NIST Technical Note 1626

ALGORITHMS FOR GENERATING LARGE SETS OF

SYNTHETIC DIRECTIONAL WIND SPEED DATA FOR

HURRICANE, THUNDERSTORM, AND SYNOPTIC WINDS

Mircea Grigoriu

Building and Fire Research Laboratory

National Institute of Standard and Technology

Gaithersburg, MD 20899-8611, and

School of Civil and Environmental Engineering

Cornell University

Ithaca, NY 14850

February 2009
TABLE OF CONTENTS

=====================================================

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Acronyms and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures..............................................................................vii

Abstract........................................................................................xi

Chapter 1

Introduction..................................................................................1

Chapter 2

Wind speed model.........................................................................3

2.1 Directional wind speeds...................................................................3

2.2 Dependent directional wind speeds..................................................4

Chapter 3

Return periods for wind speeds and demand.................................7

3.1 Wind speeds and structural demands...............................................7

3.2 Sector-by-sector method...................................................................8

Chapter 4

Wind model for hurricane winds...................................................11

4.1 Model calibration............................................................................11

4.2 Monte Carlo algorithm....................................................................13

Chapter 5

Wind model for thunderstorm and synoptic winds.......................15

Chapter 6

Numerical illustration....................................................................17

6.1 Model calibration.............................................................................17

6.2 Monte Carlo algorithm.....................................................................17

6.3 Uncertainty in Monte Carlo estimates..............................................19

Chapter 7

Conclusions.....................................................................................21

Appendix A

Generation of directional wind speeds............................................................23

i
Appendix B

Parametric bootstrapping.................................................................................25

References.........................................................................................27

ii
ACKNOWLEDGMENTS

=====================================================

This work was performed under the terms of the Intergovernmental Personnel Act. The

author wishes to acknowledge valuable contributions by Dr. J. A. Main of the Building

and Fire Research Laboratory (BFRL). Useful exchanges with Dr. E. Simiu of BFRL, who

initiated this study, are also acknowledged with thanks.

iii
.

iv
LIST OF ACRONYMS AND ABBREVIATIONS

=====================================================

MRI Mean Recurrence Interval

v
.

vi
LIST OF FIGURES

=====================================================

Figure 1 Wind speeds in mph with MRIs up to 10,000 year at Tucson,

Arizona....................................................................................................6

Figure 2 Nominal structural demands for q = 2 (left panel) and

q = 2.8 (right panel) with MRIs up to 10,000 year for

Tucson, Arizona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Figure 3 Data and model-based wind speeds in Miami for MRIs

up to 1,800 years.......................................................................................18

Figure 4 Data and model-based wind speeds in Miami for MRIs

up to 18,000 years...................................................................................18

Figure 5 One hundred replicates of model-based wind speeds

in Miami for MRIs up to 1,800 years........................................................19

Figure 6 Histograms of vR for R = 1, 786 years........................................................19

vii
.

viii
Disclaimers

Certain commercial entities, equipment, or materials may be identified in this document in

order to describe an experimental procedure or concept adequately. Such identification os

not intended to imply recommendation or endorsement by the National Institute of Standard

and Technology, nor is it intended to imply that the entities, materials, or equipment are

necessarily the best available for the purpose.

The policy of the NIST is to use the International System of Units in its technical

communications. However, in North America in the construction and building materials

industry, certain non-SI units are so widely used instead of SI units that it is more practical

and less confusing to include measurement values for customary units only.

ix
.

x
ABSTRACT

=====================================================

Probabilistic models are developed for directional wind speeds in hurricane, thunderstorm, and synoptic wind storms. The models, calibrated to data, are used to generate

synthetic directional wind speeds over periods of arbitrary length and assess the uncertainty in the resulting extreme wind speeds. The generation uses MATLAB functions,

called dir−hurricane, bootstrap−par, and bootstrap−par−ts, that are available on

www.nist.gov/wind. The synthetic data generated by the models and MATLAB functions

developed in this study provide a rational tool for constructing synthetic directional wind

speed data that are statistically consistent with existing wind records. The developments

in this study are needed because large sets of synthetic data are required to assess the

performance of tall buildings and other structural systems under directional wind speeds.

xi
.

xii
Chapter 1

INTRODUCTION

============================================

The assumption that wind effects are proportional to the square of the extreme wind

speeds, regardless of their direction, is used in building codes for the design of most rigid

structures. This assumption allows the estimation of wind effects with large mean recurrence

intervals (MRIs) from probabilistic models of extreme wind speeds calibrated to relatively

small observed data sets. However, for structures sensitive to wind directionality effects,

including special rigid structures and structures with significant dynamic response, the estimation of wind effects with large MRIs requires the use of time series of directional wind

speeds with length exceeding the length of the MRI of interest. To overcome this difficulty, it is proposed to construct synthetic directional wind speed records that (1) have any

specified length and (2) are statistically consistent with available records. The construction

of synthetic wind records involves two steps. First, probabilistic models are developed for

directional wind speeds, and these models are calibrated to data. Second, the calibrated

models are used to generate synthetic directional wind speed records.

Data available for calibrating the proposed probabilistic models for directional wind

speeds are stored in matrices with n rows and d columns. The elements in the rows of

these matrices are wind speeds recorded in d directions during distinct wind events. The

calibration of the probabilistic models for directional wind speeds poses notable difficulties

since in practice many of the entries of the wind data matrices can be zero, so that the

number of non-zero wind speeds can be much smaller than the number of wind events. This

situation is particularly severe for thunderstorm and synoptic winds since both the number

n of wind events and the number of non-zero wind speeds are relatively small. Accordingly,

the construction and calibration of probabilistic models for thunderstorm and synoptic winds

must in many if not most cases be based, in addition to data, on heuristic considerations.

