From Eq. () one can see that for U measured at two heights {z1,z2}, the roughness can be calculated by ln⁡(z0)=U(z2)ln⁡z1-U(z1)ln⁡z2U(z2)-U(z1). While one can also obtain the roughness via the shear exponent e.g., that is often used in wind energy, Eq. () does not involve approximations and directly follows from the definition of roughness. One can also use friction velocity measured in the surface layer and wind speed from one (or more) height(s) to derive roughness (e.g., z0=zexp⁡[-κU(z)/u*] from Eq. ), but doing so requires sonic anemometers, which are not yet commonly used in the wind energy industry. Thus, we use Eq. () for the observed roughness data analyzed and shown in this paper, and leave alternate z0 estimation methods for concurrent work and dissemination that focuses solely upon roughness. This choice is further supported by the focus of the present article – we are concerned here with the impact of roughness length on wind energy estimates – and because we develop and use an uncertainty-estimation framework that is generally applicable to z0, regardless of whether z0 is derived from Eq. () or via U/u*.

Even in seemingly ideal conditions – such as measuring wind profiles in the surface layer at a site where the terrain is flat and appears uniform, with non-neutral cases excluded – in practice one still observes a broad range of roughnesses. This is demonstrated in Fig. , which shows the roughness length calculated from 10 and 40 m measurements at the Danish National Wind Turbine Test Station at Høvsøre for upwind directions corresponding to flat and homogeneous surfaces (east of the meteorological measurement mast). Here we have filtered out non-neutral conditions by keeping only cases unaffected by thermodynamic stability by using z/|L|<0.001, i.e., for Obukhov lengths L much greater than the heights of measurement.

Figure  starkly demonstrates that even at a homogeneous, well-studied, and presumably simple site, roughness length has a distribution of significant width. Note that we plot the distribution of roughness length in logarithmic space; this is done because it is ln⁡(z0) that directly affects the wind profile, as in Eq. (). This also highlights the breadth of the distribution (several orders of magnitude) and that we must subsequently approach roughness uncertainty in a multiplicative (dimensionless) way and not in an additive way. We also remind that the roughness lengths generally used in wind flow modeling and resource assessment actually correspond to some geometric mean, which should be based on the z0 distribution alternately one can express wind profiles in terms of the distribution P(ln⁡z0) and corresponding arithmetic mean; cf.,; unfortunately z0 is not (yet) defined explicitly as such in typical wind engineering practice. Thus, in this paper we focus on roughness uncertainty within the “implied mean-roughness” framework implicit in standard wind engineering.

In addition to the relatively wide distribution apparent for roughnesses obtained from 30 min averages shown in Fig.  (and slightly wider for 10 min averages, not shown), one can also see some local – and nonideal – details. One sees the minor effects of a barn and a small building located roughly 800 m upwind at ∼ 80 and ∼ 110∘, respectively, as well as the larger effect of the seasonally varying marsh–fjord coastline 800–900 m to the southeast (∼ 130–135∘). Such roughness changes tend to violate the assumptions behind the logarithmic profile over a range of observation heights falling within the nonequilibrium internal boundary layer (IBL) transition region

The IBL develops downwind from a roughness change with expansion slope (z : x) of roughly 1 : 100, and the top of the associated transition region expands at a variable rate of 1 to ∼ 12–15. For the example noted here, this corresponds to the flow measured by anemometers at both 10 and 40 m being affected.

For the observation-based roughness lengths displayed in Sect.  (Fig. ), the distributions are best described (and thus plotted) as P(ln⁡z0), again consistent with both the ln⁡(z0) behavior expected within the wind profile and with the geometric (multiplicative) averaging needed to obtain a characteristic mean roughness. The width of the ln⁡(z0) distributions shown in Fig.  gives indication of the variability in ln⁡z0 over many 30 min (or 10 min) periods. In particular the P(ln⁡z0) for 30∘-wide directional sectors can be considered, that is P(ln⁡z0|φ), since sectors of this width are commonly used in resource assessment. The homogeneous 60 and 90∘ sectors at Høvsøre (Fig. ) have similar shapes, and both exhibit half-peak widths of roughly one-half order of magnitude (a factor of ∼ 3); i.e., for a given sector's background roughness z0, the width of the distribution can be seen as that defined roughly between z0/3 and 3z0.

