L Hasse, Universita¨t Kiel, Kiel, Germany
Copyright 2003 Elsevier Science Ltd. All Rights Reserved.
Introduction
For ships at sea it has been widespread practice to include weather information into ship’s logbooks to document the situation during its operations. For sail ships, wind information is most important. Beaufort adopted a scale to estimate the ‘force of the wind’ in 1805 when he was commanding officer of a man of war. The scale was formulated in terms of the effect of wind on sail ships of a certain type, but was subsequently used for other sail ships and steamers too.
Beaufort’s scale of wind force was devised for use in the marine environment. However, since 1874 the Beaufort scale has been used in the international telegram code to transmit wind information from both sea and land. While Beaufort had used the behavior of sail ships in a given wind to define for what conditions common language terms like ‘gentle breeze’, ‘moder-ate gale’, ‘whole gale’, or ‘storm’ should be applied, a different definition was required for land surfaces.
Observers used certain indicators; e.g., behavior of flags, trees or drag plates and feeling of wind in the face.
With today’s knowledge of boundary layer meteor-ology, attempts to estimate the wind force over land appear questionable. One would need to use indica-tors of a known drag coefficient at a prescribed height in an open, level area without obstructions to the flow.
Even then correction for the roughness of the under-lying terrain and for stability (e.g., day/night) would be required in order to make estimates comparable.
Obviously, anemometer operation on land is a more direct method to determine wind speed instead of estimation. Use of the Beaufort scale to determine wind speed on land is not recommended and will not be discussed in the following.
Anemometer measurements on ships are difficult due to flow distortion. Also, only relative wind speed and direction are measured. True wind needs to be calculated by vector addition of the independently measured course and speed of a ship, a source of additional error variance. Estimates of Beaufort force have therefore remained a tool for an extended period of time.
With improved types of rigging and transition from sail ships to steamers, the original definition of Beaufort forces was endangered. The appearance of
the sea, wind effects on rigging, whistling of the wind, and other phenomena may have helped to pass on a tradition from experienced observers to younger colleagues, though not coded in words. Ship officers in the transition time likely had sufficient training on sail ships to estimate the wind force even on board steamers.
In 1927 a description of the Beaufort scale in terms of sea state was formulated by Petersen as a result of many years of experience (Table 1). His description was added later to the wind code by the International Meteorological Organization (IMO), the predecessor of the World Meteorological Organization (WMO).
Obviously, IMO did not view reference to sea state as a redefinition, but rather as a written account of an existing practice. It is common habit to call the redefined scale the Beaufort scale of wind force too.
The use of Beaufort numbers for coding purposes had initiated early investigations into wind speed equivalents to Beaufort numbers, often called a
‘Beaufort equivalent scale’. Measured wind speeds presumably could ensure consistent use of Beaufort numbers over land and would help to alleviate the difficulties resulting from change of ship types with time. Also, estimated wind force numbers do not really fit into the concepts of theoretical meteorology, where wind velocity is used in the basic Navier–Stokes equations.
From these introductory remarks, we see that there are at least two tasks:
1. selection of a suitable relation between the force of wind and the wind velocity, and
2. correction for inhomogeneities in the time series of winds at sea.
Relation between Beaufort Force and Wind Velocity: A Problem of Physics and Regression Interpretation
A sail ship is not a well-defined tool to measure wind speed. The same is true for the appearance of the sea surface that depends to some degree on the history both of the wind field and the wave field and even on turbulence in the atmospheric boundary layer over the sea. In retrospect it appears wise that a coarse scale only was devised for a measure of wind force. As we can expect only a statistical relation, we need to define requirements on such a scale.
Following Lindau, an optimal Beaufort equivalent scale is required to convert Beaufort estimates into BEAUFORT WIND SCALE 189
wind speeds such that derived climatological quanti-ties like means and variances agree with respective quantities from unbiased wind measurements.
The attempts to derive a Beaufort equivalent scale differ in the meteorological setup of the experiment and in the subsequent statistical interpretation. Diffi-culties exist in both parts. It appears that misunder-standings in the interpretation of statistical methods have hampered the development of and agreement on a Beaufort equivalent scale more than questions of measurements and exposure.
