Revisiting the Analysis of the High Resolution Wind Model 

Using MADONA Field Data 

by Chatt C. Williamson, Dennis M. Garvey, and Sam S. Chang 

ARL-TR-3497 April 2005 

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Army Research Laboratory 

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ARL-TR-3497 April 2005 

Revisiting the Analysis of the High Resolution Wind Model 

Using MADONA Field Data 

Chatt C. Williamson, Dennis M. Garvey, and Sam S. Chang 

Computational and Information Sciences Directorate, ARL 

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Revisiting the Analysis of the High Resolution Wind Model Using 

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14. ABSTRACT 

Cionco and Byers (1995) conducted an evaluation of the High Resolution Wind model (HRW) performance using the 

MADONA field study data (Cionco et. al.1999). The primary indicator for model performance used in their evaluation 

was the regression coefficient obtained from fitting a linear model to the observed data versus the simulated data. The 

regression coefficients obtained in their study were remarkably high, 0.9701 and 0.8567 for wind direction and speed 

respectively. In this report, we look further into this result to try to understand the significance of the large correlations

indicated by these coefficients. We further investigate other statistical measures to evaluate the performance of the 

HRW model and compare these measures with a simple model using a homogeneous wind field. 

15. SUBJECT TERMS MADONA, HRW.

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iii

Contents 

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

Models and Data ...............................................................................................................4 

Methodology ......................................................................................................................5 

Results and Discussion ......................................................................................................7 

Conclusions ........................................................................................................................9 

References ..........................................................................................................................9

Distribution list.................................................................................................................10 

Figures 

Figure 1. Observed and HRW modeled Wind Direction for the 39 Cases listed in Table 1. 

Note that the abscissa is ordered by the M10 Wind Direction. ...........................................3 

Tables 

Table 1. The 39 cases used for the evaluation of the HRW models performance. The last 

column shows the order number for wind direction corresponding to the abscissa for 

Figure 1. ...............................................................................................................................2 

Table 2. List of the statistical values r, SEE, Bias, MAE, AAE, and RMSE for wind 

direction and wind speed for the four models considered. ..................................................7 
iv

This page is intentionally left blank. 
1

Introduction 

A number of regression studies addressing the performance of the HRW model were 

performed by Cionco and Byers (1995) using data obtained during the MADONA field 

study. These authors evaluated the model using a simple linear regression analysis, 

namely y=ax+b, where x represents the measured wind direction or speed and y the 

corresponding model result. At least two wind fields were used to initialize the model. 

In this report, we will focus on the regressions performed using the M10 station to 

initialize the wind field of the model. Table 1 lists the 39 cases used in the evaluation of 

the HRW model performance and indicates the wind direction and speed used to initialize 

HRW. Our investigation was motivated by two factors: first, the unexpectedly high 

correlation values for wind direction as reported by Cionco and Byers (1995) and, 

secondly, graphs such as that presented in figure 1. Figure 1 is a plot of the wind 

direction, both modeled and observed at the various observation stations ordered by the 

observed wind direction measured at station M10. (The observed wind direction from the 

M10 station is also included for reference.) From this plot one can see that there can be 

significant spread between the observed and modeled wind direction. The same can also 

be said for measured and modeled wind speeds. It should also be noted that there are a 

number of cases where the observed wind direction and the corresponding model wind 

direction are on opposite sides of the M10 observation, the significance of which will be 

discussed later. 

While HRW is a fairly simple model for this particular case, i.e., the MADONA 

evaluation, there is an even simpler model that we can compare against. This model can 

be termed the spatially homogeneous model (SHM). One advantage to comparing 

against this particular type of model is that the methodology for initializing the HRW 

model is to use an initial wind field that is homogeneous throughout the domain. The 

advantage here is we can estimate the value added to the estimation obtained by running 

the HRW model. 
2

Table 1. The 39 cases used for the evaluation of the HRW models 

performance. The last column shows the order number for wind direction 

corresponding to the abscissa for Figure 1. 