However, the uncertainty in determining a representative roughness length – via the appropriate (geometric) mean – is not the same as the width of the ln⁡(z0) distribution. Rather, the uncertainty in the mean roughness is the width of the distribution of expected means calculated for a given site and sector. For this purpose we use a basic “bootstrap” resampling method : simply resampling randomly from the diagnosed (30 min) roughness lengths, we synthesize a distribution of 105 values of geometric-mean roughness (〈z0〉g) per sector. This results in a log-normal distribution of mean z0 (Gaussian distribution of ln⁡z0); this distribution P(exp⁡ln⁡z0) is centered around a value equal to the geometric mean that had been found for each sector by operating directly on the wind data. The width of each (sector-wise) distribution of mean roughnesses from resampling depends on the number of resampled points used to create each mean in the synthesized distribution. For a number equivalent to 1 year's worth of data (based on the sector-wise frequency of occurrence), the mean distributions are in fact much narrower than the distributions shown in Fig. . The bootstrapped mean-roughness distribution is almost perfectly fit by a log-normal form; the half width w〈z0〉RS for this form can be simply expressed non-dimensionally (i.e., effectively normalized by the expected mean) via the standard deviation of mean ln⁡z0 from resampling (σ〈ln⁡z0〉RS) as w〈z0〉RS〈〈z0〉RS〉g=exp⁡σln⁡[z0/〈z0〉g]RS=exp⁡σ〈ln⁡z0〉RS-1.

For the Høvsøre homogeneous land sectors treated here and the bootstrapped means, each calculated from 1 year's worth of resampled data, the w〈z0〉RS of the sector-wise distributions of these means are about 5 % of the expected mean roughness length (specifically, 5.4, 4.1, and 5.3 % of the respective 〈〈z0〉RS〉 in each sector from Eq. ). Thus, considering only calculations of z0 from wind speeds measured at two (10 and 40 m) heights via Eq. () from 1 year of data, the roughness uncertainty for the three sectors shown in Fig.  is about 5 %. For longer data sets, the uncertainty decreases; for example, randomly drawing from the entire 10-year set leads to half widths of 1–2 %.

One should be reminded that there are other methods to calculate z0, such as using the surface-layer friction velocity u* and wind speed at one (or more) measurement height(s) via Eq. (), which may result in different values of estimated mean and/or characteristic roughness length. For example, repeating the analysis above using Eq. () with U and u* measured at 10 m height, we again obtain well-behaved distributions of bootstrapped mean roughness whose half widths are about 5 %; one might take this as the implied uncertainty. However, the mean values (for a given sector) can actually differ between the two methods by an amount that can greatly exceed 5 % (in these Høvsøre land sectors they can differ by a factor of ∼ 3!). This difference is related to the flow physics at increasing distances from the mast (the momentum flux footprint), the details of which are beyond the scope of this paper; we defer further discussion of such differences to Sect. .

Even for an ideal homogeneous landscape, the wind industry, which is a collection of wind engineers and companies, will as a group assign different roughnesses to characterize the surface (whether actively or inherited via acquired maps). This results in a distribution of z0 assigned to predict the wind for any given site, and in effect to an (epistemic) uncertainty, and subsequently industry-wide variation in predicted AEP, even at the most simple sites.

We provide a simple practical example of gauging such epistemic uncertainty based on a systematic exercise: we asked separate groups of wind resource assessment experts to individually evaluate the surface roughness length for two commonly encountered land surface types. The groups of participants in this exercise were polled at meetings of the Danish Wind Power Network “Vindkraft-Net” and of the Meteorology section of the Department of Wind Energy (Risø lab/campus) in the Danish Technical University; their backgrounds and foci range from wind engineering and commercial site assessment to research boundary-layer meteorology and wind resource calculation.

The participants were shown a picture containing both a grassy area and a forested area (the latter specified as having a mean tree height of 15 m) and were asked to give z0 for each of these two areas; the picture is replicated in Fig. . The raw results of the roughness survey, which consisted of 19 and 28 participants are shown in Table .