One-Sided Regressions
Derivations of Beaufort equivalent scales typically use regression techniques. For a pair of variables (x,y), two regression curves can be derived, depending on which of the variables are considered ‘independent’. A common notion of regression sounds like: ‘The regression ofy on x gives the best estimate (by the method of least squares) ofyfor a given independentx, and, similarly, the regression x on y gives the best estimate ofxas a function of independenty’. The one-sided regression implies the ‘independent’ variable as nonrandom and attributes all random variation to the other, as can be seen if one follows the derivation, e.g., for a linear fit. The same is true in nonlinear regression,
where typically means of the dependent variable are calculated for intervals of the independent variable.
The physical problem of one-sided regressions is seen in the following. If we determine an equivalent scale from data including error variances in both variables, the spread of the independent variable is increased by its random errors and the resulting regression line has too small a slope. Applying this regression to climatological variables (e.g., monthly mean wind speed and variance), the derived variances will deviate systematically from respective quantities determined from unbiased measured winds.
In the development of Beaufort equivalent scales, two approaches have been used:
1. We take the Beaufort number as well determined and calculate the mean velocities for each interval of Beaufort number; i.e., we use a nonlinear regression of wind velocity on Beaufort number.
2. We argue that the average wind velocity is well measured and the random variations stem from imperfect estimation of Beaufort numbers. In this case the regression of Beaufort force on wind velocity is the given choice.
The pitfall is seen in the following: in one-sided regression the independent variables are treated as
Table 1 Definition of Beaufort force in terms of sea state Beaufort
number
Common name Definition
0 Calm Sea like a mirror
1 Light air Ripples with the appearance of scales are formed, but without foam crests
2 Light breeze Small wavelets, still short but more pronounced; crests have a glassy appearance and do not break 3 Gentle breeze Large wavelet; crests begin to break; foam of glossy appearance; perhaps scattered white horses 4 Moderate breeze Small waves, becoming longer; fairly frequent white horses
5 Fresh breeze Moderate waves, taking a more pronounced long form; many white horses are formed (chance of some spray)
6 Strong breeze Large waves begin to form; the white foam crests are more extensive everywhere (probably some spray)
7 Near gale Sea heaps up and white foam from breaking waves begins to be blown in streaks along the direction of the wind
8 Gale Moderately high waves of greater length; edges of crests begin to break into the spin drift; the foam is blown in well-marked streaks along the direction of the wind
9 Strong gale High waves; dense streaks of foam along the direction of the wind; crests of waves begin to topple, tumble, and roll over; spray may affect visibility
10 Storm Very high waves with long overhanging crests; the resulting foam, in great patches, is blown in dense white steaks along the direction of the wind; on the whole, the surface of the sea takes a white appearance; the tumbling of the sea becomes heavy and shock-like; visibility affected 11 Violent storm Exceptionally high waves (small- and medium-sized ships might be for a time lost to view behind the
waves); the sea is completely covered with long white patches of foam lying along the direction of the wind; everywhere the edges of the wave crests are blown into froth; visibility affected 12 Hurricane The air is filled with foam and spray; sea completely white with driving spray; visibility very seriously
affected
Reproduced with permission from World Meteorological Organization (1970)The Beaufort Scale of Wind Force. Reports on Marine Science Affairs No. 3. Geneva: World Meteorological Organization.
190 BEAUFORT WIND SCALE
nonrandom and all variability from both axes is ascribed to the dependent variable.
The first approach was applied in early investiga-tions listing wind speeds averaged for each Beaufort number, although difficulties both of Beaufort obser-vations and of the exposure and calibration of wind instruments were discussed.