Wind @ M10 Case 

No. Day Month Time 

Speed Direction 

Order Number 

by Wind 

Direction 

1 14 Sept 14:10 5.9 286 34 

2 15:55 7.6 281 29 

3 17:20 8 289 36 

4 15 Sept 12:10 8.5 236 22 

5 13:55 8.1 234 21 

6 15:55 8 229 18 

7 17:40 5 231 20 

8 16 Sept 10:30 4 98 2 

9 13:55 2.7 153 7 

10 16:10 4.7 144 6 

11 17:50 3.2 179 9 

12 17 Sept 10:30 5.2 132 4 

13 14:15 3.8 206 12 

14 17:25 7.4 193 11 

15 18 Sept 10:30 4.1 285 33 

16 13:10 4.5 275 25 

17 14:55 2.2 213 13 

18 18:15 2.8 182 10 

19 19 Sept 10:30 5.8 286 35 

20 12:10 4.9 283 31 

21 14:00 4.8 282 30 

22 16:10 4.4 295 37 

23 18:25 1.4 223 15 

24 20 Sept 10:30 4.7 218 14 

25 12:40 3.3 224 16 

26 14:45 3.4 249 23 

27 15:55 2.7 230 19 

28 21 Sept 10:30 6.5 75 1 

29 14:25 4 129 3 

30 15:25 4.7 133 5 

31 18:00 3.1 161 8 

32 22 Sept 10:30 1.6 228 17 

33 13:30 3.4 336 38 

34 15:10 3.4 351 39 

35 23 Sept 10:30 6.7 279 28 

36 12:00 6.8 284 32 

37 14:15 6.9 277 26 

38 16:10 4.1 278 27 

39 17:40 2.3 258 24 
3

45

90

135

180

225

28 8 29 12 30 10 9 31 11 18

Case Index for HRW Simulation

Wind Direction (Degrees)

Observations

HRW

M10 Observation

135

180

225

270

315

14 13 17 24 23 25 32 6 27 7

Case Index for HRW Simulation

Wind Direction (Degrees)

Observations

HRW

M10 Observation

180

225

270

315

360

5 4 26 39 16 37 38 35 2 21

Case Index for HRW Simulation

Wind Direction (Degrees)

Observations

HRW

M10 Observation

180

225

270

315

360

20 36 15 1 19 3 22 33 34

Case Index for HRW Simulation

Wind Direction (Degrees)

Observations

HRW

M10 Observation

Figure 1. Observed and HRW modeled Wind Direction for the 39 Cases listed in Table 1. Note that the abscissa is 

ordered by the M10 Wind Direction. 
4

Models and Data 

The reader is referred to Ball and Johnson (1978) for a review of the basis of the HRW 

model. Slight modifications have been made to this code base (Byers 1985) though the 

underlying numerical scheme remains essentially unchanged. Briefly, the HRW model is 

a limited area diagnostic wind model that requires minimal meteorological information to 

produce a wind field estimate. Typical domains are on the order of 5 km by 5 km with a 

horizontal resolution of 100 m. Input required to initialize the model is a single wind 

speed and direction, a potential temperature profile, and an effective terrain height which 

is the combination of both the underlying terrain and the morphological features 

contained within the model domain. 

The other models being considered, the spatially homogeneous models (SHM), are very 

simple. This type of ‘model’ is often used to drive transport and diffusion models for a 

quick assessment of a hazardous plume’s fate. One cannot overemphasize the simplicity 

of the SHM. We note that there are obvious disadvantages to using simple homogeneous 

estimators should the complexity of the underlying terrain increase, should the domain of 

interest become relatively large, or should the atmospheric dynamics from larger scale 

features, such as frontal passages, influence the flow. 

For convenience, we can denote the HRW model and the different SHMs we consider as 

HRW Model: M1 ( , ) 1 1 1 = V u v , 

SHM (M10): M2 ( , ) 2 2 2 = V u v , 

SHM (μ): M3 ( , ) 3 3 3 = V u v , 

SHM (LRO): M4 ( , ) 4 4 4 = V u v . 

Here ( , ) 1 1 1 V u v is a wind vector field produced by the HRW model (M1) as briefly 

described above. The second model (M2) is a spatially homogeneous model which 

produces a wind fieldV2 equal to the observed wind vector at station M10. The field V2

is constant throughout the domain (i.e.( const, const) u2 = v2 = ). Likewise, the third 

model (M3) refers to a homogeneous wind field,V3 equal to the mean observed wind, 

i.e.,
5

. 1

, 1

1 3

1 3

∑

∑

=

=

=

=

n

k k

n

k k

v

n

v

u

n

u

Here n is the total number of observations that were used in the previous regression 

studies of the MADONA data by Cionco and Byers (1995). Typically n was 11, though 

there were two cases, Case 9 and Case 23, where n was 10 and 9, respectively. The 

fourth model considered (M4) called the least representative observation model (LRO) is 

represented by a constant wind vector fieldV4 , which is the homogeneous vector field 

corresponding to the observation which produces the lowest value of the linear 

correlation coefficient r. This coefficient will be discussed shortly. The LRO and spatial 

mean cases are used to establish a baseline of the possible worst case and best case 

scenarios, respectively. 

