Statistical downscaling prediction of sea surface winds over the global ocean

(1)

by

Cangjie Sun

B.Sc., Nanjing Institute of Meteorology, 2010

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the School of Earth and Ocean Sciences

c

Cangjie Sun, 2012 University of Victoria

(2)

Statistical Downscaling Prediction of Sea Surface Winds over the Global Ocean

by

Cangjie Sun

B.Sc., Nanjing Institute of Meteorology, 2010

Supervisory Committee

Dr. Adam H.Monahan, Supervisor (School of Earth and Ocean Sciences)

Dr. Andrew J. Weaver, Departmental Member (School of Earth and Ocean Sciences)

Dr. Julie Zhou, Outside Member

(3)

Supervisory Committee

Dr. Adam H.Monahan, Supervisor (School of Earth and Ocean Sciences)

Dr. Andrew J. Weaver, Departmental Member (School of Earth and Ocean Sciences)

Dr. Julie Zhou, Outside Member

(Department of Mathematics and Statistics)

ABSTRACT

The statistical prediction of local sea surface winds at a number of locations over the global ocean (Northeast Pacific, Northwest Atlantic and Pacific, tropical Pacific and Atlantic) is investigated using a surface wind statistical downscaling model based on multiple linear regression. The predictands (mean and standard deviation of both vector wind components and wind speed) calculated from ocean buoy observations on daily, weekly and monthly temporal scales are regressed on upper level predictor fields (derived from zonal wind, meridional wind, wind speed, and air temperature) from reanalysis products. The predictor fields are subject to a combined Empirical Orthog-onal Function (EOF) analysis before entering the regression model. 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 Gaussian model of wind speed probability density func-tion, which indicates that the wind speed is more sensitive to the standard deviations (which generally are not well predicted) than to the means of vector wind component in the midlatitude region and vice versa in the tropical region. This sensitivity of wind speed statistics to those of vector wind components can be characterized by a

(4)

simple scalar quantity θ = tan−1(µ

σ) (in which µ is the magnitude of average vector wind and σ is the isotropic standard deviation of the vector winds). The quantity θ is found to be dependent on season, geographic location and averaging timescale of wind statistics.

While the idealized probability model does a good job of characterizing month-to-month variations in the mean wind speed based on those of the vector wind statistics, month-to-month variations in the standard deviation of speed are not well modelled. A series of Monte Carlo experiments demonstrates that the inconsistency in the char-acterization of wind speed standard deviation is the result of differences of sampling variability between the vector wind and wind speed statistics.

(5)

List of Tables

Table 2.1 Information of Reanalysis Data . . . 10 Table 4.1 The predictive skills of DJF monthly-averaged w, σ and µ at

Buoys 41001, 51002 and 22001. . . 48 Table A.1 Information of Buoy Data . . . 98 Table A.2 Information of Buoy Data . . . 99

(8)

List of Figures

Figure 1.1 Mean, standard deviation fields of ERA-40 reanalysis zonal wind (first row), meridional wind (second row), and wind speed (third row) . . . 3 Figure 1.2 Geographical distribution of the buoys considered in this study 5 Figure 2.1 Correlation maps of mean zonal wind at buoy 41001 with

large-scale predictors at 850 hPa on daily, weekly, and monthly timelarge-scales: mean zonal wind (first row), mean meridional wind (second row), mean wind speed (third row), and mean air temperature (fourth row). The position of the buoy is indicated by the white cir-cle. The white boxes in the first row panels denote the domain used for the EOF decomposition of large-scale predictor fields. Predictions on different timescales are grouped by column: daily predictions (first column), weekly predictions (second column), monthly predictions(last column). . . 12 Figure 2.2 As in Figure 2.1 but for correlations of mean wind speed at buoy

41001 with the large-scale fields.. . . 13 Figure 2.3 As in Figure 2.1 but for correlations of mean zonal wind at buoy

51002 with the large-scale fields.. . . 14 Figure 2.4 As in Figure 2.1 but for correlations of mean wind speed at buoy

51002 with the large-scale fields . . . 15 Figure 3.1 Cross-validated monthly prediction skill (r2_{) for predictors from}

each pressure level of wind statistics at buoy 41001 (32.31o_N, 75.48oW): mean zonal wind (solid blue line), mean meridional wind (solid red line), mean wind speed (solid black line), monthly zonal wind standard deviation (blue dashed line), and meridional wind standard deviation (red dashed line). . . 23 Figure 3.2 As in Figure 3.1 but with buoy 51002 (17.09o_{N, 157.81}o_{W). . .} ₂₄

(9)

Figure 3.3 As in Figure 3.1 but with buoy 21002 (28.30oN, 233.85oW). . . 25 Figure 3.4 Correlation plots and polar prediction plots on monthly timescale

in DJF. Left column: the correlation plots of mean wind speed with means (solid black line) and standard deviations (dashed grey line) of vector wind components in 36 directions. The green circle denotes a correlation value of 1. Right column: the pre-diction skills plots of means (solid red line) and standard devi-ations (red dashed line) of vector wind components in 36 direc-tions, mean wind speed (blue line), wind speed standard devi-ation (dashed blue line). The black circle denotes a reference prediction skill of 0.8. . . 27 Figure 3.5 Cross-validated DJF predictive skills r2 _{on daily timescale. First}

row: best predicted vector wind component, second row: mean wind speed, third row: best predicted standard deviation of vec-tor wind component, fourth row: standard deviation of wind speed. . . 31 Figure 3.6 As in Figure 3.5 but for predictive skills on weekly timescale. . 32 Figure 3.7 As in Figure 3.5 but with predictive skills on monthly timescale. 33 Figure 3.8 The prediction skills (cross-validated r2) of the mean wind speed

relative to those of standard deviations of wind speed (first row), the best predicted standard deviations of vector wind compo-nents relative to the best predicted means of vector wind com-ponents (second row), and the mean wind speed relative to the best predicted means of vector wind components (third row). The first column is the daily timescale predictions, the second column is the weekly timescale predictions, the third column is the monthly timescale predictions. . . 34 Figure 4.1 F as a function of x . . . 40 Figure 4.2 The r2modeling skill of w on a monthly averaging timescale from

the Gaussian model (Eq. (4.16)) compared to direct calculations of w. The upper and lower plots show the results for DJF and JJA respectively . . . 41 Figure 4.3 Sensitivity of w to µ and σ as functions of θ . . . 42

(10)

Figure 4.4 (a). Modelling skill of DJF mean wind speed by the Gaussian model (Eq. (4.16)) with time-average σ. (b). The climatological distribution of θ in DJF. . . 45 Figure 4.5 As in Figure 4.4 but for JJA. . . 46 Figure 4.6 The locations of the three buoys considered in case studies. The

background map is the distribution of θ in DJF as displayed in Figure 4.4 (b). . . 48 Figure 4.7 Scatter plots of monthly-average DJF w versus µ (left column)

and σ (right column) for buoy 41001 (first row), buoy 51002 (second row), buoy 22001 (third row). . . 49 Figure 4.8 Climatological θ distributions on daily, weekly, and monthly timescales, with positions of all 52 buoys. . . 51 Figure 4.9 The correlation-based predictive skill of w relative to that of µ