Note that Table  includes not only a geometrically defined mean z0g≡[∏in(z0)i]1/n=exp⁡∑inln⁡(z0)i/n and associated dimensionless standard deviation exp⁡{σln⁡z0/〈z0〉g} that are consistent with the logarithmic definition of roughness, but also the commonly used arithmetic mean and (normalized) standard deviation of user-estimated z0. The latter statistics are included for comparison and because (in contrast to the flow physics) there is some tendency for wind engineers to think linearly rather than logarithmically. As can be seen in Table 1, the arithmetic (linear) mean of z0 is unsurprisingly larger than the properly (logarithmically) averaged z0, by ∼ 30–40 % for grass and ∼ 20–80 % for forest. Arithmetic calculation of z0 statistics subsequently tends to give a smaller normalized deviation compared to the proper log-rms statistic for the raw surveyed data, particularly as the z0 distribution is dominated by values smaller than 1 m (expected from the mathematical character of geometric [ln⁡z0] vs. arithmetic averages). Overall, the variability in polled roughness lengths for the two cases is on the order of but larger than the expected roughness length itself, i.e., by a factor of ∼ 1.1–1.3 times the estimated mean z0 for grass or ∼ 1.1–1.6 times the mean for the forest case. This might be taken as an estimate for uncertainty in z0 for such cases.

The variability in the user data differs between the polled groups and might be affected by the limited sample size. Due to the limited distributions of polled roughness lengths (not shown) gathered from each of the two expert groups, an alternate estimate of collective user uncertainty (i.e., industry-wide) is provided by again applying a resampling method to the distribution of surveyed z0. Following the averaging of expert-elicited z0 and the uncertainty characterization of the previous section, bootstrapping is used to resample the elicited z0 values and construct a distribution of the means. Calculating each mean from n nonunique random data samples and repeating 106 times, we generate distributions of z0‾ for the two cases. For n≳3, we find log-normal distributions for the bootstrapped geometric mean z0‾=〈z0〉g, as expected from the central limit theorem (ln⁡z0‾ becomes Gaussian). In the limit of the sample data set being perfectly representative of wind industry practices, the bootstrapped distribution for a given n is equivalent to the P(z0‾) expected when any given wind engineer uses n values to calculate the mean roughness for a site such as the grass or forest case used here. The means of the resampled distributions are the same as for the raw roughness samples in Table 1, regardless of n. The deviation, however, decreases with n. For n=1, the deviations converge to those in Table 1, while the values of the effective geometric deviation w〈z0〉RS behave as approximately 1+n-0.53 (the deviations fall slightly more rapidly than n-1/2 due to the slightly irregular sample or survey). As an example, Table  shows the geometric means and deviations for these mean distributions using n=3 for the two groups and cases considered.

From Table , one infers the seemingly obvious result that for users taking an average of three industry-accepted roughness estimates (assuming that they span the sample taken) – instead of just one – the expected (industry-wide) uncertainty is reduced; we point out that such a conclusion depends on having reasonably representative roughness values to choose from.

To summarize, in this subsection we saw that the equivalent (normalized logarithmic) standard deviation from surveys of engineer or user-assigned roughness is of the order of the expected roughness itself, as shown in Table . In terms of Eq. (), we expect an uncertainty equal to the half width of the (expected user input) distribution of ln⁡z0 to be approximately w〈z0〉g∼〈z0〉g. In the following section we would like to show, in general, how uncertainty in z0 – whether due to user input or measurement – propagates into wind speed and AEP estimates.

By using the logarithmic wind profile (Eq. ) at both measurement and prediction locations, along with the forward and reverse geostrophic drag-law forms, Eqs. () and (), one can write the predicted wind speed Upred in terms of the prediction-site roughness z0,2 and geostrophic wind G. The geostrophic wind is further expressible in terms of the measured wind Uobs, measurement height zobs, and background roughness z0,1 for the measurement site. The resulting expression for Upred can be differentiated with respect to any of {Uobs,zobs,zpred,z0,1,z0,2} in order to find the sensitivity of predicted wind speed Upred to these quantities. We would like to know the effect of roughness uncertainty upon Upred; taking its derivative with regard to the roughness lengths at observation and prediction heights and rearranging, we obtain the useful expressions ∂ln⁡Upred∂ln⁡z0,1≃1ln⁡(zobs/z0,1)1-1ln⁡[G/(fz0,2)]-A0×1-Uobs/Gln⁡(zobs/z0,1)2ln⁡κUobsfz0,1-A0ln⁡zobsz0,1-1 and ∂Upred∂ln⁡z0,2≃cGGκA0+ln⁡zpredf/Gln⁡G/(fz0,2)-A02. Here we have made the expression compact by writing G(Uobs,zobs,z0,1) simply as G. Inspection of the two sensitivity expressions, Eqs. () and (), reveals that Upred is more sensitive to the background roughness at the observation site (z0,1) than the roughness z0,2 at the prediction site. Furthermore, it is seen that Upred also has some sensitivity to observation height zobs, while z0,1 dominates.