The data used in this analysis was previously presented in Cionco and Byers (1995). We 

do further analysis of the observed-simulated data pairs in this report. The domain of 

interest was Porton Down, UK. The terrain could be described as gentle to moderate 

rolling hills. 

Methodology 

To investigate the relatively high linear regression correlation coefficients previously 

reported, we would like to establish a benchmark which can be compared against on a 

relative basis. This initial benchmark will use the linear regression correlation 

coefficient. We compare the regression coefficients for both wind speed and wind 

direction as computed by Cionco and Byers (1995) to those produced by computing the 

linear regressions for the SHM realizations described in the previous section. With this 

initial benchmark we hoped to be able to estimate the value added by running the HRW 

model for the MADONA case study. 

A simple linear regression analysis can be expressed as 

y = mx + b.

Here x is the observed value of either wind speed or wind direction. The variable y is the 

corresponding value produced by one of the four models M1, M2, M3, or M4. There are 

four measures that one can consider when using linear regression techniques to evaluate 

models. These four measures are m – the slope of the linear model, b – the y-intercept of 

the linear model, r – the correlation coefficient of the model, and SEE – the standard error 

estimate of the model. If the linear model were a perfect representation of the 

correspondence between the measured and modeled values, we would have the 
6

following: 

0.

1, and

0,

1,

=

=

=

=

SEE

r

b

m

Typically, when using linear regression, we are concerned with m, r, and SEE. The key 

indicator used by Cionco and Byers (1995) was the correlation coefficient r. This is 

defined as follows: 

( )( )

∑ ( ) ∑ ( )

∑

= =

=

− ⋅ −

− − = n

i i

n

i i

n

i i i

x x y y

x x y y

r

1

2

1

2

1 . 

The standard error estimate is defined as: 

( ) . 2

1

1

Model 2 Model Linear ∑ = − − = n

i i i y y

n

SEE

We also look at other traditional statistical measures used in model evaluation studies 

such as bias, maximum and average absolute error, and the root mean square error 

(RMSE). They provide a more direct comparison of the differences between the 

measurements and the model outputs and are independent of the regression analysis, used 

to evaluate the model performance. 

We define the Bias of a model to be the following: 

∑ ( ) = = − n

i i i y y

n

Bias

1

1 Model Obs

Note that the Bias gives a gross estimate of the model’s tendencies. For example when 

modeling wind direction, we can estimate whether the model tends to favor being 

clockwise or counter-clockwise of the observed value. Knowing this characteristic, a 

modeler could then either correct the model as appropriate or inform users of the known 

bias. We define the Maximum Absolute Error MAE as: 

Model Obs max 1 i i

n

i MAE = y − y =

This particular error estimate is a measure of the worst case that a modeler could expect 

and could possibly establish a bound on whether or not to use the information produced 

by or derived from the model output. We define the Average Absolute Error AAE as 

follows: 

∑= = − n

i i i y y

n

AAE

1

1 Model Obs
7

This error estimate is similar to the bias estimator described above and can be used in a 

similar manner. Finally, the Root Mean Square Error RMSE is defined as: 

( )2

1

1 Model Obs ∑= = − n

i i i y y

n

RMSE

This is the traditional root mean square error and is generally thought of as a good 

estimator for model performance. 

Results and Discussion 

Table 2 lists the values of r, SEE, Bias, MAE, AAE and RMSE for the wind direction and 

wind speed for the four models. As can be seen in table 2, even the homogeneous model 

using the Least Representative Observation (LRO) yields a regression coefficient that is 

quite high, indicating two things. First, this is an indicator that the underlying terrain, 

while not flat, is in a sense fairly benign, confirming our estimate earlier that the terrain 

could be described as gentle to moderate rolling hills. Second, this indicates that the 

correlation coefficient is a relatively weak measure in determining the validity of the 

model. It should also be noted from the table above that the model, in terms of the 

correlation coefficient performed, worse when compared to the homogeneous field used 

to initialize it. 

Table 2. List of the statistical values r, SEE, Bias, MAE, AAE, and RMSE for wind direction and 

wind speed for the four models considered. 

 r SEE Bias MAE AAE RMSE 

Wind Direction 

 HRW 0.9701 16.3 2.97 60 12.2 17.1

 SHM (M10) 0.9811 12.5 1.98 64 9.8 13.3

 SHM(μ) 0.987 10.3 -0.01 72.6 7.3 10.4

 SHM(LRO) 0.9666 16.7 1.88 110 10.9 17.6

Wind Speed 

 HRW 0.8567 1.02 0.33 3.6 0.85 1.13

 SHM (M10) 0.9149 0.77 0.17 3.1 0.64 0.86

 SHM(μ) 0.9357 0.65 0 2.9 0.53 0.69

 SHM(LRO) 0.8454 0.83 -0.29 3.1 0.86 1.13
8

Since the correlation coefficient is a fairly weak estimator in measuring the model 

performance for the MADONA studies, it is appropriate to look at other measures to 

ascertain the model performance. These other measures, as described previously, are the 

Bias, MAE, AAE, and RMSE. The values for these measures for both wind speed and 

wind direction are also reported in table 2. As is indicated in table 2, in each case the 

statistical measure becomes slightly worse once the homogeneous field is relaxed through 

the running of the HRW model, except for the wind direction maximum absolute error. 