(left column) and σ (right column) on daily (first row), weekly (second row), and monthly averaging timescales (third row). The color of the data points denotes the value of θ. One to one lines are given in solid blue. . . 53 Figure 4.10The predictive skill of µ relative to that of u in high θ regimes

(θ ≥ 1) in relationship to γ (as indicated by the colour of the data points). . . 56 Figure 4.11Kernel density estimates of the probability distribution (across

all 4 seasons and 52 buoys) of the correlation between µ and w on daily timescales. . . 58 Figure 4.12(a). Cross-validated predictive skill of w directly from predictors

(r2_{(w)) versus that of reconstructed w (reconstructed r}2_(w)). (b). Predictive skill of reconstructed w versus that of the best predicted vector wind component. . . 59 Figure 4.13The relationship between the observed wind speed predictive skill

(corr(wp, w)[obs]), and the modelled value from Eq. (4.37) . . . 62 Figure 4.14G as a function of θ . . . 64 Figure 4.15(a). The modeling skill of DJF sub-monthly timescale σw from

the Gaussian model (Eq. 4.42). (b). The same plot as displayed in Figure 4.2 for comparison to (a). . . 66

(11)

Figure 4.16(a). As in Figure 4.15 (a), using σw obtained from the non-Gaussian model with non-zero vector wind skewness (Eq. 4.48). (b). As in Figure 4.15(a), using σw obtained from the non-Gaussian model with non-zero vector wind skewness and kurtosis (Eq. 4.52). . . 69 Figure 4.17Results of Monte Carlo Experiment One: Modeling skills r2 _of

w and σw by the Gaussian models (Eq. (4.16) and (4.42)) for different rsand rm, and for M = 500 (left column), M = 20 (right column). . . 73 Figure 4.18Sensitivities of σw to µ and σ with a function of θ. . . 74 Figure 4.19The r2 modeling skill of σw as a function of the signal-to-noise

ratio (Eq. (4.60)). . . 78 Figure 4.20The SNR as a function of rs and M. . . 78 Figure 4.21Autocorrelation function of the zonal wind component (u) with

time lag up to 3 days at two locations: (a): 142.5o_{E, 22}o_{N . (b):} 260o_{E, 25}o_{S. The acf is estimated from Reanalysis 2 data. . . .} ₈₀ Figure 4.22(a). The map of e-folding decay time Te obtained from Eq.

(4.63). (b). Spatial distribution of the effective number of de-grees of freedom per month of surface winds. . . 81 Figure 4.23(a). Spatial distribution of the quantity SNR as defined by Eq.

(4.60). (b). As in Figure 4.16 (b), displayed for comparison to (a). 84 Figure 4.24Sampling variability (“noise”) as a function of M. . . 85 Figure 5.1 Correlation r2 _{modelling skill of 95}th _{percentile of wind speed}

from the WPDF model. . . 89 Figure 5.2 Wind predictions for those buoys in blue were carried out using

ERA-40 data, while those in red used ERA-Interim. . . 90 Figure 5.3 As in Figure 4.9 but with predictors calculated from ERA 40

data and ERA Interim data. . . 91 Figure A.1 Cross-validated predictive skills r2 _{on daily timescale for spring}

(MAM). First row: best predicted vector wind component, sec-ond row: mean wind speed, third: best predicted standard devi-ation of vector wind component, fourth row: standard devidevi-ation of wind speed. . . 100 Figure A.2 As in Figure A.1 but with predictive skills on weekly timescale. 101

(12)

Figure A.3 As in Figure A.1 but with predictive skills on monthly timescale. 102 Figure A.4 Cross-validated predictive skills r2 on daily timescale for summer

(JJA). First row: best predicted vector wind component, second row: mean wind speed, third: best predicted standard deviation of vector wind component, fourth row: standard deviation of wind speed. . . 103 Figure A.5 As in Figure A.4 but with predictive skills on weekly timescale. 104 Figure A.6 As in Figure A.4 but with predictive skills on monthly timescale. 105 Figure A.7 Cross-validated predictive skills r2 on daily timescale for autumn

(SON). First row: best predicted vector wind component, second row: mean wind speed, third: best predicted standard deviation of vector wind component, fourth row: standard deviation of wind speed. . . 106 Figure A.8 As in Figure A.7 but with predictive skills on weekly timescale. 107 Figure A.9 As in Figure A.7 but with predictive skills on monthly timescale. 108 Figure A.10The r2 _{modeling skill of w on daily, weekly and monthly}

aver-aging timescale from the Gaussian model (Eq. (4.16)) compared to direct calculations of w as calculated from buoy data . . . . 109 Figure A.11The r2modeling skill of w on a monthly averaging timescale from

the Gaussian model (Eq. (4.16)) compared to direct calculations of w. The upper and lower plots show the results for spring and autumn. . . 110 Figure A.12(a). Modelling skill of MAM mean wind speed by the Gaussian

model (Eq. (4.16)) with time-average σ. (b). The climatological distribution of θ in MAM. . . 111 Figure A.13(a). Modelling skill of SON mean wind speed by the Gaussian

model (Eq. (4.16)) with time-average σ. (b). The climatological distribution of θ in SON. . . 112

(13)

ACKNOWLEDGEMENTS I would like to thank:

Dr. Adam H. Monahan for being a great supervisor with incredible insight. I appreciate his helpful guidance and support throughout my graduate studies. I could not fathom a better supervisor for he encouraged every endeavour of mine.

My committee members for their perceptive suggestions and comments. My parents for their support and love.

the NSERC Collaborative Research and Training Experience Program for funding my graduate research.

(14)

Introduction

Sea surface winds influence the exchange of heat, momentum, and mass between the ocean and the atmosphere (e.g. Garratt, 1992; Thompson et al, 1983; Isemer and Hasse, 1991; Wanninkhof, 1992). To accurately characterize the boundary conditions of ocean-atmosphere coupled climate models a good knowledge of sea surface winds is required. Furthermore, sea surface winds are important with regard to the wind generated ocean currents and waves.

An overview of the climatological distribution of sea surface winds on a global scale is presented in Figure 1.1, which shows the mean and standard deviation fields of 6-hourly zonal wind, meridional wind and wind speed from the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 Reanalysis 10-metre winds. Considering the mean sea surface zonal wind, it can be seen that easterly trade winds dominate the tropical regions while westerlies prevail in the mid-latitude regions. The magnitude of the mean meridional wind is generally smaller than that of the mean zonal wind. The few regions with strong meridional winds are mostly tropical and subtropical regions (e.g. along the western coast of South America, Australia and eastern coast of Asia) beneath the descending branch of Hadley cell. The standard deviation fields of zonal wind and meridional wind are mutually consistent. Both display minimum variability in the tropics and subtropics and maximum variability in the mid-latitude storm tracks (particularly in the Northern Hemisphere). In the middle latitudes of the Southern Hemisphere, the variability of sea surface winds is weaker than that in the storm tracks in the Northern Hemisphere.