The probabilistic model for hurricane wind speeds is sufficiently general to describe

directional wind speeds in thunderstorms and synoptic storms. Nevertheless, if the number

of observations available for these storms is small, the hurricane model may not be useful for

modeling their wind speeds. For hurricane winds, the probabilistic model is based on translation vectors, discussed later in this report, and accounts in an approximate but conservative

manner for the correlation between directional wind speeds. For thunderstorm and synoptic

winds, the correlation between directional wind speeds is not modeled because the available

records are insufficient to estimate it reliably. The arrival rate of hurricane, thunderstorm,

and synoptic events is modeled by a homogeneous Poisson processes with mean rate inferred

from observations. The proposed probabilistic models calibrated to data are used to develop

MATLAB codes that (1) generate directional wind speeds for hurricane, thunderstorm, and

synoptic winds over specified periods and (2) assess the uncertainty in the generated wind

speeds using bootstrapping methods.

The report is organized as follows. Basic considerations on the probabilistic models

for directional wind speeds for hurricane, thunderstorm, and synoptic winds are presented

in Chapter 2. This chapter also explores effects of correlation between directional wind

speeds. Chapter 3 defines wind speeds with specified MRIs, presents calculations based on

1
nominal structural demands with various MRIs, and examines the sector-by-sector method

for calculating wind speeds with given MRIs. Probabilistic wind speeds for hurricane winds

are defined and calibrated in Chapter 4. This chapter also outlines the steps of the Monte

Carlo algorithm used to generate directional wind speeds in hurricanes. Chapter 5 defines

and calibrates probabilistic models for directional wind speeds in thunderstorm and synoptic winds, and presents a Monte Carlo algorithm for generating directional wind speeds

for thunderstorm and synoptic events. Chapter 6 demonstrates the application of the MATLAB functions for hurricane winds, including the construction of replicates for predicted wind

speeds with various MRIs obtained by bootstrapping. Concluding remarks are presented in

Chapter 7. Two appendices provide detailed information on the use of the MATLAB functions for generating directional wind speeds and for uncertainty assessment. These computer

codes, named dir−hurricane, bootstrap−par, and bootstrap−par−ts, are available on

www.nist.gov/wind.

2
Chapter 2

WIND SPEED MODEL

============================================

Let T1 < T2 < · · · denote the sequence of random arrival times of hurricane or other

extreme wind storms at a site. It is assumed that the random times {Tk, k = 1, 2, . . .} are

the jump times of an homogeneous Poisson process {N(t), t ≥ 0} of intensity λ > 0, so that

the average number of extreme wind events during a time t > 0 is equal to the expectation

E[N(t)] = λ t of N(t). The intensity of N can be estimated from data. For example, suppose

we have observed 30 extreme wind events at a site during a 10 year period. An estimate of

λ is λˆ = 30/10 = 3 events/year. The time arrival model of extreme wind events can easily

be generalized to account for the possible seasonality of extreme wind events by letting λ to

be time dependent.

Let V

(k)

be a d-dimensional wind speed vector whose coordinates {Vk,i, i = 1, . . . , d}

are wind speeds recorded in d directions during the k

th extreme wind event, that is, the

wind speed vector at time Tk. It is assumed that the wind speed vectors {V

(k)

, k = 1, 2, . . .}

are independent copies of a d-dimensional random vector V with joint distribution F. In

view of the above considerations and assumptions, the proposed model for extreme wind

events is completely characterized by (1) the intensity λ of the Poisson model N and (2) the

distribution of V . As previously mentioned, the intensity λ of the Poisson process N can

be estimated from data by elementary calculations. However, the selection of the joint

distribution F of V is much more difficult because there are no general models for arbitrary

non-Gaussian joint distributions and the available data is insufficient for constructing reliable

empirical distributions for V .

Two different models are used for the distribution of V . For hurricanes, we assume that

V is a translation vector, that is, a nonlinear transformation of a d-dimensional Gaussian

vector. Translation vectors account in an approximate manner for the correlation between

directional wind speeds. Data on hurricanes available, for example, in the NIST database

(www.nist.gov/wind), are sufficient to calibrate this model of V , which can also be used

for thunderstorm and synoptic winds if sufficient data were available for these types of

storms. On the other hand, translation vectors cannot be used to describe thunderstorm

and synoptic winds if, as is the case for the data available for this study (that is, Automated

Surface Observing System (ASOS) data for Newark airport, NJ, LaGuardia airport, NY,

and Kennedy airport, NY), the number of observations is insufficient. For these types of

storms and data, we assume therefore that directional wind speeds are independent random

variables.

2.1 DIRECTIONAL WIND SPEEDS

Suppose the coordinates {Vi} of the wind speed vector V are defined by

Vi = F

−1

i



Φ(Gi)



, i = 1, . . . , d, (1)

where {Fi, i = 1, . . . , d} denote arbitrary distributions, Φ denotes the distribution of the

standard Gaussian variable with mean 0 and variance 1, and {Gi, i = 1, . . . , d} are correlated

3
standard Gaussian variables with covariance matrix ρ = {ρij = E[Gi Gj

], i, j = 1, . . . , d}.

We note that (1) {Fi} are the distributions of the coordinates {Vi} of V , (2) Eq. 1 establishes

a one-to-one correspondence between the wind speed vector V and the Gaussian vector

G = (G1, . . . , Gd), (3) the correlation coefficients between (Gi, Gj ) and (Vi, Vj ) are similar

for positive values of ρij , and (4) V , referred to as a translation vector, is completely defined

by the marginal distributions {Fi, i = 1, . . . , d} and the covariance matrix ρ of G ([1],

Section 3.1.1).

Under the translation model in Eq. 1, the resulting model for directional extreme winds

is completely defined by the intensity λ of the Poisson process N, the covariance matrix ρ

of the Gaussian image G of V , and the marginal distributions {Fi} of V . These parameters

can be estimated from data and used subsequently in a Monte Carlo algorithm to generate

virtual wind time series of any length. As previously stated, we use the representation in

Eq. 1 for hurricane winds.

For thunderstorm and synoptic winds, we set ρ equal to the identity matrix, so that

the coordinates of G are independent standard Gaussian variables. Accordingly, the random

variables {Vi} defined by Eq. 1 are independent and follow the distributions {Fi}.