However, there are good arguments to use the second approach, the regression of estimates to measured speeds. We know that Beaufort numbers are estimates only, which are influenced by different external conditions and by uncertainties in the ob-server’s judgment. Also the quantization error is larger for estimates due to the larger intervals of the Beaufort scale compared to speeds in meters per second. It is not unreasonable to infer the error of Beaufort number estimation to be much larger than the uncertainty of anemometer measurements. Ko¨ppen in 1888 already advocated the averaging of Beaufort estimates for intervals of measured speeds in order to establish an equivalent scale.
IMO adopted a scale for official use, corresponding to today’s WMO code 1100, that was established in 1906 by Simpson, Meteorological Office, London, using regression of Beaufort estimates to measured speeds as recommended by Ko¨ppen.
Two-Sided Regressions
It is evident that both Beaufort force estimates and velocity measurements contain random errors, and a two-way regression would be the appropriate tool.
The best relation between two random variables lies between the two one-sided regressions. For linear regressions, in the absence of information on the respective error variances the best choice would be the bisector of the angle between the two one-sided regression lines. It minimizes the orthogonal distances of observed points from the line. This is equivalent to assuming the error variances to be the same fraction of total variance for each of the variables. This so-called orthogonal regression is certainly better than a one-sided regression, when both variables are subject to random errors. However, in a given case, the fraction of error in the total variance of each variable need not be the same and an improved technique is required.
For illustration consider the observations at the Ocean Weather Station (OWS) K inFigure 1. The two one-sided regression lines and the orthogonal regres-sion line are plotted together with the monthly means of anemometer measurements at OWS K and Beaufort estimates from ships in the vicinity. The fit through the monthly means deviates from the orthogonal regres-sion in the direction towards the one-sided regresregres-sion of Beaufort numbers on measured speeds. This implies
that the uncertainty of Beaufort estimates is a larger fraction of the total variance than that of anemometer measurement.
The orthogonal regression yields a better Beaufort equivalent scale than any of the one-sided regressions – in the sense to better reproduce the means of wind speed from Beaufort estimates. The agreement is even better than one would expect from error estimates of wind speed measurements and Beaufort estimates.
The reason for this is seen in the natural variability between measurements co-located in space and time that enters as ‘unexplained’ variance in the regression too.
An Optimized Beaufort Equivalent Scale
Wind speed measurements at Ocean Weather Stations (OWSs) and Beaufort estimates from passing volun-tary observing ships (VOSs) were used in several attempts to derive an improved equivalent scale. Some averaging is needed to reduce errors of measurements in respective observations and also to account for the natural variability that enters because VOSs pass OWSs at some distance.
Lindau developed a sophisticated method to determine effective variances. From simultaneous
30
25
20
15
10
5
Wind speed (kn)
2 3 4 5 6 7
Beaufort
Figure 1 Relation between Beaufort force and wind speed (kn), based on observations from OWS K and ships within 500 km from OWS K, period 1960–71. One-sided and ‘orthogonal’ linear regression (thin lines) are compared with monthly means. The linear fit of monthly means (full line) deviates from the best linear regression calling for explicit consideration of error variances.
(Reproduced with permission from Lindau R (1994) Eine neue Beaufort-A¨ quivalentskala. Kiel: Berichte Institut fu¨r Meereskunde Kiel, No. 249.)
BEAUFORT WIND SCALE 191
observations of pairs of VOSs, differences were obtained and variances calculated as a function of distance between the ships. Extrapolating to fictitious zero distance, the error variances of VOSs were obtained. Using the same technique on pairs of OWSs and VOSs, the error variances of OWSs were also determined.
Knowing the variances, influence of errors and natural variability could be reduced by averaging in time for OWSs and in space for VOSs. A suitable area around an OWS was selected to contain the same variance in space from VOS than at the OWS in time for 1 day. This radius and the appropriate number of VOS observations were determined separately for each OWS and each season.
With the error variance at OWS and VOS reduced by averaging the appropriate number of observations, and the natural variability being assured to be the same, the ‘orthogonal’ regression yields the correct relation. Instead of straight lines, Lindau used the method of cumulative frequencies in order to admit nonlinearities. Anemometer heights at weather ships are near 25 m. Wind measurements at OWSs were reduced to 10 m height using a diabatic wind profile.