(15)

Wind speed is a positive quantity so the field of mean wind speed is positive everywhere. The strongest mean wind speeds are found in the mid-latitude regions of both Northern and Southern Hemispheres. Minimum mean wind speeds are found in the tropical regions. The standard deviation of wind speed is somewhat smaller than that of zonal wind and meridional wind, but with a similar geographical distribution. While they represent the best tools available for studying large-scale climate vari-bility, the output of global climate models is generally not directly relevant to infer-ences about the local climate. Coarse resolution and approximate parameterizations of sub-grid scale processes both potentially limit the accuracy of the representation of local variability, especially in the planetary boundary layer. In particular, sea sur-face winds measured at ocean buoys 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 that on large scales. Dynamical downscal-ing approaches this problem usdownscal-ing nested high-resolution Regional Climate Models (RCMs), or even higher resolution mesoscale models (MM). In contrast, statistical downscaling (SD) through statistical models is a complementary strategy employed to downscale simulations from global climate models to local scale variability. Though dynamical downscaling has the merits of being physically based and not assuming a stationary climate, potentially significant drawbacks such as possible errors asso-ciated with imperfect parameterizations of key processes (e.g. clouds and boundary layer processes), systematic biases (e.g. climate drift), coarse spatial resolution, and extremely high computational demands constrain the use of pure dynamical down-scaling. The essence of SD is to identify synchronized behavior on large and small scales. A basic weaknesses of SD is that it assumes statistically stationary relation-ships based on historical observations and physical connections between large-scale variables and local variables. As a result, neither dynamical nor statistical downscal-ing is expected to be error-free. In fact, a comparison between the results from two methods can shed light on the uncertainty in predictions.

(16)

Mean Wind Speed (m/s)

2 4 6 8 10

Mean Meridional Wind (m/s)

−5 0 5

Mean Zonal Wind (m/s)

−5 0 5

Standard Dev. of Zonal Wind (m/s)

2 4 6

Standard Dev. of Meridional Wind (m/s)

1 2 3 4 5 6

Standard Dev. of Wind Speed (m/s)

1 2 3 4

Figure 1.1: Mean, standard deviation fields of ERA-40 reanalysis zonal wind (first row), meridional wind (second row), and wind speed (third row)

(17)

This study considers the statistical relationships between small-scale ocean winds and large-scale free-tropospheric circulation based on historical data. Extrapolation of this statistical model to future climate is not considered (although this could be the subject of a future study, to which the present study is directly relevant). Fol-lowing earlier work by Monahan (2012a), SD is used to investigate the prediction and predictability of sea surface winds measured at buoys over the global oceans on daily, weekly (10 days) and monthly timescales. As sea surface wind variability at the timescale of 7 days is close to the variability at the timescale of 10 days, for convenience, the weekly timescale in our study is defined as 10 days average.

A downscaling model based on multiple linear regression is employed to ex-plore the relationship between surface buoy winds and upper level large-scale atmo-spheric fields (obtained from reanalysis products). With appropriate cross-validation, multiple linear regression produces statistically robust prediction models (Monahan, 2012a). The number of statistical degrees of freedom in our study for building the statistical model can be fairly small (e.g. 12 years of 3 months within a given season at a typical buoy), and a simple statistical model is less likely to suffer overfitting (which occurs when a statistical model describes random error instead of the true underlying relationship between predictor and predictand).

The geographic distribution of the buoys over the global oceans is shown in Figure 1.2. The spatial coverage of the buoys is: along the eastern and western coast of North America (about 26 buoys), in the Northwest Pacific (4 buoys), and in tropical Pacific and Atlantic (a total of 21 buoys). Those buoys in the Indian Ocean and Southern Ocean for which data were readily available were only deployed in the past 5 years and do not have enough data to construct statistically robust downscaling models. These buoys are therefore excluded from this study.

Monahan (2012a) investigated the predictability of surface winds in the subarctic Northeast Pacific off of Western Canada, while Culver (2012) studied the statistical predictability of historical land surface winds over central Canada. Other studies (e.g. van der Kamp et al., (2011); Salameh et al., (2009)) investigated the predictability of the projections of the vector winds in regions of complex topography. In contrast to land surface winds, sea surface winds are less influenced by stationarily local fea-tures (e.g. topography, fixed surface inhomogeneities). Therefore, the connection between sea surface winds and upper level large-scale atmospheric fields is expected

(18)

Figure 1.2: Geographical distribution of the buoys considered in this study

to be stronger 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). As is shown in Figure 1.1, there are three different major wind regimes over the ocean: moderate NH midlatitude westerlies with strong vari-ability, strong tropical easterly trade winds with weak varivari-ability, and the Southern Ocean with strong westerly winds with moderate variability. Thus, consideration of statistical downscaling of sea surface winds allows for a relatively idealized setting over a relatively large parameter range.

This study has five primary goals:

1. To characterize the predictive information that upper-air large-scale predictors carry for local sea surface winds across a range of wind climates.

2. To explore the predictive skills of different surface-wind statistics on different temporal scales. Monahan (2012a) focused on the predictive skills of the statistics of two vector wind components (zonal wind, meridional wind) and wind speed. In this study, the predictive skill of wind components (vector wind projection in 36 directions around the compass) and wind speed are investigated.

(19)

3. To relate the predictability of wind speed to that of the statistics of vector wind components on daily, weekly and monthly timescales over the global oceans, in the context of the idealized Gaussian model of the wind speed probability distribution function introduced by Monahan (2012a).

4. To explore the predictability of wind speed reconstructed from the predictions of vector wind components relative to direct predictions of wind speed.

5. To assess the predictability of the sub-monthly standard deviation of wind speeds across the global ocean.

Along with the absolute predictive skills of surface-wind statistics, the predictabil-ity of the statistics of wind speed relative to those of the vector wind components will be considered in this study. While prediction of wind speed is often what is needed in practice (e.g. for wind power estimation, surface flux calculation), gridscale-averaged vector wind components are what is produced by NWP models or GCMs (e.g. Bates and Merlivat, 2001; Jones and Toba 2001; Donelan et al. 2002).

The idealized Gaussian model used in this study is based on the probability den-sity function (pdf) of the wind speed. Many previous studies have considered the wind speed pdf as parameterized empirically by the two-parameter Weibull distribu-tion (e.g. Justus et al. 1978; Pryor et al., 2002; Conradsen et al. 1984; Deaves and Lines 1997; Pang et al. 2001). This study makes use of a hierarchy of expressions for the pdf of wind speed in terms of the statistics of vector wind components intro-duced in Monahan (2006a). In this way, explicit relationships between wind speed and vector wind components are established.

While the predictability of the first order statistics (i.e. means) of wind speed is the primary goal in this study, higher order statistics (i.e. standard deviation and 95th percentile) of wind speed are also explored with the idealized Gaussian model. It is well known that sampling variability generally increases with higher-order moments. This study will consider the influence of this sampling variability on these higher-order quantities.