The following section examines dependence of wind speed maxima over wind directions

and elementary structural demands on the correlation between the coordinates {Vi} of the

wind speed vector V . These considerations are relevant only for models of hurricane winds.

2.2 CORRELATED DIRECTIONAL WIND SPEEDS

Let G0and G00 be d-dimensional standard Gaussian vectors with coordinates of mean

0, variance 1, and covariance matrices ρ

0 = {ρ0

ij = E[G0i G0j

]} and ρ

00 = {ρ00

ij = E[G00i G00j

]}

such that ρ

0

ij ≤ ρ

00

ij , i, j = 1, . . . , d. Then

P



∩

d

i=1 {G

0

i ≤ ξi}



≤ P



∩

d

i=1 {G

00

i ≤ ξi}



(2)

by a corollary to the normal comparison lemma ([5], Corollary 4.2.3). This inequality holds

for the special case in which G0has independent coordinates and G00 has positively correlated

coordinates, that is, ρ

0

ij = 0, i 6= j, and ρ

00

ij ≥ 0 for i, j = 1, . . . , d.

The inequality in Eq. 2 extends directly to the wind speed vector X defined by Eq. 1,

since the events {Vi ≤ xi} and {Gi ≤ Φ



Fi(xi)



} have the same probability, ΦFi(xi)



,

i = 1, . . . , d, are monotonically increasing functions, and the correlation coefficient between

two distinct coordinates (Vi, Vj ) of V is an increasing function of the correlation coefficient

ρij between the coordinates (Gi, Gj ) of G ([1], Section 3.1.1).

Let V

0 = {V

0

i } and V

00 = {V

00

i } be wind speed vectors defined by Eq. 1 with G0

and G00

in Eq. 2 in place of G. Recall that G0and G00 are standard Gaussian vectors with covariance

matrices such that ρ

0

ij ≤ ρ

00

ij , i, j = 1, . . . , d. Then

P



∩

d

i=1 {X

0

i ≤ xi}



≤ P



∩

d

i=1 {X

00

i ≤ xi}



, (3)

where xi = F

−1

i



Φ(ξi)



, i = 1, . . . , d. Let F

0

max and F

00

max be the distributions of V

0

max =

max1≤i≤d{V

0

i } and V

00

max = max1≤i≤d{V

00

i }, respectively. The MRIs R0 and R00 for a wind

4
speed v irrespective of direction under the models V

0

and V

00, respectively, of a wind speed

vector V satisfy the inequality

R

0

(v) = 1

1 − F0

max(v)

≤

1

1 − F00

max(v)

= R

00(v), (4)

showing that the MRI of a wind speed v is shorter under the model with weaker correlation

between directional wind speeds. In other words, if the actual correlation of a wind speed

vector V is underestimated, the resulting wind speeds are conservative since they have

shorter MRIs than those corresponding to the correct correlation between directional wind

speeds.

Consider the special case in which G0, and, therefore, V

0

has independent coordinates,

that is, ρ

0

ij = 0, i 6= j, and V

00 has positively correlated coordinates, that is, ρ

00

ij ≥ 0. Then,

a relationship as in Eq. 4 holds with R0, so that the assumption of independence between

directional wind speeds is conservative if the actual correlation between directional wind

speeds are positive and the wind speed can be modeled by a translation vector.

For illustration, suppose the coordinates of the Gaussian image G of V in Eq. 1 are

equally and positively correlated, that is E[Gi Gj] = ρ ≥ 0, i, j = 1, . . . , d, i 6= j. The vector

G can be defined by

Gi =

√

ρ W +

p

1 − ρ Wi, i = 1, . . . , d, (5)

where W and Wi, i=1,. . . ,d, are independent standard Gaussian variables with mean 0 and

variance 1. That the variables Gi have the stated properties follow by direct calculations.

The coordinates of the image V of G are also equally correlated ([1], Section 3.1.1).

We present two examples illustrating the dependence of Fmax and wind speeds with

various MRIs on the correlation coefficient ρ of G, assumed to be positive.

Example 1. Records of yearly wind speed maxima observed in d = 8 directions at Tucson,

Arizona, from 1959 to 1970 yield means and standard deviations of wind speeds in these

directions of {μi, i = 1, . . . , 8} = {30.40, 35.93, 42.13, 44.37, 39.90, 37.97, 41.40, 35.73} in mph

and {σi, i = 1 . . . , 8} = {7.85, 11.34, 7.91, 9.26, 7.34, 4.31, 9.25, 6.74} in mph. Let V , d = 8,

denote these directional wind speeds assumed to follow the Extreme Type I distributions

Fi(x) = exp − exp(−αi (x − ui)), x ∈ R, (6)

where μi = ui+0.577216/αi and σ = π/(

√

6 αi). We are interested in properties of the largest

wind speeds Vmax irrespective of direction, that is, the largest coordinate of V . Figure 1

shows wind speeds Vmax in Tucson with MRIs up to 10,000 years under the assumption that

the coordinates of V are independent and equally correlated with correlation coefficients

ρ = 0.7 and ρ = 0.999. Differences between wind speeds corresponding to independent and

correlated coordinates of V are negligible for ρ = 0.7. These differences become significant

for wind speed vectors with strongly correlated coordinates, for example, directional wind

speeds corresponding to ρ = 0.999.

We note that (1) the independence assumption yields conservative approximations in

agreement with our previous theoretical arguments and (2) Vmax provides a crude measure

of demand at a cross section in a structure if characterized by the same internal forces

irrespective of wind direction.