The resulting equivalent scale is thus applicable to 10 m height. The results are given in Table 2 and depicted inFigure 2.
Discussion of Heights for Equivalent Wind Velocities Beaufort estimatesper sehave no natural height above the sea surface. However, equivalent scales give wind speeds. Because of the approximately logarithmic wind profile in the atmospheric boundary layer, for applications of the Beaufort equivalents the corre-sponding height needs to be known.
Code 1100 in use from 1926 to 1946 was believed to refer to 6 m height. Since the required standard height of anemometers had changed from 6 to 10 m, a slightly changed code 1100 was introduced in 1946 by WMO, supposedly applicable to 10 m height. Both versions of code 1100 were based on data sets into which observations from the Scilly Islands entered in a relatively large number. However, wind speeds were
Table 2 Selection of Beaufort equivalent scales, given in m s!1. Often knots (1 kn50.5144 m s!1; 1 m s!151.944 kn) are used as the unit. The last line gives the reduction factor to compare equivalent scales for reference heights of 25 and 10 m, derived for the ensemble of measurements at North Atlantic OWSs
Beaufort 0 1 2 3 4 5 6 7 8 9 10 11 12 Source
IMO (1926) 6 m 0.0 1.1 2.5 4.3 6.3 8.6 11.1 13.8 16.7 19.9 23.3 27.1 a
Code 1100 10 m – 0.8 2.4 4.3 6.7 9.4 12.3 15.5 18.9 22.6 26.4 30.5 B34.8 a
Lindau 10 m 0.0 1.2 2.7 4.6 7.2 9.7 12.1 14.6 17.3 20.2 23.4 27.1 31.4 b
CMM-IV 18 m 0.8 2.0 3.6 5.6 7.8 10.2 12.6 15.1 17.8 20.8 24.2 28.0 B32.2 a
Lindau 25 m 0.1 1.2 2.8 4.9 7.7 10.5 13.1 15.9 18.9 22.2 26.0 30.3 35.4 c
Kaufeld 25 m 0.4 1.9 4.1 6.4 8.7 11.0 13.4 15.9 18.7 21.8 25.1 28.6 32.4 d
Reduction factor 25 to 10 m
0.96 0.94 0.94 0.92 0.92 0.92 0.92 0.91 0.90
aReproduced from World Meteorological Organization (1970)The Beaufort Scale of Wind Force. Reports on Marine Science Affairs No. 3.
Geneva: World Meteorological Organization.
bReproduced from Lindau R (2001)Climate Atlas of the Atlantic Ocean. Berlin, Heidelberg: Springer Verlag.
cReproduced from Lindau R (1995) A new Beaufort equivalent scale.Proceedings of the International COADS Winds Workshop. Kiel:
Berichte aus dem Institut fu¨r Meereskunde Kiel, No. 265 (available from Institut fu¨r Meereskunde, 24105 Kiel, Germany or NOAA, Environmental Research Laboratories-CDC, Boulder, CO 80303, USA).
dReproduced from Kaufeld L (1981) The development of a new Beaufort equivalent scale.Meteorologische Rundschau34: 17–23.
0 1 2 3 4 5 6 7 8 9 10 11 12
Beaufort 35
30
25
20
15
10
5
0 Wind speed (ms_1)
Code 1100 CMM-IV Kaufeld Lindau (25 m)
Figure 2 Different Beaufort equivalent scales: code 1100 (10 m reference height), crosses; CMM-IV (25 m height), circles; Kaufeld (25 m height), dashes; Lindau (25 m height), dots. Code 1100 determined by regression of Beaufort on wind speed, CMM-IV by wind speed on Beaufort, Kaufeld and Lindau from cumulative frequencies.
192 BEAUFORT WIND SCALE
taken at the small island of St Mary’s that features several heights reaching 30–50 m above mean sea level. It is uncertain to what height in undisturbed flow over water the measurements would correspond.