Chapter 2 describes the data and methodology used in this study. In particular, detailed descriptions of the buoy data and upper-level large scale reanalysis data are provided. A statistical downscaling model based on cross-validated multiple linear regression is then introduced. Chapter 3 displays the prediction results of surfacewind

(20)

statistics. The prediction results with predictors based on pressure levels from 1000 hPa up to 500 hPa are first presented. Case studies using 850 hPa pressure level predictors (from which all subsequent predictions follow) are then discussed. General results regarding the predictability of the surface-wind statistics across 52 buoys are then displayed. Chapter 4 introduces the idealized Gaussian model for vector wind components that leads to the wind speed probability density function. The relation-ship between the mean wind speed and the statistics of vector wind components is explored through theoretical analysis from the Gaussian model and sensitivity exper-iments with reanalysis data. In particular, the relationship between the predictability of the mean wind speed and that of the statistics of vector wind components is inves-tigated using insights from the idealized Gaussian model. Furthermore, the strategy of reconstructing near-instantaneous wind speed directly from predictions of vector wind components is studied both empirically and in terms of analytical results intro-duced in Monahan (2012b). Finally, the Gaussian models performance in modelling the standard deviation of wind speed is investigated. Chapter 5 summarizes the gen-eral results in this study, and offers more insights on the broader implications (e.g. relating to the predictability of long-term climate variability) and limitations of our results (resulting from the simplifications and assumptions made in this study). This Chapter finishes with a discussion of further directions of future studies.

In Chapter 2 and Chapter 3 the lower case “u” and “v” denote zonal wind com-ponent and meridional wind comcom-ponent respectively. However, in the subsequent chapters, u and v denote two generic orthogonal vector wind components(unless oth-erwise specified)). Furthermore, following the notational convention of Culver (2012), x denotes the mean of the quantity x on an averaging timescale (e.g. daily, weekly, or monthly), and σx denotes the standard deviation of x on sub-averaging timescales. The notation mean(x) and std(x) denote the climatological statistics. For instance, mean(u) on daily timescales during DJF represents the mean value of daily-averaged u of all the days in DJF across all the years considered.

(21)

Chapter 2 Data and Methodology

2.1 Data

This study assesses the degree of cross-validated statistical predictability of the histor-ical sea surface wind observation data (predictands) from a total of 52 moored ocean buoys using free-tropospheric large-scale circulation data (predictors) from global re-analysis products.

2.1.1 Buoy Data

The duration of the data at buoys considered in this study ranges from 8 years to 28 years (Table A.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. The 52 buoys considered were obtained from four sources.

1)10-minute averaging reports of wind speed and direction data (measured at 3-4 m above mean sea level) from 5 buoys in the tropical Atlantic were obtained from the Prediction and Research Moored Array in the Atlantic project (PIRATA, downloaded from

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

(22)

mean sea level) from 12 buoys in the tropical Pacific were obtained from the Tropical Atmosphere Ocean project (TAO/TRITON, downloaded from

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

3)Hourly reports of wind speed and direction data (approximately 5 m above mean sea level) from 31 buoys off of the west and east coast of North America were obtained from the National Data Buoy Center (NDBC, downloaded from

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

4)Three-hourly reports of wind speed and direction data (approximately 5 m above mean sea level) from 4 buoys in the Northwest Pacific were obtained from the Japan Meteorological Agency (JMA, downloaded from

http://www.data.kishou.go.jp/db/vesselobs/datareport/html/buoy/buoyNoS2e.html) These data were then used to calculate means of vector wind components (wind

speed projection in 36 directions around the compass) and wind speed on daily, weekly and monthly timescales. Standard deviations of these quantities on sub-averaging timescales were also calculated. These calculations were applied to each of the four calendar seasons (DJF, MAM, JJA, SON) respectively. Missing values were removed from the buoy time series and no other preprocessing was carried out on these datasets.

2.1.2 Global Reanalysis Products

Free-tropospheric large-scale circulation predictor fields (zonal wind U, meridional wind V, and temperature T) were obtained from three global reanalysis products. Wind speed fields (W) were computed from the zonal and meridional wind component fields. The reanalysis products are listed in (Table 2.1).

An intercomparison of the prediction and predictability of sea surface winds in the different reanalysis products was carried out to make sure that the established statistical relationship between buoy data and large scale predictors is not an artifact of the selection of one particular reanalysis product. As there is a mismatch in the time span between buoy data and individual ECMWF reanalysis data products, ERA-40 data and ERA-Interim were used together to do predictions across all the 52 buoys while NCEP Reanalysis 2 data was used alone. ERA-40 data ranges from mid-1957

(23)

Table 2.1: Information of Reanalysis Data P P P P P P P P P P P P P P_P Para-meters Reanalysis

Products _ERA-40 _{ECMWF-Interim} _{NCEP Reanalysis 2}

Variables zonal wind, meridional wind, air temperature zonal wind, meridional wind, air temperature zonal wind, meridional wind, air temperature Duration(year) 1958-2002 1979-2011 1979-2011 Pressure level(hPa) 1000,925,850,_775,700,600, 500 1000,925,850, 775,700,600, 500 1000,925,850, 700,600,500 Sea Level(meter) 10 10 10 Resolution(degree) 2.5*2.5 2.5*2.5 2.5*2.5

Data Frequency 6 hourly 6 hourly 6 hourly

to 2002 while ERA-Interim starts from 1979 until 2011. Buoy data that range from mid-1970s to early 2000s were predicted by ERA-40 dataset while buoy data ranging from late 1990s to late 2010s were predicted by ERA-Interim.

2.2 Downscaling Model Based on Multiple Linear

Regression

2.2.1 Construction of Large-Scale Predictors by the

Com-bined EOF Analysis

Large scale circulation data mean zonal wind (U), mean meridional wind (V), mean wind speed (W) and mean air temperature (T) on daily, weekly and monthly timescales were chosen to be the predictors in this study.

Several criteria were used to choose the above four predictors. A strong relation-ship between large-scale predictors and the local variables was required: the

(24)

predic-tors should be statistically important contribupredic-tors to the variability in predictands. Furthermore, the large-scale predictors should be well described by NWP models or GCMs. Simulations from GCMs and NWP models are generally most accurate in the free troposphere (Benestad et al., 2008). The predictive information carried by the above four predictor fields on sea surface winds should have horizontal scales that are potentially predictable by global-scale climate models. Correlation fields of surface mean zonal wind (u) and mean wind speed (w) with large-scale U, V, W, and T at 850 hPa on daily, weekly and monthly timescales during winter season are displayed in Figure 2.1, 2.2, 2.3, 2.4. Two buoys in different locations (one buoy off the east coast of North America and one buoy in tropical Pacific) are shown. At both buoys, u is strongly correlated with U at 850 hPa: positive correlations are found locally while negative correlation fields are found to the north. At other sites (not shown), there are always local strong positive correlation though the negative correlation is sometimes found to the south ( e.g. west coast of North America) or absent (e.g. off the coast of South America). It is observed that the horizontal scales of correlation fields increase as the averaging timescales increase: on daily and weekly timescales, the spatial scales of the correlation fields are smaller than those on monthly timescale. This feature is consistent with the fact that on the synoptic (weather) scale the influ-ence exerted on surface winds by large scale circulation is more local while on longer timescales (climate scale) large-scale teleconnection patterns become more important. In contrast to the predictive information for surface zonal winds, the predictive information for w is greatly different between the two buoys considered. For the buoy off the east coast of North America, the spatial scales of correlation fields and the absolute correlation values of surface wind speed are much smaller than those of the zonal wind. On the other hand, the predictive information carried by large-scale predictors for surface wind speed at the tropical Pacific buoy is similar to that of the surface zonal wind at this location. At the tropical Pacific buoy, where there is positive correlation field for zonal wind there is negative correlation field for wind speed. In these locations, because the winds are easterly, higher wind speeds are associated with more negative zonal winds so it is expected that the correlation between surface wind speed and zonal wind aloft should be negative.