5
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

60

70

80

90

100

110

120

MRI (years)

Wind speed (mph)

ρ = 0.0

ρ = 0.7

ρ = 0.999

Figure 1: Wind speeds in mph with MRIs up to 10,000 year at Tucson, Arizona

Example 2. Consider the wind environment in the previous examples and a structural

demand assumed to be dominated by wind speeds from a single direction, say direction 1,

so that P(V1 > x)  P(Vi > x) or, equivalently, F1(x)  Fi(x) ' 1 for i = 2, . . . , 8. If we

view wind speed as a measure of structural demand, the distribution of Vmax is

P(Vmax ≤ x) = Y

8

i=1

Fi(x) ' F1(x), (7)

and

P(Vmax ≤ x) = P



∩

8

i=1 {Vi ≤ x}



= P



V1 ≤ x, α2 V1 + β2 ≤ x, . . . , α8 V1 + β8 ≤ x



= P



V1 ≤ x, V1 ≤ (x − β2)/α2, . . . , V1 ≤ (x − β8)/α8



' P(V1 ≤ x) = F1(x) (8)

under the assumptions that the coordinates of V are independent and perfectly correlated,

respectively, where Vi = αi V1 + βi, i = 2, . . . , 8, for perfectly correlated directional wind

speeds and (αi > 0, βi) are constants such that x  (x−βi)/αi. In this case, the dependence

between directional wind speeds has a negligible effect on nominal structural demand.

6
Chapter 3

MEAN RECURRENCE INTERVALS FOR

WIND SPEEDS AND NOMINAL DEMANDS

============================================

Let X1, X2, . . . be a sequence of independent identically distributed real-valued random

variables with distribution Fx and N a homogeneous Poisson process of intensity λ. Consider

the random series taking the values X1, X2, . . . at the jump times 0 < T1 < T2 < · · · of N.

The average time between the jump times of N is 1/λ.

Suppose we retain from the above time series only those values exceeding an arbitrary

threshold a. The random times 0 < T0

1 < T02 < · · · of the non-zero values of the resulting

series define a new homogeneous Poisson process of intensity λ(a) = λ



1 − Fx(a)



. If λ is

measured in jumps/year, then 1/λ(a) gives the average number of years between consecutive

values of the original series exceeding a. The solution aR of

R =

1

λ



1 − Fx(aR)

 (9)

is a level exceeded on average once every R years, and is referred to as threshold or level

with MRI R.

3.1 WIND SPEEDS AND STRUCTURAL DEMANDS

The joint distribution of V is

F(x) = P



∩

d

i=1 {Xi ≤ xi}



= P



∩

d

i=1 {Gi ≤ ξi}



, (10)

where ξi = F

−1

i



Φ(xi)



denotes the image of xi, i = 1, . . . , d, in the Gaussian space (Eq. 1).

Since the events {Vmax = max1≤i≤d{Vi} ≤ x} and ∩

d

i=1{Vi ≤ x} have the same probability,

the distribution Fmax of the largest wind speed Vmax irrespective of direction is

Fmax(x) = P



Vmax ≤ x



= P



∩

d

i=1 {Vi ≤ x}



= P



∩

d

i=1 {Gi ≤ ξ}



, (11)

where ξ = F

−1

i



Φ(x)



.

Distributions of the type defined by Eq. 11 can be used to calculate wind speeds with

specified MRIs. For example, let V

(1)

,V

(2)

, . . . be a wind speed series consisting of independent copies of V arriving at random times 0 < T1 < T2 < · · · defined by the jumps of

a homogeneous Poisson process of intensity λ jumps/year. Suppose we are interested in the

sequence V1,max, V2,max, . . . of extreme wind speeds irrespective of direction. The wind speed

vR of this series of MRI R years is the solution of (Eq. 9)

R =

1

λ



1 − Fmax(vR)

. (12)

Similar results hold for structural demands under wind loads. Let D

(1)

s , D(2)s , . . . be a

real-valued series of demands at a critical point s of a structure corresponding to a wind

7
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

MRI (yrs)

Demand

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1

2

3

4

5

6

7

x 105

MRI (yrs)

Demand

Figure 2: Nominal structural demands for q = 2 (left panel) and q = 2.8 (right panel) with

MRIs up to 10,000 year for Tucson, Arizona (non-dimensional)

speed series V

(1)

,V

(2)

, . . . with time step equal to the jump times of a homogeneous Poisson

process of intensity λ events/year. The construction of the demand series involves two

steps. First, design interaction formulas are used to calculate demands {D

(k)

s,i , i = 1, . . . , d},

corresponding to the coordinates {V

(k)

i

, i = 1, . . . , d} a wind speed vector V

(k)

. Second, the

demand sequence of interest in design is the series D

(k)

s = max1≤i≤d{D

(k)

s,i }. The demand dR,s

of MRI R years at point s is the solution of (Eq. 9)

R =

1

λ



1 − FDs(dR,s)

 , (13)

where FDs denotes the distribution of Ds.

Example 3. Nominal structural demand for rigid and flexible structures is proportional

to wind speed at power 2 and, say, 2.8, respectively ([7], p. 177). Accordingly, structural

demands at a point s in a structure caused by directional wind speeds {Vi} are assumed

to have the order of magnitude Ds,i = V

q

i

, where q = 2 and q = 2.8 for rigid and flexible

structures, respectively.

Figure 2 shows quasi-static and dynamic responses with MRIs up to 10,000 years generated by the directional wind speed in Example 1, that is, equally correlated Extreme Type 1

directional wind speeds calibrated to data in Tucson, Arizona. The solid, dashed, and dotted

lines correspond to correlation coefficients ρ = 0.0, 0.7, and 0.999. As for extreme winds in

Fig. 1, demands for both rigid and flexible structures are not sensitive to correlation, unless

the correlation is very strong, that is, the independence model is conservative.

3.2 SECTOR-BY-SECTOR METHOD

The sector-by-sector method defines structural demands with an MRI R as the largest

8
of the directional structural demands with this return period. Let {dR,si, i = 1, . . . , d} denote

structural demands with MRI R at point s in a structure corresponding to directional winds.

These demands are solutions of

R =

1

λ



1 − FDs,i (dR,si)

, (14)

where FDs,i is the distribution of demand at location s under wind speed Vi

from direction

i. It is assumed that structural demand with MRI R is d

∗

R,s = max1≤i≤d dR,si.

