now ask the question: might the

poor simulation of sw result from different sampling

variability of m, s, and sw? At any location, within each

month, the wind fluctuations have on the order of 15–30

statistical degrees of freedom as the surface winds gen-

erally 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 m and s

influence the Gaussian model’s performance in model-

ing w and sw.

By construction in these idealized calculations the

vector winds were Gaussian, uncorrelated, and isotropic

so the wind speed population statistics are exactly re-

lated 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 in-

vestigate the role of sampling variability.

In our first experiment, we let

u5m0(11 rmd1) and (12)

s5s0(11 rsd2) , (13)

where d1 and d2 are random numbers with a uniform

distribution on (2½, ½). We will interpret m0 and s0 as



the climatological mean and standard deviation of the

vector winds, while the random numbers d1 and d2 de-

scribe month-to-month fluctuations in these statistics

with strengths scaled by rm and rs, respectively. Note that

fluctuations of u 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 5 120 realizations of u and s

each representing a separatemonth.Within eachmonth,

we randomly sampled M days (the number of inde-

pendent wind realizations within each month) of vector

wind components (u, y) from the Gaussian model with

the mean (u, 0) 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 y, as well as the sample m and s. The

sample m and s were then used to compute thew and sw

from theGaussianmodel using Eqs. (1) and (9).We then

correlated the observed and modeled wind speed mo-

ments, to characterize the modeling skills of w and sw.

This procedure was repeated for different values of rmand rs, withm ranging from 1 to 10m s21 and s kept fixed

at 3m s21 to assess sampling variability under different u

regimes. An ensemble of 300 estimates of the modeling

skill was computed.

We consider bothM5 500 andM5 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 w and sw are

plotted as functions of u in Fig. 10, from which the fol-

lowing can be observed:

(i) The modeling skill of w 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 w for

different sizes of the true signal strength.

(ii) Themodeling skill of sw can be quite poor for small

values of rs (the true month-to-month variability

of s). When rs 5 0, the modeling skill of sw is

FIG. 10. Monte Carlo experiment derived modeling skills r2 of w and sw by the Gaussian models

[Eqs. (1) and (9)] for different values of rs and rm, and for (left) M 5 500 and (right) M 5 20.



substantially poorer than that with rs 5 0.45. For

instance, for M 5 500, when rm 5 0.45, rs 5 0 (Fig.

10a), and the r2 modeling skill of sw is about 0.6. In

contrast, when rm 5 0, rs 5 0.45 (Fig. 10c), and the

modeling skill of sw is close to 0.95. The value of rm(the true month-to-month variability in m) does not

substantially influence the modeling skill of sw (not

shown).

(iii) The modeling skill of sw increases with M. Under

the same set of rm and rs values, the modeling skill

of sw is better withM5 500 than withM5 20. For

M5 20, themodeling skill of sw is low even when rsis relatively large.

When rs 5 0, the population vector wind standard

deviation remains the same from month to month so

all fluctuations of s are produced by sampling fluctu-

ations alone. In this case fluctuations in sw modeled

by Eq. (9) differ significantly from the true fluctua-

tions in the wind speed submonthly standard de-

viation. 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 sw.

Similarly it is observed that the modeling skill of sw

increases asM 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 sw is de-

termined by a SNR that is related to the size of the true

month-to-month fluctuations of s (characterized by rs)

and the number of independent wind realizations M

within the month. Sampling fluctuations in sw are dis-

tinct from those of m and s, so variability in sw 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 andM (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

SNR5

24 std(~s)

mean(~s)(rs,M)2

std(~s)

mean(~s)(rs 5 0,M)

35

std(~s)

mean(~s)(rs 5 0,M)

5

264 C(~s)(rs,M)

C(~s)(rs 5 0,M)

3752 1, (14)

where C(~s)5 std(~s)/mean(~s), mean(~s) is the ensemble

mean value of the sampled vector wind standard de-

viation over all 120 months, and std(~s) is the corre-

sponding standard deviation. As discussed above, it is

expected that the size of the variability in ~s will depend

on the true signal strength rs and the number of degrees

of freedom M. The SNR defined by Eq. (14) charac-

terizes the month-to-month fluctuation of s (the signal)

in the dataset relative to the sampling variability (noise)

given by [std(~s)/mean(~s)](rs 5 0,M). The SNR can be

computed fromMonte Carlo simulations and compared

with the modeling skill of sw.