(25)

Daily U Weekly Monthly V W T −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 2.1: Correlation maps of mean zonal wind at buoy 41001 with large-scale predictors at 850 hPa on daily, weekly, and monthly timescales: mean zonal wind (first row), mean meridional wind (second row), mean wind speed (third row), and mean air temperature (fourth row). The position of the buoy is indicated by the white circle. The white boxes in the first row panels denote the domain used for the EOF decomposition of large-scale predictor fields. Predictions on different timescales are grouped by column: daily predictions (first column), weekly predictions (second column), monthly predictions(last column).

(26)

Daily U Weekly Monthly V W T −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 2.2: As in Figure 2.1 but for correlations of mean wind speed at buoy 41001 with the large-scale fields..

(27)

Daily U Weekly Monthly V W T −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 2.3: As in Figure 2.1 but for correlations of mean zonal wind at buoy 51002 with the large-scale fields..

(28)

Daily U Weekly Monthly V W T −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 2.4: As in Figure 2.1 but for correlations of mean wind speed at buoy 51002 with the large-scale fields

(29)

Combined EOF Analysis

The raw data from the four large-scale fields are not used directly as predictors in the regression model. Instead, a combined Empirical Orthogonal Functions (EOFs) analysis is carried out on the large-scale predictors at each pressure level. All of the predictor fields considered are spatially autocorrelated, and the fields themselves are mutually correlated. The purpose of the operation of EOFs is to express the data in a way which would optimally repackage the variance in the data, and reduce the dimensionality and redundant information of the atmospheric multivariate dataset. First, the temporal means for each variable are removed from the time series at all grid points. Then Empirical Orthogonal Functions (EOFs) of each field were computed respectively, obtaining a series of EOFs and corresponding Principal Components (PCs). The PCs for each field were normalized by the square root of their spatial-mean variance. As the four fields are not totally independent from each other, the resulting four sets of normalized PCs were concatenated into a single field which was subject to a second EOF analysis. The resulting PC time series optimally repackage the information within and across the four large-scale fields. In particular, the leading PCs account for a majority of variance in the large-scale fields and will enter the multiple linear regression as predictors. Predictions on monthly timescale were made using leading 6 PCs as predictors; predictions on weekly timescale were made using leading 16 PCs as predictors; while predictions on daily timescale were made using leading 26 PCs as predictors. The criterion for the number of PCs included in the regression model is the minimum number explaining 85 percent of the variance. The prediction results are not sensitive to reasonable changes in the number of leading PCs included in the model. The statistical degrees of freedom increase as the averaging timescales decrease. As a result, more predictors can be included in the linear regression model on daily timescales than monthly timescales. The spatial scale of the correlation maps on daily timescale is smaller than on monthly timescale, which also indicates that more EOF modes are potentially needed to capture most variance in the large-scale dataset.

The combined EOFs analysis was carried out on chosen domains (squared areas) specified to capture the regions of strongest correlation between the predictands and the predictor fields, instead of the entire reanalysis domain. Regions of low correlation carry irrelevant variances and potentially degrade the predictive skills; and so were not included in the predictor domain. Selecting the domains was flexible as prediction

(30)

skills were found to be insensitive to reasonable changes to the boundaries of the do-mains. As long as most strongly correlated regions were included by chosen domains, the leading EOFs capture the dominant features of large-scale variability. Predictions on daily, weekly and monthly timescales at the same buoy shared a common domain while domains may vary from buoy to buoy.

2.2.2 Construction of Surface-Wind Predictands

The statistics of both wind speed and vector wind components were predicted on 3 different timescales (daily, weekly, monthly) in this study. Each of the four sea-sons, i.e. winter (DJF), spring (MAM), summer (JJA), autumn (SON) was respec-tively predicted to reduce potential complications in the predictability caused by non-stationarities related to the seasonal cycle in predictors and predictands. For example, on daily timescales during DJF, surface winds of all the days of DJF were predicted from daily averaged large-scale predictors aloft with the same time span. On monthly timescale during DJF, surface winds of all the months of DJF were pre-dicted from monthly averaged large-scale predictors aloft with the same time span. A summary of the constructed predictands are:

Predictands:

1. w — Mean wind speed.

2. σw — Standard deviation of wind speed.

3. ˜u · ˜e — Mean vector wind components at 36 directions around compass. ˜u · ˜e = u · cos(θ) + v · sin(θ). u and v are two orthogonal vector wind components, e.g. zonal wind and meridional wind. θ increases from 0o _{to 360}o _{with a 10 degree} increment. By construction projections separated by 180 degrees are the same (up to a sign).

4. σu·˜˜e — Standard deviation of the vector wind components at 36 directions around compass.

5. µ =√u2+ v2 — Amplitude of the average vector wind. 6. σ =r 1

2(σ 2

(31)

The statistics µ and σ will be introduced in the context of an idealized Gaussian model of the vector wind probability density function in Chapter 4.

2.2.3 Downscaling Method

Multiple Linear Regression

Multiple linear regression (MLR) was used to model the linear relationship be-tween surface-wind predictands and the large-scale predictors. With appropriate cross-validation to be discussed below, MLR provides statistically robust prediction results. For many calculations, number of data points available for fitting the model was small. Especially on monthly timescales, the typical number of data points was 18 years of 3 months each (within a given season) at each buoy. For such small datasets, using a simple statistical model such as MLR can avoid overfitting. Nonlinear mod-els (e.g., neural network, binary classification trees) are generally more subject to overfitting as bigger datasets are potentially needed to estimate the great amount of parameters in more sophisticated models. On daily timescale, the datasets are much bigger which could be fit to more complex statistical models. However, to make sure that the method used for predictions is consistent among different timescales, MLR is the only statistical model implemented in this study. When enough data are available, more complicated statistical models could possibly give better prediction results. As such, the predictions in this study could be representing a lower bound on the predictability of surface-wind statistics at buoys considered.