Since

FDs(ξ) = P



∩

d

j=1 {Ds,j ≤ ξ}



≤ P



∩

d

i=1 {Ds,i ≤ ξ}



= FDs,i (ξ) (15)

and

1 − 1/R = FDs,i (dR,si) = FDs

(dR,s) ≤ FDs,i (dR,s), (16)

we have dR,s ≤ dR,si, i = 1, . . . , d, by the monotonicity of distribution functions, implying

dR,s ≤ d

∗

R,s = max1≤i≤d dR,si. Hence, R-year demands defined by the sector-by-sector method

over-predict actual R-year demands, showing that the method is unconservative.

9
.

10
Chapter 4

WIND MODEL FOR HURRICANE WINDS

============================================

The proposed model for directional extreme wind speeds is completely specified by a

Poisson process N of intensity λ describing the time arrival of extreme wind events, and a

translation random vector V defining directional wind speeds in individual extreme wind

events. The defining parameters of the model are the intensity λ of N, the distributions {Fi}

of the coordinates {Vi}, i = 1, . . . , d, of V , and the covariance matrix {ρij}, i, j = 1, . . . , d,

of the Gaussian image G of V .

Our objectives are to (1) calibrate the wind model to data, (2) develop a Monte Carlo

algorithm for generating synthetic extreme wind speed records of any length that are consistent with the available information, and (3) assess the sensitivity to statistical uncertainty

in wind speeds and structural demands with large MRIs.

4.1 MODEL CALIBRATION

Suppose that wind data consist of records of extreme directional wind speeds in d

directions recorded during n distinct events occurring over an ny year period. We use these

data to estimate the defining parameters of the wind model, that is, the intensity λ of N

and the probability law of V .

The average number λ of extreme wind speed events per year can be estimated by the

ratio λˆ = n/ny. This estimator of λ is unbiased and has small uncertainty for the available

data since n is of the order 1,000. It is assumed that λˆ = n/ny is the actual value of λ.

The calibration of the probability law of V is less simple and consists of two steps.

First, we estimate the the distributions {Fi} of the directional wind speeds {Vi}. Second, we

estimate the covariance matrix ρ of the Gaussian image G of V . We proceed with the first

step. Since various directional wind speed data are zero, we model the marginal distributions

{Fi} of the directional wind speeds {Vi} by

Fi(x) = qi 1(x ≥ 0) + (1 − qi) F ̃

i(x), i = 1, . . . , d, (17)

where 1(A) denotes the indicator of set A, a function equal to 1 and 0 if A is true and

false, respectively, qi = P(Vi = 0) is the probability that Viis 0 in an arbitrary extreme

wind event, and F ̃

i

is a proper distribution characterizing the non-zero values of Vi. The

functional forms of the distributions F ̃

i are assumed known. We need to estimate both the

probabilities qi = P(Vi = 0) and the parameters of the distributions F ̃

i

, i = 1, . . . , d.

The probability qi can be estimated by the ratio ˆqi = ni/n, where niis the number of

observed zero wind speeds in direction i = 1, . . . , d. This estimator of qiis unbiased, that

is, its mean is qi, with variance qi (1 − qi)/n and coefficient of variation p

(1 − qi)/qi/

√

n.

For the data set under consideration, n = 1, 000 and the probabilities qi are usually in the

range 0.4 to 0.9 so that its estimators are typically accurate. For example, for n = 1, 000 and

qi = 0.5, the coefficient of variation of the estimator of qiis 1/

√

n = 0.0316. The relatively

small uncertainty in the estimators of qi suggest to approximate qi by their estimates ˆqi.

11
We now estimate the parameters of the distributions {F ̃

i} assumed to have known

functional forms by using only the non-zero readings of directional wind speeds {Vi}. For

example, suppose {F ̃

i} are reverse Weibull distributions with parameters (αi

, ηi, ci), and let

Y be a Weibull random variable with parameters α > 0, ξ ∈ R, and c > 0, distribution

F(y) =







1 − exp −



y−ξ

α

c

, y > ξ

0 y ≤ ξ,

(18)

and density

f(y) = c

α



y − ξ

α

c−1

exp −



y − ξ

α

c

, y > ξ. (19)

Values of distribution F(y) and solutions of F(yp) = p ∈ [0, 1] can be calculated by, for

example, the MATLAB functions F(y) = cdf(’wbl’, y−ξ, α, c) and yp−ξ = icdf(’wbl’, p, α, c).

The random variable X

d= −Y has the distribution

F(x) = P(X ≤ x) = P(Y > −x) = exp − (

η − x

α

)

c



, x ≤ η = −ξ, (20)

and is called reverse Weibull variable. The expression of F in Eq. 20 coincides with that

in [8] (Equation A1.65a) depending on the parameters (σ, μ, γ) denoted here by (α, η, c).

Moments of any order of Y can be obtained from moments E



Y ̃ q



= Γ(1 +q/c) of the scaled

random variable Y ̃ defined by Y = ξ + α Y ̃ ([3], Chapter 20). For example, the mean μy,

variance σ

2

y

, and skewness γy,3 of Y are

μy = ξ + α Γ(1 + 1/c)

σ

2

y = α

2



Γ(1 + 2/c) − Γ(1 + 1/c)

2



γy,3 =

Γ(1 + 3/c) − 3 Γ(1 + 1/c) Γ(1 + 2/c) + 2 Γ(1 + 1/c)

3



Γ(1 + 2/c) − Γ(1 + 1/c)

2

3/2

. (21)

Let vi,1, vi,2, . . . , vi,n−ni be non-zero wind speed data in a direction i = 1, . . . , d recorded

at a site assumed to follow a reverse Weibull distribution F ̃

i

, where niis the number of

zero wind speeds in direction i observed in n extreme wind speed events. Our objective

is to estimate the parameters (αi, ηi, ci) of F ̃

i

. The method of moments, the method of

maximum likelihood, the method of probability-weighted moments, and other methods can

be used to estimate the parameters of this distribution ([4], Chapter 22). Extensive numerical

studies suggest that the method of moments delivers satisfactory estimators for the unknown

parameters of F ̃

i

, in contrast to, for example, the maximum likelihood method that can

produce unstable estimators [6]. These features of the method of moments and its simplicity

are our reasons for using it to estimate the parameters (αi, ηi, ci) of the distributions {F ̃

i}.