From Eqs. ()–(), which follow from Eqs. ()–() (see Appendix A for details), we arrive at an (implicit) expression relating the uncertainty in predicted hub-height wind speed to the uncertainty in background roughness at the observation site (Δz0,1): ΔUpredUpred|Δz0,1≃exp⁡1.11+zobs80m-1/7×li(zobs/a1z0,1)-1/7-li(z1/z0,1)-1/7}, where li(x) is the log-integral function (e.g., ; see appendix also). In Eq. (), a1 is the fractional uncertainty in observation-site background roughness length, a≡z0+Δz0z0=1+Δz0/z0, evaluated at z0=z0,1. Thus, roughness uncertainties can be described geometrically (as they should be): for a given background roughness, we then have a range of log roughness described by ln⁡(z0,1)±ln⁡(a) and corresponding roughness lengths ranging from z0,1/a to az0,1.

Just as Eq. () was derived above for variations in roughness at the measurement site, we similarly derive the uncertainty in predicted wind speed due to uncertainty in the prediction-site roughness z0,2 from Eq. (): ΔUpredUpred|Δz0,2≃1-ln⁡aln⁡(zpred/z0,2)1+ln⁡aA-ln⁡[G/(fz0,2)]. This follows from Eq. (), which includes details of the derivation (Appendix A).

The sensitivity of hub-height (predicted) wind speed to z0,1, via Eq. (), is shown in Fig.  for the case of zobs= 60 m observation height and a hub height of 100 m. Similarly, the uncertainty in predicted wind speed due to uncertainty in prediction-site roughness z0,2, via Eq. (), is displayed in Fig. .

The estimated relative uncertainty in predicted wind speed (ΔUpred) is first plotted vs. fractional roughness uncertainty a for a number of different measurement-site background roughnesses (z0,mast), and then it is also plotted against z0,mast for different relative roughness uncertainty (Δz0,mast/z0,mast=a-1), expressed as a percentage. For small background roughnesses one can see less effect on predicted wind speed for a given roughness error or uncertainty, with a nearly linear dependence of relative wind speed uncertainty upon z0,mast for measurements taken over smooth land or water (z0,mast<∼ 1 cm). For larger magnitudes of roughness uncertainty, as expected, one sees larger expected uncertainty in wind speed as well; this effect is reduced for smooth measurement sites (in conjunction with the previous statement). Also, for higher background roughnesses, the sensitivity of wind speed to (relative) roughness error is amplified, as shown by the green lines in panel a or the right-most (high z0) part of panel (b) in Figs. –. Comparing Figs. –, one also sees that the effect of a given change (or uncertainty in) z0,2 has the opposite sign of the corresponding effect due to an equal change in z0,1, but with the measurement or mast location's roughness z0,1 having a larger effect than the prediction site roughness z0,2. That is, the magnitudes of ΔUpred(Δz0,1) in Fig.  are larger than the magnitudes of ΔUpred(Δz0,2) displayed in Fig. .

Above we saw that wind speeds predicted via the GDL (Eq. ) with roughness-affected (logarithmic) wind profile (Eq. ) can be more sensitive to z0,1 than to z0,2. Thus, for an overall bias in roughness estimates, we should expect a net bias in wind speed predictions via wind atlas methods. In other words, for roughnesses that are systematically overestimated (or underestimated) by the same factor abias at measurement and prediction sites, we then expect a corresponding bias in predicted mean wind speed. This effect is shown by Fig. , which displays the fractional change in predicted wind speed as a function of fractional change in measurement and prediction-site z0 for combinations of {z0,1,z0,2} that span typical application (colored lines).

As one might expect, for measurement and observation sites with similar background roughness, the change ΔUpred/Upred is relatively small, especially for systematically underestimated roughness lengths (abias<1). Figure  also shows that for small biases (abias→1), the wind speed prediction error is larger when the roughnesses at measurement and prediction sites are dissimilar. However, for roughness errors of a factor of ∼ 2 or more, the nonlinearity of Eq. () with () complicates the dependence of ΔUpred on abias. In addition to the typical range of z0 used in wind resource estimation (colored lines), Figure  also shows the gross effect of measurement over forest (or effectively more complex terrain, i.e., with effective roughness z0,1= 1 m; denoted by grey lines); one can see the corresponding increase in ΔUpred/Upred for such cases when there is overestimation of z0, even if z0,2 and z0,1 are both 1 m.

