On the Application of Linear Regression to Surface-Layer Wind Profiles for Deducing Roughness Length and Friction Velocity

Abstract

In a steady, spatially homogeneous neutral stability flow dominated by mechanical turbulence, the friction velocity ($u_{*}$) and the roughness length (z₀) can be estimated by applying linear regression to measurements of wind speed (u) at several heights (z). Surface-layer wind profile plots in the scientific literature normally have ln(z) on the vertical axis and u on the abscissa, which might suggest a linear regression, such as the ordinary least-squares method, which minimizes the vertical residuals, i.e., the ln(z) deviations, from the fitted line. Here, we show that ordinary least-squares fitting of the profile data is sensitive to the choice of the variable whose residuals are being minimized, and that the linear regression should be computed as u versus ln(z), i.e., minimizing the u deviations. This is equivalent to ln(z) being the independent variable and u the dependent variable. The differences in the estimated values of $u_{*}$ and z₀ compared to those resulting from the ln(z) versus u linear regression can be expressed as a function of the coefficient of determination (r²) of the wind-profile data. Applying the ordinary least-squares method while minimizing the deviations of ln(z) leads to systematic overestimation of the $u_{*}$ and z₀ values. Using these values as input into the atmospheric dispersion model AERMOD leads to increased shear-induced turbulence and consequently enhanced dilution of the plume.

Appendix: Derivation of the Differences of \( u_{*} \) and z0 from OLS(u|z) and OLS(z|u)

Let $\xi_{{u_{*} }}$, the difference in the estimates of $u_{*}$, be defined as

$$ \xi_{{u_{*} }} = \frac{{u_{*(z|u)} - u_{*(u|z)} }}{{u_{*(u|z)} }} . $$

(14)

Inserting the expressions for $u_{*}$ (Eqs. 6 and 9) in Eq. 14, the difference can be expressed as a function of the slopes of the linear regressions

$$ \xi_{{u_{*} }} = \frac{1}{{{\beta_{1}}_{(u|z)} {\beta_{1}}_{(z|u)} }} - 1 . $$

(15)

A relation between the slopes in Eq. 15 is found as follows: the slope β₁ obtained from the OLS(u|z; v) method (Eq. 4a) is equivalent to the ratio of the covariance of u and ln(z) to the variance of ln(z), which can be rewritten as

$$ \beta_{1(u|z;v)} = r \frac{{s_{u} }}{{s_{z} }} , $$

(16)

where s_z and s_u are the sample standard deviations of the ln(z) and u data, respectively, and r is the Pearson’s correlation coefficient, defined as the covariance of ln(z) and u divided by the product of their standard deviations, that is, cov(ln(z), u)/(s_zs_u).

The line obtained from the ordinary least-squares regression of u versus ln(z) while considering the minimization of the horizontal residuals between the data and the regression line will have the slope (Isobe et al. 1990)

$$ \beta_{1(u|z;h)} = \frac{1}{r} \frac{{s_{u} }}{{s_{z} }} . $$

(17)

Switching the u and ln(z) axes after performing the linear regression is equivalent to a simple reflection across the line u = ln(z), thus

$$ \beta_{1(z|u;v)} = \frac{1}{{\beta_{1(u|z;h)} }} . $$

(18)

Hence, the product of $\beta_{1(u|z;v) }$ and $\beta_{{1(z|u;{\text{v}})}}$ using Eqs. 16–18 gives

$$ \beta_{1(u|z;v) } \beta_{1(z|u;v)} = r^{2} . $$

(19)

Therefore, the difference between the two estimates of $u_{*}$ depends on the coefficient of determination between u and ln(z)

$$ \xi_{{u_{*} }} = \frac{1}{{r^{2} }} - 1 . $$

(20)

By replacing the definitions of the slopes of the regression lines (Eqs. 6 and 9) in Eq. 19, one obtains

$$ {u_{*}}_{(z|u)} = \frac{{{{u_{*}}}_{(u|z)} }}{{r^{2} }} , $$

(21)

which explains readily why the $u_{*}$ computed from OLS(z|u) is always greater or equal than the one from OLS(u|z). Regarding the roughness length, let $\xi_{{z_{0} }}$, the difference in OLS estimates of z₀, be defined as

$$ \xi_{{z_{0} }} = \frac{{z_{0(z|u)} - z_{0(u|z)} }}{{z_{0(u|z)} }} . $$

(22)

Substituting for $z_{0}$ (Eqs. 7 and 10) in Eq. 22, gives

$$ \xi_{{z_{0} }} = { \exp }\left[ {{\beta_{0}}_{(z|u)} + \frac{{{\beta_{0}}_{(u|z)} }}{{{\beta_{1}}_{(u|z)} }}} \right] - 1 . $$

(23)

For a regression line, the y-intercept is defined by Eq. 4b. Applied to OLS(u|z) and OLS(z|u), this gives

$$ {\beta_{0}}_{(u|z)} = \bar{u} - {\beta_{1}}_{(u|z)} \overline{{\ln \left( {\frac{z}{{z_{ref} }}} \right)}} , $$

(24a)

and

$$ {\beta_{0}}_{(z|u)} = \overline{{{ \ln }\left( {\frac{z}{{z_{ref} }}} \right)}} - {\beta_{1}}_{(z|u)} \bar{u} , $$

(24b)

respectively, with the overbar representing the mean value of the underlying quantities. After replacing in Eq. 23 the expressions for β₀ in Eq. 24a, b, and using the expression of the correlation of determination (Eq. 19), the expression for ${u_{*}}_{(u|z)}$ (Eq. 6), and the wind profile (Eq. 2), Eq. 23 simplifies after some algebraic manipulations to

$$ \xi_{{z_{0} }} = \left( {\frac{{\bar{z}}}{{z_{0(u|z)} }}} \right)^{{1 - r^{2} }} - 1 . $$

(25)