The following 3 step algorithm can be used to estimate the parameters (αi, ηi, ci).

Step 1. Construct the record (y1 = −vi,1, y2 = −vi,2, . . . , yn−ni = −vi,n−ni

). Since

(vi,1, vi,2, . . . , vi,n−ni

) are independent samples of a reverse Weibull distribution F ̃

i with

parameters (αi, ηi, ci), then (y1, . . . , yn−ni

) are independent samples of a Weibull distribution with parameters (α = αi

, ξ = −ηi, c = ci).

12
Step 2. Calculate estimates ˆμy, ˆσ

2

y

, and ˆγy,3 for the mean μy, variance σ

2

y

, and skewness

coefficient γy,3 from the sample (y1, . . . , yn−ni

). For example, ˆμy =

Pn−ni

i=1 yi/(n − ni),

mˆ q =

Pn−ni

i=1 

yi − μˆy

q

/(n − ni) are estimates of the central moments of order q ≥ 2,

σˆ

2

y

= ˆm2, and ˆγq = ˆmq/σˆ

q/2

y for q = 3, 4.

Step 3. Estimates the parameters (α, ξ, c) from Eq. 21. First, find an estimate ˆc for c

from the last equality in Eq. 21 with ˆγy,3 in place of γy,3. This nonlinear equation needs

to be solved by iterations. Second, find an estimate ˆα for α from the second equality

in Eq. 21 with (σ

2

y

, c) replaced by (ˆσ

2

y

, cˆ). Third, find an estimate ˆξ for ξ from the

first equality in Eq. 21 with (μy, α, c) replaced by (ˆμy, α, ˆ cˆ). If ˆξ > min1≤j≤n−ni

{yj},

then set ˆξ = min1≤j≤n−nj

{yj} and calculate (ˆα, cˆ) from the first equalities in Eq. 21

with (ˆμy, σˆ

2

y

) in place of (μy, σ2

y

). Set (αi = ˆα, ηi = −ˆξ, ci = ˆc). Repeat these steps

for all directions i = 1, . . . , d to determine completely the parameters of the marginal

distributions of the wind speed vector V .

The last ingredient of the probability law of V under the translation model in Eq. 1 is

the covariance matrix ρ of G. The covariance matrix ρ can be estimated from the Gaussian

image of the available directional wind speed data defined by the mapping Gi = Φ−1



Fi(Vi)



,

i = 1, . . . , d, in Eq. 1. A direct use of this mapping causes numerical problems since the

distributions {Fi(x)} have discontinuities at x = 0. We eliminate this computational problem

by replacing zero readings in the data set with random noise of small intensity. Specifically,

we replace zero wind speeds with independent samples of a uniform random variable with

support (0, ε), where ε > 0 is a parameter smaller than all non-zero readings. Accordingly,

the mapping Vi7→ Gi = Φ−1



Fi(Vi)



is replaced by Vi7→ G∗

i = Φ−1



F

∗

i

(V

∗

i

)



, where

F

∗

i

(x) = qi



x

ε

1(0 ≤ x ≤ ε) + 1(x > ε)



+ (1 − qi) F ̃

i(x), i = 1, . . . , d, (22)

F ̃

i

is as in Eq. 1, V

∗

is the modified wind speed vector, and G∗is the Gaussian image of V

∗

.

The covariance matrix ρ

∗ of G∗

can be estimated from the Gaussian image of the modified

wind speed data by using standard statistical tools.

There are two sources of uncertainty in the estimated probability law of V , the marginal

distributions {Fi} and the covariance matrix of the Gaussian image of V . Since we postulate

the functional forms of {Fi}, the uncertainty in these distributions is limited to the uncertainty in their estimated parameters. The construction of the Gaussian image of the wind

speed data involves modifications of the available wind record, so that there are modeling

errors. However, these errors are not relevant for design since they relate to very small wind

speeds. The modified and original wind data coincide for wind speeds relevant for design.

We use parametric bootstrapping to quantify approximately effects of the uncertainty in

the parameters of the probability law V on wind speeds and structural demands of various

MRI’s.

4.2 MONTE CARLO ALGORITHM

Once the probability law of the proposed directional wind speed model has been calibrated to a wind data recorded at a site, we can apply the following algorithm to generate

13
directional wind speed records of any length that are consistent the date set. Suppose our

objective is to generate directional wind speed at a site under consideration over τ years. If

λ is the average number of extreme wind events in a year, the average number of extreme

wind events in τ years is nτ = [λ τ ] where [a] denotes the largest integer smaller than a.

Accordingly, the extreme wind environment of the site is described by nτ events.

The following three step algorithm can be used to generate extreme wind environments

at a site that are consistent with the available site data.

– Step 1. Generate samples g

∗ = (g∗

1

, . . . , g∗

d

) of a d-dimensional standard Gaussian vector

G∗ with covariance matrix ρ

∗

. Various methods can be used to generate independent

samples of G∗([2], Section 5.2.1).

– Step 2. Calculate the image v

∗ = (v∗

1

, . . . , v∗

d

) of g

∗ defined by the mapping

v

∗

i =



F

∗

i

−1

Φ(g

∗

i

)



, i = 1, . . . , d. (23)

– Step 3. Repeat Steps 1 and 2 nτ times to generate a time series of directional extreme

wind speeds over τ years. The time arrivals of these extreme wind events are samples

of a Poisson process N of intensity λ.

14
Chapter 5

WIND MODEL FOR THUNDERSTORM AND SYNOPTIC WINDS

============================================

Available data for thunderstorm and synoptic winds can be insufficient to estimate the

dependence between directional wind speeds and even the three parameters of the reverse

Weibull distribution we used for hurricane winds. It is assumed that directional wind speeds

are independent and have distributions defined by Eq. 17, where F ̃

i are the Extreme Type I

distributions

F ̃

i(x) = exp 

− exp − αi (x − ui)

 , x ∈ R, i = 1, . . . , d. (24)

The means {μi} and standard deviations {σi} of directional wind speeds are related to the

defining parameters (αi, ui) by the relationships

αi =

π

√

6 σi

ui = μi −

0.577216

αi

. (25)

The defining parameters of the probabilistic model for thunderstorm and synoptic winds

are the mean arrival rate λ of these wind events and the parameters {(αi, ui), i = 1, . . . , d}.