Model Equation

In MLR, the predictand variable is expressed as a linear function of predictor variables and an error term:

yi = b0+ b1xi,1+ b2xi,2+ ... + bpxi,p+ ei, i = 1, ..., n (2.1)

xi,j = value of jth predictor at ith temporal point b0 = regression constant

(32)

p = number of predictors

yi = predictand at ith temporal point ei = error term

Parameter estimates are made in a way that the sum-of-squares errors is mini-mized. The resulting prediction equation is

ˆ

yi = ˆb0+ ˆb1xi,1+ ˆb2xi,2+ ... + ˆbpxi,p (2.2) where the caret ˆ denotes estimated values. To solve for the coefficient bi, it is more convenient to rewrite the equation in the matrix form:

Y = X ∗ B + E (2.3) where Y =       y1 y2 .. . yn       , B =       b0 b1 .. . bp       , X =       1 x11 · · · x1p 1 x21 · · · x2p .. . ... ... ... 1 xn1 · · · xnp       , E =       e1 e2 .. . en       (2.4)

where n is the number of observations. The estimated coefficient matrix is given by: ˆ B =       ˆ b0 ˆ b1 .. . ˆ bp       = (X0X)−1X0Y (2.5)

where X0 is the transpose of matrix X. After the model has been fitted, the regression residuals are obtained as

ˆ

ei = yi− ˆyi (2.6)

where ˆyi denotes the predicted value of the predictand yi at the ith temporal point from Eqn. (2.2).

(33)

ESS = Σn_i=1eˆi2, Error sum of squares (2.7) T SS = Σn_i=1(yi− y)2, Total sum of squares (2.8) RSS = Σn_i=1( ˆyi− y)2, Regression sum of squares (2.9) It follows from the fact that the prediction and the error are uncorrelated that TSS = RSS + ESS. The predictive skill of the MLR is summarized by the “r-squared” value as:

r2 = RSS

T SS = corr(yi, ˆyi) 2

(2.10) where corr(yi, ˆyi) is the correlation between the predictands and the predicted values. The value of r2 _{represents the “fraction of variance accounted for” by the regression} model, and will be used as the measurement of prediction skills in this study.

Cross validation

A Cross-validation strategy is employed in the Multiple Linear Regression model to prevent model overfitting. Cross-validation partitions the datasets into a series of training sets (from which model parameters are estimated) and validation sets (on which model predictions are assessed). In this study, each year of data was used successively as the validation data, and the remaining years data were used to estimate the parameters of the model. For example, predictions of the first year were obtained from the regression model built with data from all following (i.e. only excluding first years dataset). Then the predictions of the second year were obtained in a similar way (only the second year’s dataset was taken out when estimating the model parameters). When the predictions for all the years were obtained, the r2value (i.e. square of correlation between predictions and observations) is computed. The cross-validation strategy carried out by this study is sometimes referred as leave-one-year-out validation (done in blocks to account for the presence of autocorrelation in the time series). The advantage of this method is that all observations are used for both training and validation.

(34)

Chapter 3 Prediction Results

3.1 Prediction Skills with Predictors From

Differ-ent Pressure Levels

3.1.1 Predictions from 1000 hPa to 500 hPa

To assess the dependence of surface wind predictability on the pressure level of the pre-dictors, the vertical structure of the cross-validated prediction skills was explored at three buoys in different locations (Northwest Atlantic, Northwest Pacific and tropical Pacific). The prediction skills (r2 _{value) for each of the mean zonal wind, meridional} wind, wind speed, and standard deviations of zonal wind and meridional wind on a monthly timescale for each of the four seasons (DJF, MAM, JJA, SON) are shown as functions of the pressure level of the predictor fields in Figure 3.1, 3.2 and 3.3.

Several features of the predictions common across all three buoys and all four seasons can be observed from the vertical predictive skill plots:

• The prediction skills of the surface-wind statistics are not in general strongly dependent on pressure levels from 1000hPa up to 500 hPa.

• At least one of the mean zonal wind and mean meridional wind is always better predicted than other surface-wind statistics. In particular, predictions of either u or v have higher skills than those of w. In other words, the prediction skill

(35)

of wind speed is generally bounded above by that of the best predicted mean vector wind component.

• The sub-monthly standard deviations are generally less well predicted than the monthly means.

• Prediction skills of surface-wind statistics at each buoy display clear variability across the year. For example, at buoy 41001 in the Northwest Atlantic, the prediction skills of mean zonal wind peak in DJF and SON, fall off in MAM and JJA, while the mean wind speed predictions peak in MAM and SON and fall off in DJF and JJA.

The uniformity of prediction skills throughout free troposphere allows us to focus on predictors at one pressure level. Predictors at the 850 hPa pressure level (which is generally above the boundary layer over the ocean) were used to make the predictions considered after this point.

The implications of the rest of the results will be explored further in Chapter 4 in which insight will be provided on the relative predictability of the statistics of wind speed and vector wind components.

(36)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

P re ssur e Le vel s ( h P a )

Buoy 41001 (DJF)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 41001 (MAM)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 41001 (JJA)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Prediction Skill r

2 P re ssur e Le vel s ( h P a )

Buoy 41001 (SON)

u

v

σ

w

Figure 3.1: Cross-validated monthly prediction skill (r2) for predictors from each pressure level of wind statistics at buoy 41001 (32.31o_{N, 75.48}o_{W): mean zonal wind} (solid blue line), mean meridional wind (solid red line), mean wind speed (solid black line), monthly zonal wind standard deviation (blue dashed line), and meridional wind standard deviation (red dashed line).

(37)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 51002 (MAM)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 51002 (JJA)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 51002 (SON)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 51002 (DJF)

u

v

σ

w

Prediction Skill r

2

Figure 3.2: As in Figure 3.1 but with buoy 51002.

(38)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 22001 (DJF)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

P res s u re Lev e l( hpa)

Buoy 22001 (MAM)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

P ress u re Level (hpa)

Buoy 22001 (SON)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925 1000

Buoy 22001 (JJA)

u v σ σ w

Prediction Skill r

2

(39)

3.1.2 Case Study with Predictors at 850 hPa Pressure Level

The previous analysis shows that in general there are differences in predictability be-tween the statistics of zonal and meridional wind components. Previous studies have shown that the best-predicted component may not be aligned in either the east-west or north-south direction (Culver, 2012; van der Kamp et al., 2011 ). The predictors at 850 hPa were used to predict further statistics of surface-wind predictands. In ad-dition to the predictions of means and standard deviations of zonal wind, meridional wind and wind speed as considered in the last section, statistics of vector wind com-ponents (both means and standard deviations) in 36 directions around the compass, and the standard deviation of the wind speed, were also predicted.

The winter season results (DJF) on monthly timescales at the same three buoys considered in Section 3.1.1 are shown with polar plots in Figure 3.4. The first column shows the correlation plots of mean wind speed with the means (solid black line) and standard deviations (dashed grey line) of the vector wind components in the 36 directions. The second column shows the predictive skills (r2) of each of the surface-wind statistics. It is observed that both the predictions of vector surface-wind components and the correlation of mean wind speed with the vector component statistics are generally anisotropic. The speed prediction is isotropic by construction (wind speed is scalar quantity).