Here, in this one case, we see a slight improvement. Recalling the caveats of the simple 

SHM model presented earlier, we investigated the relatively large MAE for both the 

HRW model and the SHM. These values were corresponding values for station M12 for 

Case 27 of the model runs. An inspection of the five minute averaged data revealed that 

the reason for these high values was most likely a frontal boundary with a wind shift 

propagating through the domain from west to east. Further analysis showed that the next 

largest MAE for the SHM was 48 degrees (Case 9), while there were three cases (9, 26 

and 27), where the MAE for HRW was 48 degrees or greater. This further analysis also 

showed that there were 7 cases (9, 12, 18, 23, 26, 27, and 32) where the SHM and HRW 

both had an MAE greater than 30 degrees. There were an additional 9 cases where the 

HRW MAE exceeded 30 degrees while the SHM MAE did not. 

It was noted earlier in our discussion of figure 1 that for a number of the observations, the 

corresponding model value produced by HRW was on the opposite side of the M10 

observation. Specifically, of the 426 regression data pairs, 164 or slightly more than 1 in 

3, exhibited this behavior. Recalling that HRW is initialized with a homogeneous wind 

field equal to the station M10 observation {i.e., SHM(M10)}, we see that HRW actually 

‘corrected’ the wind field in the wrong direction more than a third of the time. Slightly 

less than half, specifically 196 cases, were relaxed or ‘corrected’ in the appropriate 

direction. The remaining cases consisted mostly of those observation-model data pairs 

associated with a station M10 observation. Here the model relaxed the vector field away 

from the observation. 

There were other measures of model performance that were investigated that are not 

reported here. These included looking at each case individually to see whether the model 

captured the range (max, min) of the observations for the wind speed and wind direction, 

as well as the mean, the median, and the quartiles. There was no discernable pattern to 

make any significant conclusion, though the mean values were typically within reason. 
9

Conclusions 

By these yardsticks, there is little to no value added by running HRW to estimate the 

wind field in this particular case. The correlation coefficients derived from the linear 

regression model for the Spatially Homogeneous Model for both the station M10 

observation and the observations spatial mean for each case were slightly better than 

those that were previously reported. The reason for the high correlations previously 

reported is associated with the already high correlation of the observed winds at all of the 

sites with the homogeneous field used to initialize HRW for any one run. We can see 

that, statistically, the model performance is actually slightly worse than the homogeneous 

estimate used to initialize the model, and this holds true for all statistical measures that 

we have considered. As a measure of the HRW model performance, the correlation 

coefficient by itself and in combination with the standard error is quite weak and is not a 

good indicator of the validity of the model. 

References 

Ball, J. A. and S. A. Johnson, 1978: Physically-based High-Resolution Surface Wind and 

Temperature Analysis for EPAMS, Report ASL-CR-78-0043-1. 

Byers, J. H., 1985: High Resolution Wind (HRW) Program for Wind, Temperature, and 

Turbulence Analysis, Report submitted to ASL under Contract DAAD07-82-C-0009. 

PSL, NMSU. 

Cionco, R. M. and J. H. Byers, 1995: Initializing the HRW Model with Meteorological 

Inputs from Different Sites to Optimize Model Output. Proceedings of the 11th

Symposium on Boundary Layers and Turbulence, AMS, Boston, Mass. 

Cionco, R. M., W. aufm Kampe, C. Biltoft, J. H. Beyers, C. G. Collins, T. J. Higgs, A. R. 

T. Hin, P.-E. Johansson, C. D. Jones, H. E. Jorgensen, J. F. Kimber, T. Mikkelsen, K. 

Nyren, D. J. Ride, R. Robson, J. M. Santabarbara, J. Streicher, S. Thykier-Nielsen, H. 

van Raden, and H. Weber, 1999: An Overview of MADONA: A Multinational Field 

Study of High Resolution Meteorology and Diffusion over Complex Terrain, BAMS 

Vol. 80, No. 1, January 1999. 
10

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