In contrast to a possible bias in roughness assignment, one can imagine a worst case scenario as having a negative error in z0,1 and a positive error in z0,2 (or vice versa), e.g., a1=1/a2. In this scenario the result resembles the plots in Fig. , but rotated 45∘ with the y axis stretched by a factor of 2: cases with z0,1=z0,2 no longer have small error, but all the lines show a large uncertainty for a far from 1 (e.g., ±40 % at a1±1=a2∓1=0.1 for z0,1= 1 cm and z0,2= 1 m, corresponding to the solid green line), and all lines have ΔUpred=0 for a=1.

A more general situation is that of independent errors in roughness assignment at different sites. In this limit, one foresees a distribution of ΔUpred, given uncertainties in z0,2 and z0,1 (basically P(z0,2) and P(z0,1)). Two examples of this are given in Fig. . The figure shows P(ΔUpred/Upred) for the cases of winds observed over grass but predicting winds over grass or forest, where the grass and forest z0 have log-normal distributions P(a) with means and widths given for the combined samples in Table . Following the earlier examples, the observation height is taken as 60 m and prediction (hub) height is 100 m.

In the figure, one can see the combined effect of different roughness distributions and uncertainties, particularly for the case of grass to forest (panel b in Fig. ). For this case, the half width of the grass P(z0,1) corresponds to 117 % (where 〈z0,1〉= 4 cm) and that for the forest corresponds to 141 % of 〈z0,2〉= 0.85 m following Table . The combined effect gives wider error distributions P(ΔUpred) for the grass-to-forest case than for the grass–grass case, as expected from Fig. , for example; the standard deviations corresponding to the z0-induced mean-wind error distributions in Fig.  are 1 and 4 % for the predictions over grass and forest, respectively (and both error distributions are nearly Gaussian, with skewnesses of 0.02 and -0.2). To be yet more conservative, if we follow Sect.  using a gross estimate of observational z0 uncertainty equivalent to a half width (roughness uncertainty factor) of w〈z0〉RS∼ 3, the uncertainty σΔUpred/Upred (distribution widths) for the two cases shown in Fig.  grow to 8.6 and 14 %. Towards practical consideration for wind engineers, we also point out that for prediction over water (again from zobs= 60 to zpred= 100 m with z0,1= 4.1 cm) using the conservative roughness-uncertainty estimate a=w〈z0〉RS∼ 3 again leads to uncertainty in Upred that exceeds 6 % and an error distribution that is somewhat non-Gaussian (skewness ≈ 0.6, plot not shown); we provide this number to demonstrate the roughness-induced uncertainty expected when using land-based measurements for offshore predictions.

The uncertainty in background roughness can also be translated into AEP uncertainty by employing a relation between wind speed and AEP, i.e., via a turbine (or perhaps wind farm) power curve. The propagation of z0 uncertainty to AEP follows that derived for wind speed above, but with some assumptions. First, we assume Weibull-distributed winds, which is standard practice in wind energy and also facilitates analytical derivation of a bulk relation between AEP and mean wind speed 〈U〉. Because power curves in practice do not have a “kink” at rated wind speed, but rather a smooth transition from the ideal 〈U〉3 regime to the maximum (rated) power regime of operation e.g.,, we can derive an analytical effective power-curve form, expressible as a function of 〈U〉/Vrat, i.e., Eq. () (shown in Appendix ). To accomplish analytical integration and readily relate mean wind speed (or Weibull-A parameter) per turbine-rated speed, some mathematical approximations are used, wherein we also assume that the Weibull-k parameter is close to a value of 2 (within 10–20 %). The analytical power-curve form PC(U/Vrat) then leads to a power-law relation between normalized AEP and wind speed: AEPnorm∝〈U〉/Vratp, where the power-law exponent p is also a function of 〈U〉/Vrat, as shown in Appendix .