The correlation plots show the sensitivity of mean wind speed to the statistics of vector wind components in each direction. It is observed that at buoy 41001, the mean wind speed is more correlated to the standard deviations than to the means of the vector wind components in all directions. In particular, the correlation between mean wind speed and the standard deviation of vector wind component in the NW-SE direction is the highest. Not surprisingly, the prediction skill of mean wind speed is similar to that of the standard deviation of vector wind component in this direction, while there was no such correspondence with the predictability of the mean vector wind in this direction. The result that the mean wind speed had a similar prediction skill to that of the standard deviation of the vector wind component in one particular direction was characteristic of a number of buoys considered.

(40)

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

Buoy 41001

Buoy 21002

Buoy 51002

Figure 3.4: At the left column are the correlation plots of mean wind speed with means (solid black line) and standard deviations (dashed grey line) of vector wind components in 36 directions. At the right column are the prediction skills plots of means (solid red line) and standard deviations (red dashed line) of vector wind components in 36 directions, mean wind speed (blue line), wind speed standard deviation (dashed blue line). The black circle denotes a reference prediction skill of 0.8.

Figure 3.4: Correlation plots and polar prediction plots on monthly timescale in DJF. Left column: the correlation plots of mean wind speed with means (solid black line) and standard deviations (dashed grey line) of vector wind components in 36 directions. The green circle denotes a correlation value of 1. Right column: the prediction skills plots of means (solid red line) and standard deviations (red dashed line) of vector wind components in 36 directions, mean wind speed (blue line), wind speed standard deviation (dashed blue line). The black circle denotes a reference prediction skill of 0.8.

(41)

In contrast, at buoy 51002, the mean wind speed generally has a higher corre-lation with the mean vector wind components than with the vector wind standard deviations. In particular, the correlation is strongest in an approximately ENE-WSW direction, and the prediction skill of the mean wind speed is close to that of the mean vector wind component in this direction. The prediction skills of the standard devia-tions of vector wind components at buoy 51002 are generally quite small. Buoy 51002 is also representative of a class of buoys at which the mean wind speed predictability is closely related to that of the (best-predicted) mean vector wind component.

Unlike buoys 41001 and 51002, where the mean wind speed showed a strong correlation with the statistics of vector wind components in one particular direction, at buoy 21002 the mean wind speed was not strongly correlated to either of the means or standard deviations of vector wind components. Correspondingly the mean wind speed predictability does not have a simple connection with that of the vector wind components statistics in any direction. Buoy 21002 represents a third class buoys.

Some general results from these three representative buoys can be summarized as follows:

1. The best predicted mean vector wind component is generally better predicted than the best predicted vector wind component standard deviation. This result is in agreement with the results in the last section, where the predictions of standard deviations were generally less skillful than the predictions of means of zonal wind and meridional wind.

2. At many buoys, the prediction skill of the mean wind speed was consistent with one of the mean or standard deviation of vector wind components in some particular direction. At other buoys (e.g. buoy 21002), the prediction skill of mean wind speed can be only moderately correlated with the vector wind component statistics.

3. The predictions of the statistics of vector wind components are generally anisotropic. The second of these general results will be explored through a model of the de-pendence of mean wind speed on the statistics of vector wind components in Chapter 4. The anisotropic character of the vector wind components predictions is not well understood. In particular, we were not able to determine a clear control on the direc-tion of the best predicted vector wind components. Previous studies have suggested

(42)

that the maximum prediction skill of vector wind components is aligned with topo-graphic features in mountainous areas (van der Kamp et al., 2011), although vector prediction anisotropy is also observed in regions with little topographic variability (Culver, 2012). For the buoys considered in this study, no clear relationship between prediction anisotropy and surface inhomogeneity was observed.

3.2 Predictability of Surface Wind Statistics

3.2.1 Wind Statistics Predictability Map

The DJF prediction skills of the best predicted mean vector wind components (max(˜u · ˜e)), the mean wind speed (w), the best predicted standard deviations of vector wind com-ponent (max(σu·˜˜e )), and the standard deviation of wind speed (σw) on daily, weekly, and monthly timescales are shown in Figure 3.5, 3.6 and 3.7. Corresponding maps for the other calendar seasons are displayed in Appendix A. Clearly, the prediction skills of the wind statistics depend on season, averaging timescale and geographic location. Several general results follow from these prediction maps.

1) As discussed in the case studies considered in Section 3.1, 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, all four seasons, and all three averaging timescales. The theoretical basis for the difference between the prediction skills of mean wind speed and the best predicted mean vector wind component will be further explored in the next section. Predictions of vector wind quantities are also generally anisotropic (not shown).

2) The buoys which have relatively high prediction skills of mean wind speed are generally located in tropical regions. Off of the eastern and western coasts of North America, the prediction skills of mean wind speed are generally considerably lower. This result will also be examined in later sections in more detail.

3) The sub-averaging timescale standard deviations of both vector wind compo-nents and wind speed are generally poorly predicted at all geographic locations and all seasons. As the predictors were large-scale mean flow fields (mean zonal wind, mean meridional wind, mean wind speed and mean air temperature at 850 hPa), it

(43)

is perhaps not surpring that these do not carry much information on the standard deviations of surface winds. The use of potential predictor fields such as the stan-dard deviation of zonal wind and meridional wind at 850 hPa, were analyzed in an exploratory analysis. This analysis found that including the standard deviations of the large-scale fields as predictors does not improve the predictions of surface-wind standard deviations very much, while at the same time it decreases the prediction skills of mean vector wind components and mean wind speed.

The first and third of these general results can also be seen in the scatterplots (Figure 3.8) comparing the relative predictability of the vector wind and wind speed statistics. The scatterplots were done across the four calendar seasons and the 52 buoys, thus, a total of 208 data points were obtained. Each data point represents the prediction skill of the specified surface-wind statistics in one season at one buoy. In general, we see that mean quantities are better predicted than standard deviations at all averaging timescales, and that the best predicted vector wind component is generally better predicted than the mean wind speed.

(44)

Lat

it

ude (

o

N)

0

50 100 150 200 250 300 350

90

67.5

45

30

0 -30

L

a

ti

tu

d

e

(

o

N)

0

50 100 150 200 250 300 350

90

67.5

45

30

0 -30

L

a

ti

tu

d

e

(

o

N)

0

50 100 150 200 250 300 350

90

67.5

45

30

0 -30

Longitude (

o

E)

Lat

it

ude (

o

N)

0

50 100 150 200 250 300 350

90

67.5

45

30

0 -30

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

max

u · e)

w

max(σ

·

)

σ

Figure 3.4: DJF predictions on daily timescale. First row: max u · e), second row: w , third row: max(σ · ) , fourth row:σ

Figure 3.5: Cross-validated DJF predictive skills r2 on daily timescale. First row: best predicted vector wind component, second row: mean wind speed, third row: best predicted standard deviation of vector wind component, fourth row: standard deviation of wind speed.