The mean arrival rate is estimated following the procedure for hurricanes. The relationships

in Eq. 25 can be used to find {(αi, ui), i = 1, . . . , d} from estimates of directional wind speed

means and standard deviations.

For the data sets used in the report, most wind directions have only zero readings, so

that the resulting estimates of qi and μi are zero. In the absence of additional information, the

user may choose to assume that these estimates are caused by insufficient data, and that qi

and μi on these directions should be similar to neighboring non-zero estimates. Accordingly,

regressions are developed for {μi} and {qi}. For the directions with a sufficiently string of

non-zero wind speeds, the estimates of qi and μi calculated from data are retained. For

the other directions qi and μi are those given by regressions. The coefficient of variation

corresponding to the direction with the largest number of non-zero readings is used in all

directions. With these assumptions the model for directional wind speeds in thunderstorm

and synoptic events is completely defined. We note that these considerations are heuristic.

Alternative procedures can be used to assign non-zero values to qi and μiin directions with

only zero readings.

Samples of directional wind speeds for thunderstorm and synoptic events can then be

obtained by elementary calculations. First, we generate d independent random numbers

(r1, . . . , rd) uniformly distributed in (0, 1). If ri ≤ qi, the wind speed in direction i is set

zero. Otherwise, the wind speed in this direction is (Eq. 24)

x = ui − ln − ln(r)



/αi (26)

where r is a random number uniformly distributed in (0, 1).

15
.

16
Chapter 6

NUMERICAL ILLUSTRATION FOR HURRICANES

============================================

The directional extreme wind speed series at Miami, milepost 1450, consists of n = 999

events with wind speeds measured in d = 16 directions. This data set is used to calibrate

the proposed model for directional extreme wind speeds, calculate wind speeds over a broad

range of MRIs irrespective of direction, and assess the uncertainty in the generated wind

speeds by parametric bootstrapping. The defining parameters of the directional extreme

winds are the mean arrival rate λ of events per year, the probabilities {qi = P(Vi = 0)} of

directional wind speeds being zero, the parameters of the distributions {F ̃

i} of the non-zero

values of directional wind speeds {Vi}, and the covariance matrix ρ

∗ of the Gaussian image

G∗of the modified wind speed vector V

∗

.

6.1 MODEL CALIBRATION

The mean arrival rate of hurricane at this milepost is λ = 0.56 events per year, so that

n = 999 hurricanes span on average over a period of 999/0.56=1784 years. The estimates

of the probabilities qi = P(Vi = 0), i = 1, . . . , 16, are in the range [0.4140, 0.9450], so that

the number of non-zero values of directional wind speeds {Vi} in the data set is in the range

[55, 585]. Accordingly, the estimated parameters of the distributions {F ̃

i} of the non-zero

values of {Vi} will have different accuracies.

The Gaussian image G∗of the modified wind speed data set at Miami has been obtained

from the mapping defined by Eq. 22 with ε = 0.1, a value much smaller than values of wind

speeds relevant for design.

6.2 MONTE CARLO SIMULATION

The Monte Carlo simulation algorithm outlined in a previous section has been applied

to generate directional wind speed sequences with independent and dependent coordinates.

Both the independent and dependent wind speed vectors have the same marginal distributions. The dependence between directional wind speeds correspond to that between the

coordinates of the Gaussian vector G∗.

Results are reported for the series of the largest wind speeds {Vk,max, k = 1, 2, . . .}

irrespective of direction. Similar results can be obtained for the sequence of structural demands {D

(k)

s , k = 1, 2, . . .} at an arbitrary structural point s caused by a series of directional

extreme winds {V

(k)

, k = 1, 2, . . .}.

Figure 3 shows with heavy solid and dotted lines wind speeds vR of MRIs up to 1,800

years in Miami obtained by the proposed Monte Carlo algorithm with dependent and independent directional wind speeds. As previously stated, the calculations are based on Eq. 12.

The assumption of independence between directional wind speeds is conservative in agreement with results in Fig. 1. The thin solid line in the figure gives estimates of vR obtained

directly from data. Both model-based estimates of vR exceed those obtained directly from

17
0 200 400 600 800 1000 1200 1400 1600 1800

0

20

40

60

80

100

120

140

160

180

MRI (years)

Wind speed (mph)

Figure 3: Data and model-based wind speeds in Miami for MRIs up to 1,800 years

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

0

20

40

60

80

100

120

140

160

180

200

MRI (years)

Wind speed (mph)

Figure 4: Data and model-based wind speeds in Miami for MRIs up to 18,000 years

data. That model-based estimates of vR are larger than those obtained directly from data

may be explained by the way in which the dependence between directional wind speeds is

represented. The dependence between directional wind speeds is ignored in the independent

model and is likely to underestimate the actual dependence in the dependent model since we

replace all zero readings in the data set with independent random variables. Figure 4 shows

similar plots as in Fig. 4 for larger MRIs. Since available data is limited, wind speeds of

large MRIs cannot be obtained directly from data, as illustrated by the thin solid line in this

figure which cannot be continued beyond approximately 1,800 years. On the other hand, the

18
0 200 400 600 800 1000 1200 1400 1600 1800

0

20

40

60

80

100

120

140

MRI (years)

Wind speed (mph)

Figure 5: One hundred replicates of model-based wind speeds in Miami for MRIs up to 1,800

years

proposed Monte Carlo algorithm can be used to generate wind speeds of any MRI.