Figure  shows an example of AEP sensitivity to fractional roughness uncertainty of the observation site (ratio of estimated to actual z0,1, i.e., a=Δz0,1/z0,1+1 as in Eq. ) for the case of 〈U〉=0.7Vrat; the latter translates to a power exponent of p≃ 1.85 for the analytical power-curve form elucidated in Appendix  (see Fig. b). For Fig.  we consider the same situation as used for Figs. – (zobs= 60 and zpred= 100 m). As one might expect, the AEP uncertainty – due to uncertainties in z0,1, z0,2, or their combined effect with a common bias – simply resembles the wind speed uncertainty plots shown in Figs. –: the vertical axis of the plot appears stretched by a factor of p (≃1.85). An analogous plot of the distribution of AEP error follows similarly; for a given value of p (here 1.85), the horizontal (x-) axes in the plots of Fig.  are stretched by a factor of p to give the distribution of ΔAEP.

The relative uncertainties implied by roughness lengths calculated via surface-layer wind speed measurements were outlined in Sect.  for the seemingly ideal grassy terrain east of Høvsøre. The half widths of the roughness distributions for the homogeneous sectors were found to be on the order of a factor of 3 times 〈z0〉, while the uncertainty in obtaining a mean (representative) roughness was found through bootstrap resampling to be much smaller, about 5 %; this result came whether z0 was calculated from speeds at multiple heights in the ASL or from sonic anemometer measurements of U and u* in the ASL. However, despite similar distribution widths and similar apparent uncertainty in mean-estimation, the 〈z0〉 themselves differed by roughly one-half order of magnitude, i.e., a factor of ∼ 3 when determined in these two different ways. Thus, we first consider (conservatively) a relative uncertainty of a∼3±1 for z0, for the typical resource-assessment heights (zobs= 60, zpred= 100 m) used in Figs. –. As seen in Fig. , for systematic (bias) overestimates of a≡Δz0/z0 and a mean roughness length at the observation site of 1 cm, this translates into wind speed uncertainty values of less than 1 % when predicting 100 m winds over the same roughness and gives ΔUpred of {2,-2,-6,-10%} for predictions over roughnesses of z0,2={0.2 mm, 3 cm, 30 cm, 1 m}. For the same magnitude of systematic underestimate (a∼1/3) the corresponding ΔUpred/Upred are {-1,2,5,9%} for these z0,2, with an uncertainty of 1 % for 100 m winds predicted over the same roughness as the measurement site. Thus, we see about 1 % uncertainty in Upred for these typical heights and the same observation and prediction roughness. Meanwhile, using such observations to predict winds over nearby forested land, for example, incurs higher uncertainties, with magnitudes of 5–10 %, without yet considering modeling the flow over such terrain. To get estimates of ΔU(Δz0) for other observation and prediction heights and roughnesses, we remind the reader that these can be obtained from Eqs. ()–().

For the uncertainties inherent in user-provided roughness lengths, we address the two cases treated in Sect. . The grass case is similar to that considered in the Høvsøre analysis above, with a mean roughness of about 4 cm. If we take the half width of the expected user-input distribution of z0, i.e., exp⁡σln⁡z0/〈z0〉g from Table , then we can again arrive at estimates for the wind-speed uncertainty (this is also a bit conservative because it gives larger uncertainties than the bootstrap-derived half width). Again assuming typical application heights (zobs=60, zpred= 100 m) for predictions over site roughnesses, z0,2={0.2 mm, 1 cm, 30 cm, 1 m} and a z0-bias of 2.2±1 (±120 % from Table ), we obtain Upred uncertainties of +{3,1,-3,-6%} and -{2,0.4,-3,-5%}. These roughly correspond to (a proxy of) the industry-wide uncertainty in predicted wind speeds (with this zobs, zpred) for observations over a background roughness like the grass in Fig. . For the surveyed forest roughness in that figure, we get corresponding ΔUpred following Table  for the case of all-site biases (±141 % →a≈2.4±1 applied to both z0,1 and z0,2). For predictions from observations over such a site, applied to turbine sites with z0,2= {1 cm, 10 cm, 1 m} we get ΔUpred≈(+){11,9,3%} for systematic overestimates and (-){6,4,0.3%} for systematic z0 underestimation. The latter finding is rather significant as it implies that an underestimation of forest roughness lengths is safer than overestimating z0 when using EWA-based methods for wind resource estimates (e.g., WAsP and similar methods). This is consistent with common practice: while recent evidence from direct lidar scans of forests suggests that z0 should be at least several meters there , industrial practice has been to use z0 of 1 m or less e.g.,.