(45)

Lat it u de ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30 L a ti tu d e ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30 Lat it u de ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30

Longitude (

o

E)

L

a

ti

tu

d

e

(

o

N)

0 50 100 150 200 250 300 350

90

67.5

45

30

0 -30

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

max

u · e)

w

max(σ

_·

)

σ

Figure 3.4: As in Figure 3.4 but with predictions on weekly timescale.

(46)

Lat it ude ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30 L a ti tu d e ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30 Lat it ude ( o N) 0 50 100 150 200 250 300 350 90 67.5 45 30 0 -30

Longitude (

o

E)

Lat

it

ude (

o

N)

0 50 100 150 200 250 300 350

90

67.5

45

30

0 -30

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

max

u · e)

w

max(σ

_·

)

σ

Figure 3.4: As in Figure 3.4 but with predictions on monthly timescale.

(47)

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1 Daily Weekly Monthly

σ

w w w

σ · u · e u · e u · e

w

u · e u · e u · e

Figure 3.8: The predictions skills (cross-validated r2) of mean wind speed versus standard deviations of wind speed (first row), best predicted standard deviations of vector wind components versus best predicted means of vector wind components (second row), mean wind speed versus best predicted means of vector wind components (third row). The first column is the daily timescale predictions, the second column is the weekly timescale predictions, the third column is the monthly timescale predictions.

Figure 3.8: The prediction skills (cross-validated r2) of the mean wind speed relative to those of standard deviations of wind speed (first row), the best predicted standard deviations of vector wind components relative to the best predicted means of vector wind components (second row), and the mean wind speed relative to the best pre-dicted means of vector wind components (third row). The first column is the daily timescale predictions, the second column is the weekly timescale predictions, the third column is the monthly timescale predictions.

(48)

Moreover, the prediction maps contain much information which will be examined in later chapters. For example, from Figure 3.7, it can be observed that the prediction skill of the best predicted vector wind component on monthly timescale at buoy 51001 (23.45o_{N , 197.72}o_{W ) is similar to that of buoy 51003 (19.35}o_{N , 199.38}o_{W ).} However, the prediction skill of mean wind speed at buoy 51001 (r2_{=0.1) is much} smaller than that of buoy 51003 (r2_{=0.56). As these two buoys are very close to} each other (approximately 500 km apart), why should there be such a large difference between the predictability of mean vector winds and mean wind speeds? As an added complication, this difference is restricted to DJF: during the other seasons (Appendix A), there is no strong gradient of mean wind speed predictions between these two buoys. In DJF, buoy 51001 is at the interface between tropical easterly and mid-latitude westerlies, while buoy 51003 is well within the tropical easterlies. During the other seasons, both buoy 51001 and 51003 are within the tropical Easterlies. Wind speed variability in easterly winds is not inherently more predictable; other buoys within tropical easterlies demonstrate poor mean wind speed predictability. Furthermore, although differences of the mean wind speed predictions at the above two buoys are strong on monthly timescales during DJF, they are not very strong on daily or weekly timescale. An understanding of the dependence of predictive skill of mean wind speed on geographic locations, seasons and averaging timescales is explored in Chapter 4.

(49)

Chapter 4 Interpretation and Extension of

Prediction Results

4.1 A Gaussian Statistical Model of the Wind Speed

Probability Density Function (WPDF)

4.1.1 Derivation and Validation of WPDF Model

In the last Chapter, the predictions of surface-wind statistics (w, σw, σu·˜˜e, ˜u · ˜e ) on three averaging timescales across four seasons and 52 buoys were displayed. In par-ticular, the results showed that the prediction skills of mean wind speed are generally smaller than the prediction skills of the best predicted vector wind components. In this chapter, the predictions of wind speed relative to that of the vector wind nents will be explored by relating wind speed with statistics of vector wind compo-nents using an idealized wind speed probability density function (WPDF) statistical model. Although the winds are non-Gaussian (Monahan, 2007), their approximation as Gaussian is a good first-order approximation for understanding the relationship between vector wind moments and wind speed moments. Considering two orthogonal vector wind components u and v, the PDFs of u, v can be written as follows:

(50)

pu(u) = 1 √ 2πσu exp −(u − u) 2 2σ2 u (4.1) pv(v) = 1 √ 2πσv exp −(v − v) 2 2σ2 v (4.2)

In Monahan (2006a, 2007), it was demonstrated from observations that it is rea-sonable to model the vector wind components as uncorrelated except in a few regions in the Indian Ocean and along the equator. The extent of the domain over which where u and v is strongly correlated is small enough to be neglected as a first approx-imation. Thus, the joint PDF of u, v can be obtained under the assumption that u and v are independent as follows:

puv(u, v) = 1 2πσuσv exp −(u − u) 2 2σ2 u − (v − v) 2 2σ2 v (4.3)

We introduce wind speed and direction through a transformation from Cartesian coordinates to polar coordinates:

u = wcosΘ (4.4)

v = wsinΘ (4.5)

where Θ is the wind direction measured in degrees counterclockwise from east. As described in Monahan (2006a), the probability expression in both coordinate systems must be the same:

pwΘ(w, Θ) dw dΘ = puv(u, v) du dv (4.6) with

w dw dΘ = du dv (4.7)

(51)

pwΘ(w, Θ) = w puv(u, v) (4.8) Recall Eq. (4.3), and replace u and v with wind speed w and Θ :

pwΘ(w, Θ) = w puv(wcosΘ, wsinΘ) = w 2πσuσv exp −(wcosΘ − u) 2 2σ2 u − (wsinΘ − v) 2 2σ2 v (4.9) By integrating over Θ, we get the marginal WPDF:

pw(w) = Z π −π pwΘ(w, Θ) dΘ (4.10) It follows that w = Z ∞ 0 w pw(w) dw (4.11)

is an expression for the mean wind speed in terms of the vector wind statistics. It is also a reasonable first-order approximation that the vector wind fluctuations are isotropic (Monahan, 2006a, 2007), so we define:

σu = σv = σ (4.12)

For observed vector winds, which may not have exactly isotropic variance, we estimate σ as σ = r 1 2(std(u) 2_{+ std(v)}2₎ _(4.13)

Finally, defining the magnitude of the mean vector wind as

Statistical downscaling prediction of sea surface winds over the global ocean

Contents

List of Tables

List of Figures

Introduction

Chapter 2

Data and Methodology

2.1

Data

2.1.1

Buoy Data

2.1.2

Global Reanalysis Products

2.2

Downscaling Model Based on Multiple Linear

Regression

2.2.1

Construction of Large-Scale Predictors by the

Com-bined EOF Analysis

2.2.2

Construction of Surface-Wind Predictands

2.2.3

Downscaling Method

Chapter 3

Prediction Results

3.1

Prediction Skills with Predictors From

Differ-ent Pressure Levels

3.1.1

Predictions from 1000 hPa to 500 hPa

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925

1000

Buoy 41001 (DJF)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925

1000

Buoy 41001 (MAM)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925

1000

Buoy 41001 (JJA)

0

0.2

0.4

0.6

0.8

500

600

700

775

850

925