Lindau’s improved equivalent scale settled the problem of reference height by using anemometer measurements of known heights and reducing winds to heights of 10 m individually with the aid of the diabatic wind profile. He could also show that for the WMO code 1100 of 1946 a reference height of 10 m is reasonable. The anemometer measurements at OWSs are not corrected for flow distortion. One can hope that the exposure of instruments at OWSs and the mode of ship operation at the station will make this an acceptable error.
Often a reduction of velocity measurements from anemometer height to reference height is necessary.
Typically, a constant reduction factor, derived from the neutral wind profile, is used. However, slightly unstable conditions prevail at most parts of the oceans, approaching near-neutral conditions at higher wind speeds. For the mix of stabilities at the North Atlantic OWS, a reduction factor of 25–10 m decreases with wind speeds fromB0.94 toB0.90; seeTable 2.
Discussion of Other Scales
The matter of Beaufort equivalent scales found renewed interest in the second half of the twentieth century. Especially within the wave modeling com-munity, still in 1990 the opinion prevailed that the WMO Beaufort equivalent scale (code 1100) is in error. For example, the WMO Commission of Marine Meteorology produced a scale – known as CMM-IV scale – using regression of anemometer measurements on Beaufort estimates, similar to the scale of Cardone.
Regrettably, in the new derivations the variable with small error variances was regressed on a variable with obviously much larger error variances, leading to biased scales.
Fortunately, the governing bodies of WMO adhered to code 1100, though admitted application of CMM-IV and similar scales for scientific purposes, so-called scientific equivalent scales. In retrospect, the scientific scales are the wrong choice. They give biased climato-logical means. The use of ‘scientific’ scales in wave modeling is even more questionable since this practice is at variance to the use of code 1100 in operational weather analysis and forecasts. Seen in the light of the correct derivation by Lindau, it turns out that the insight of Ko¨ppen and Simpson around 1900 resulted in a scale, WMO code 1100, that is less biased than some scientific scales of 70 years later. (It might be noted that the description of Ko¨ppen’s method in the WMO Report on marine science affairs of 1970 is
reversed to what Ko¨ppen advocated and used: i.e., regression of Beaufort on wind speed.)
The regression technique of cumulated frequencies applied by Lindau had been used by Kaufeld before him. It has the advantage to account for non-uniform distribution of observations at the tails of the frequen-cy distribution. This technique can be seen as the nonlinear equivalent to the ‘orthogonal’ two-sided regression. Though the different error variances of the two variables were not accounted for, Kaufeld’s scale is certainly preferable to the so-called scientific scales derived by one-sided regression of wind speed on Beaufort force.
Kent and Taylor reviewed a selection of Beaufort equivalent scales and the techniques used in their derivation, and concluded that the Beaufort equiva-lent scale of Lindau is to be preferred when creating a homogeneous monthly mean wind data set from anemometer and visual winds in COADS.
Time Dependence of Beaufort Estimates
Increasing interest in long time series of climate data inevitably leads to the question of whether the reported winds from the oceans are homogeneous in time. We mentioned the slow change of observing practices. There have been changes in coding practices too. Originally, Beaufort forces were used in trans-mitting data. Effective 1948, WMO changed from Beaufort forces to transmit wind speeds in knots (1 knot equals 1 nautical mile per hour or 0.51 m s!1).
For a short time of transition, the erroneous use of codes may have influenced reported winds, but no long-term trend is expected.
There is an increasing number of ships carrying anemometers. Determination of true wind from ane-mometer measurements requires vector subtraction of course and speed of the ship, certainly an additional source for errors and inaccuracies. Also the code indicating the data as either measured (anemometer) or estimated (Beaufort scale) is known to be less reliable. Peterson found frequency distributions of estimated winds to show significant secular changes and ships carrying anemometers to report higher estimatedwinds than ships without anemometers. The latter is most embarrassing, since no single reason and simple cure can be given.
On the other hand, a trend towards higher wind speeds could well be an indication of climate change.
Growing amounts of greenhouse gases change the radiation balance. At the oceans surplus energy can fuel atmospheric circulations; e.g., midlatitude and BEAUFORT WIND SCALE 193