Consistent with the qualitative analysis described

earlier, the modeling skill of sw is determined by the

SNR as shown in Fig. 11a: as the SNR increases, sw is

better modeled. To obtain a model r2 skill better than

0.9, the SNR has to achieve a value above 3. The re-

lationship 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 5 0.2,

the number of independent realizationsM has to exceed

1000 to get a signal-to-noise ratio of 3. When rs 5 0.9,

a SNR of 3 can be obtained with M below 50.

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 s has to be relatively

large to result in a sufficiently high SNR, to obtain

a good modeling skill of sw with the IPM. We will now

estimate the SNR from the NCEP/DOE Reanalysis 2

surface wind dataset and compare it with the distribu-

tion of modeling skill of sw previously shown in Fig. 9a.

To estimate SNR, we need to have the estimates of

submonthly ~s and the value of M at each grid point.

The number of independent wind realizations within

a month can be estimated as follows:

M5N

Te

, (15)

where N is the duration of a month and Te is the auto-

correlation time scale. In computing Te, the autocorre-

lation function of the vector wind components was

modeled as a decaying exponential:



af(t)5 exp

�2t

Te

�, (16)

where t is the time lag. The value ofTewas 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 an-

isotropic (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 val-

ues of [std(~s)/mean(~s)](rs 5 0,M) at each grid point

from the Monte Carlo simulation. The value of

[std(~s)/mean(~s)](rs,M) was estimated from the ob-

served 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 sw (Fig. 9a) demon-

strates 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

sw in these regions is relatively good. In contrast, ex-

tratropical regions have smaller SNR values, which

correspond with the poor modeling skill of sw 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 sw

is strongly related to sampling variability asmeasured by

the SNR given by Eq. (14). It follows that while the IPM

provides a useful tool for describing variability of w in

terms of that ofm ands, it will not generally be useful for

doing so with sw, and presumably for other wind speed

statistics comparably sensitive to sampling fluctuations.

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 predictedmean vector

wind component is generally higher than that of the

mean wind speed. Furthermore, the mean quantities

are generally better predicted than the subaveraging

FIG. 11. (a) The r2 modeling skill of sw 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).



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 w and

the statistics of vector wind components. This model

indicates that the predictability of w relative to the

magnitude of the mean vector wind m and the vector

wind standard deviation s can be characterized by

the scalar quantity u5 tan21(m/s), which is depen-

dent on season, geographic location, and averaging

time scale. The quantity u 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 w was found to be determined by that of s

for low values of u and by m for high values of u.

3) The subaveraging time scale variabilitys was generally

found to be poorly predicted by large-scale predictors.

Therefore, in the low u regimes, the predictive skill of

w (which is determined by that of s) is generally lower

than that of the best predicted vector wind component.

On the other hand, in the high u regimes, the predictive

skill of w is determined by that of m. The predictive

skill of m relative to that of the best predicted vector

wind u is determined by the quantity g5mean(u)/

std(u). When g � 1, the predictive skill of m relative

to that of the vector winds is greatest. For smaller

values of g, the predictive skill of m is much lower

than that of the vector wind components. Correspond-

ingly, the predictive skill of w is bounded above by

that of the best predicted vector wind component, and

can be much lower, even in high u regimes.

4) The IPM generally fails to capture month-to-month

variations of the subaveraging time scale standard

deviation of wind speed, sw, in terms of variations in

m and s. With a series of Monte Carlo experiments,

we demonstrated that this can be understood to be

a result of differences in sampling variability be-

tween 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 sw.

The first three of these conclusions are consistent with

earlier results by Monahan (2012a) and Culver and

Monahan (2013). However, these earlier studies con-

sidered the SD predictive skill of surface winds in

a much more limited range of u values. The u 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 inter-

esting direction of future study.

In the idealized setting of the Monte Carlo experi-

ments, 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 thatmonth-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 sw in-

dicates 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 freedomM will increase,

the true variability of the subaveraging time scale stan-

dard 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 tech-

niques. A more detailed investigation of alternative SD

approaches is an interesting direction of future study.

Furthermore, previous studies have shown that the an-

isotropy 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 sur-

face, 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 ori-

entation 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 in-

teresting 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 Pro-

gram in Interdisciplinary Climate Science.

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Statistical Downscaling Prediction of Sea Surface Winds