The magnitude of z0-induced AEP uncertainty for typical simple sites depends in general on the ratio of 〈U〉/Vrat (for classically behaved turbines) because the relationship between 〈U〉 and AEP depends on this ratio; this dependence is most simply expressed via the exponent p=ln⁡(AEP)ln⁡〈U〉for a power-law relationAEP=〈U〉p, detailed in Appendix B. As mentioned in the previous section, with regards to uncertainty in the background roughness of either the observation or prediction site (or for a bias across both sites), the sensitivity plots of Δ〈U〉 per given roughness error are simply translated into analogous AEP-sensitivity figures via stretching the vertical axes by a factor p (as was done to get Fig.  from Fig. a); similarly the horizontal (ΔU) axis in Fig.  is stretched by a factor p. Since p basically varies between ∼ 0.8 and 2.5 (over the reasonable range of 〈U〉/Vrat∼ 0.5–0.9), then the mean wind speed uncertainties quoted in the previous subsection can be simply multiplied by a factor of ∼ 0.8–2.5, depending on the expected turbine power curve and subsequent p.

For most general practical use, we ultimately consider roughness error distributions and the consequent AEP error distributions, such as those shown in Fig. . For independent roughness error distributions at measurement and prediction sites, and assuming log-normal distributed Δz0 (as demonstrated in Sect.  and  for measured and user-estimated distributions), via Eqs. () and () we can obtain distributions of ΔAEP. The uncertainty in z0 can be expressed in terms of the dimensionless width wz0/〈z0〉; for a given width we can synthesize distributions of z0,1 and z0,2, and then find the standard deviation of the resulting distribution of ΔAEP. We do such Monte Carlo simulations over the range of dimensionless widths from 5 to 500 % for the same {z0,1,z0,2} pairs and observation and prediction heights as used in Fig. ; the results are shown in Fig. .

Figure  shows AEP uncertainty vs. roughness uncertainty; the latter is expressed as the dimensionless width wz0/〈z0〉 of the z0 distribution, calculated via the standard deviation of ln⁡(z0/〈z0〉) as in Eq. (). Following the previous subsection's analysis (where zobs= 60 m, zpred= 100 m, and p=1.85), Fig.  shows that for actual measurement-site roughness z0,1≈ 1–10 cm, given a relative z0 uncertainty of 100 % (corresponding to wz0∼z0 as in Sect. ), the GDL/z0-induced AEP uncertainty ranges from ∼ 5 % (for 〈z0,1〉= 1 cm, predicting over water) to 15 % (for 〈z0,1〉= 10 cm, 〈z0,2〉= 1 m). For the statistical uncertainty example of (mostly) homogeneous flat farm and/or grassland shown previously in Fig.  (Sect. ), taking the relative background roughness uncertainty factor to be equivalent to the width of the ln⁡(z0) distribution (centered around ∼ 1.4 cm for wind directions from ∼ 45 to 120∘), i.e., a∼3±1, leads to a similar AEP uncertainty range, roughly 6–16 % for prediction sites ranging from water to forest or urban. However, such an uncertainty estimate seems large and may be explained considering Table . For industrial use, wind engineers (e.g., in medium or large companies) in effect assign a kind of ensemble-average roughness length for any given land-use type. Consider, for example, the case of taking three community-accepted values for the grass site as in Table , i.e., a relative z0 uncertainty of roughly 50 %, one can see from Fig.  that the AEP uncertainty drops to 4–10 %. One is reminded that these AEP uncertainty values correspond to the case of observation and prediction heights of 60 and 100 m, respectively: the slight dependence of ΔAEP on zobs and zpred modifies the uncertainty for other heights. Aside from the weak dependence on measurement and prediction heights, one also sees a basic power-law form emerging for the AEP estimates: σAEP〈AEP〉∝∼wz0〈z0〉6/7, particularly for relative roughness uncertainties (widths of the distribution P(ln⁡z0/ln⁡〈z0〉)) that are wz0/〈z0〉<∼ 1–2.

We also note again that we have focused here on the AEP uncertainty caused by uncertainty in background roughness rather than the z0 uncertainty itself. Further details of the latter are the subject of ongoing work and another paper, and here we point to Fig.  as the significant result: for a given uncertainty in z0, one can find the corresponding uncertainty in AEP due to use of the GDL–EWA method.