6.3 UNCERTAINTY IN MONTE CARLO ESTIMATES

We assess the statistical uncertainty in our estimates of the wind speeds vR by bootstrapping. Classical bootstrapping using empirical distributions to generate replicates of

wind speed series is inadequate since vR is likely to be outside the range of data and all

replicates would take values in the range defined by data. To overcome this limitation, we

use parametric bootstrapping, that is, replicates are generated from probabilistic models for

directional wind speeds calibrated to data rather than empirical distributions given by data.

Accordingly, we use the Monte Carlo algorithm outlined in a previous section to generate

replicates of the directional extreme wind speeds at a site. Figure 5 shows 100 replicates of

the heavy solid line in Fig. 3, that is, wind speeds vR with return periods up to 1,800 years.

A histogram of the vR for an MRI R of approximately 1,786 years is shown in Fig. 6, and is

based on 100 replicates wind speeds vR produced by the proposed Monte Carlo algorithm.

The range of vR in these 100 replicates is approximately [110, 138] mph. Larger number of

replicates can be used to obtain ranges including vR with specified probabilities.

19
105 110 115 120 125 130 135 140

0

0.02

0.04

0.06

0.08

0.1

0.12

Wind speed (mph)

Figure 6: Histogram of vR for MRI= 1, 786 years

.

20
Chapter 7

CONCLUSIONS

============================================

Directional wind speeds for periods much longer than commonly available records are

needed to estimate wind effects in a variety of structural systems including tall buildings.

Since wind records of the requisite size are not available, we propose to generate synthetic

directional wind speed records of the requisite length from probabilistic models calibrated

to data.

It has been shown that probabilistic models can be constructed for directional wind

speeds observed in hurricanes, thunderstorm, and synoptic winds, and that these models can

be calibrated to data. Since available records for thunderstorm and synoptic winds can be

relatively short and have numerous zero-readings, a simplified probabilistic model has been

used to describe their directional wind speeds. The model is based on the assumption that

the non-zero directional wind speeds are independent Extreme Type I variables. On the other

hand, the probabilistic model for directional wind speeds in hurricanes (or thunderstorms and

synoptic winds for which sufficient observations are available) accounts for the dependence

between directional wind speeds.

The probabilistic models for directional wind speeds in hurricanes, thunderstorm, and

synoptic winds have been used to develop MATLAB functions for (1) generating directional

wind speeds in extreme wind events spanning arbitrary time intervals and (2) assessing the

uncertainty in the generated wind speeds by bootstrapping. Appendices A and B outline

the use of these MATLAB codes and their main features.

21
.

22
Appendix A

GENERATION OF DIRECTIONAL WIND SPEEDS

============================================

A MATLAB function dir−hurricane has been developed to generate wind speeds in a

specified number ns of extreme wind events that are consistent with data for hurricane, thunderstorm, and synoptic winds. The input and output variables for dir−hurricane are defined

in this function, and their definition is not repeated. We only mentioned that the generated

wind speeds are collected in (ns, d)-matrices called w−speed−MC and w−speed−IM−MC.

– For hurricane winds, the wind speeds in w−speed−MC and w−speed−IM−MC correspond to models accounting for and disregarding the dependence between directional

wind speeds, respectively.

– For thunderstorm and synoptic winds, only independent directional wind speeds are

generated because of the limited number of data. The output matrices w−speed−MC

and w−speed−IM−MC are identical for these winds.

Directional wind speed data are loaded via a dialog allowing to select wind types and

sites. The steps for this selection are explained in the code.

A.1 MODEL FOR HURRICANE WINDS

It is assumed that directional wind speeds in hurricane winds follow reverse Weibull

distributions. Translation random vectors are used to capture the dependence between directional wind speeds. The calibration of this model in the MATLAB function dir−hurricane

follows the steps outlined in Sec. 4.1. The sample generation follows the algorithm in Sec. 4.2.

The d-dimensional output vector qv gives estimates of the directional probabilities

{qi} in Eq. 17. The d-dimensional output vectors alw, cw, and xiw give estimates of the

parameters {αi}, {ηi}, and {ci} of the directional reverse Weibull distributions defined in the

context of Eqs. 18 and 19. The (d, d) matrix covgxnew−est is an estimate of the covariance

matric ρ

∗ of the standard Gaussian vector G∗

defined in Sec. 4.1. The (ns, d)-output matrices

w−speed−MC and w−speed−IM−MC have already been defined.

A.2 MODEL FOR THUNDERSTORM AND SYNOPTIC WINDS

It is assumed that directional wind speeds in thunderstorm and synoptic follow a distribution of the type in Eq. 17 but they are independent and F ̃

i are the Extreme Type 1

distribution defined by Eqs. 24 and 25.

The input parameters cmin, cmax, nc, epsi, epsi−1 are not needed when dealing with thunderstorm and synoptic winds; any values can be used for these parameters.

The d-dimensional output vector qv gives estimates of the directional probabilities {qi}.

The d-dimensional output vectors alw and xiw give estimates of the parameters {αi} and

{ui} of the distributions {F ̃

i} in Eq. 24. The output parameters cw and covgxnew−est

23
have no meaning. As previously state, the (ns, d)-output matrices w−speed−MC and

w−speed−IM−MC with generated directional wind speed data coincide.

24
Appendix B

PARAMETRIC BOOTSTRAPPING

============================================

Two MATLAB functions bootstrap−par and bootstrap−par−ts are used to assess our confidence in hurricane and in thunderstorm/synoptic wind speeds generated by

dir−hurricane. The input and output for bootstrap−par and bootstrap−par−ts are

defined in these functions, and their definitions are not repeated. We only mention that

w−speed−boot is a three-dimensional array of dimension (n, d, nb), where n = the number

of extreme wind events, d = the number of wind events, and nb = the number of replicates.

To run bootstrap−par and bootstrap−par−ts, the output of dir−hurricane needs

to be stored in the file boot for hurricane winds and the file boot−ts for thunderstorm

or synoptic winds. The boot file includes the vectors qv, alw, cw, xiw and the matrix

covgxnew−est. The boot−ts file includes the vectors qv, alw, xiw.

25
.

26
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Boston, 2002.

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27