Simultaneous estimation of roughness length (Z0) & zero-plane displacement height (d) from single-level eddy covariance measurements

The code modifies and implements the 3 approaches used in Graf et al.,2014 Boundary-Layer Meteorol.

• FP-RE-1: from logarithmic wind profile theory: WS / Ustr = 1 / k ( ln((zm - d) / Z0) - beta * ((zm - d) / Lm )
• FP-RE-2: from logarithmic wind profile theory: WS = Ustr / k ( ln((zm - d) / Z0) - beta * (zm - d) / Lm )
• FV-RE-1: from flux-variance similarity theory: std.Uz / Ustr = C1 * (1 - C2 * (zm-d) / Lm ) ^ (1/3); std.Uz = k * C1 * WS / ln((zm - d) / Z0)

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####[Desccription] The goal is to estimate Z0 and d from continuous measurements of wind and turbulent statistics, i.e., WS, Ustr, std.Uz. All input variables are originally measured at a 30-min time step, but are pre-filtered according to data quality & required meteorological conditions. i.e., not every 30 min is used in model fitting... While estimating Z0 & d, the time series is re-grouped to a daily step, i.e., treating each 30 min within a day as a sampling replicate. Z0 and d are estimated at a daily time step, as they are both functions of canopy structures (height, leaf area...), assuming no changes within a day. The time series of Z0 or d is assumed as an autoregressive process, and it's assumed there's correlation (cor) between the time series of Z0 & d

• Z0[t+1] = Z0[t] + Z0.err
• Z0.err ~ N(0 , sigma.Z0);
• d[t+1] = d[t] + sigma.d /sigma.Z0 * cor * ( Z0[t+1] - Z0[t] ) + d.err
• d.err ~ N(0 , sigma.d.iid)

####[Input variables]

• WS: wind speed (m/s)
• Ustr: friction velocity (m/s)
• zm: measurement height (m)
• Lm: Monin–Obukhov length (m)
• std.Uz: standard deviation of vertical wind velocity (m/s)
• hc: vegetation canopy height (m) only used in filtering data
• ZL: Monin–Obukhov stability (unitless) only used in filtering data

####[Constants]

• k: 0.4 von karman constant
• C1: 1.3
• C2: 2.0 estimates for the universal constants, (Panofsky & Dutton 1984)
• beta: 6.0

####[Output variables]

• Z01[DOY]: roughness length estimated for each day (DOY) by using FP-RE-1 model
• d1[DOY]: displacement height estimated for each day (DOY) by using FP-RE-1 model
• Z02[DOY]: roughness length estimated for each day (DOY) by using FP-RE-2 model
• d2[DOY]: displacement height estimated for each day (DOY) by using FP-RE-2 model
• Z03[DOY]: roughness length estimated for each day (DOY) by using FV-RE-1 model
• d3[DOY]: displacement height estimated for each day (DOY) by using FV-RE-1 model