Reference Evapotranspiration ET₀

Evapotranspiration is not measured. It is computed per plot and hour from four meteorological variables using the Penman-Monteith equation in its FAO form (Allen et al. 1998, Irrigation and Drainage Paper 56).

The result is reference evapotranspiration ET₀: the evapotranspiration of a hypothetical, well-watered, 12 cm tall grass surface with an albedo of 0.23. It is a weather index, not a measure of the actual water use of a pine or beech stand. For stand evapotranspiration see the LWF-BROOK90 simulations; ET₀ enters the climatic water balance as the atmospheric demand term.

Inputs

Variable	Code	Unit in the DB	Use
Air temperature	AT	°C	required
Relative humidity	RH	%	required
Wind speed	WS	m/s at 10 m	required
Global radiation	SR	W/m²	required, × 0.0036 → MJ m⁻² h⁻¹
Atmospheric pressure	AP	hPa	optional, else derived from elevation

If one of the four required variables is missing, nothing is written for that hour.

Which sensors count. ET₀ is defined for open-field reference conditions. Only instruments with code_location = 'F' are used; only plots that never had an open-field sensor for a variable fall back to whatever is available. Where several instruments exist, values are averaged.

The computation

Everything is computed per hour. Equation numbers refer to FAO-56.

Solar position. From the day of year $J$ follow the declination (Eq. 24) and the inverse relative Earth-Sun distance (Eq. 23):

$$\delta =0.409\sin (\frac{2\pi J}{365}-1.39),d_r=1+0.033\cos \left(\frac{2\pi J}{365}\right)$$

The solar hour angle at the midpoint of the hour follows Eq. 31 with the seasonal correction $S_c$ (Eq. 33). $L_m$ is the longitude (east positive), ${\mathrm{\Delta }}_{\mathrm{UTC}}$ the timezone offset of the timestamps:

$$\omega =\frac{\pi }{12}[(t+0.5+0.06667(L_m-15{\mathrm{\Delta }}_{\mathrm{UTC}})+S_c)-12]$$

The bounds of the hour ${\omega }_{1,2}=\omega \mp \pi /24$ are clipped to sunrise and sunset (Eq. 29/30), giving extraterrestrial radiation (Eq. 28):

$$R_a=\frac{12\cdot 60}{\pi }{G}_{sc}{d}_r[({\omega }_2-{\omega }_1)\sin \phi \sin \delta +\cos \phi \cos \delta (\sin {\omega }_2-\sin {\omega }_1)]$$

If the hour lies entirely outside daylight, $R_a=0$.

Radiation balance. Clear-sky radiation (Eq. 37), net shortwave radiation with the grass albedo 0.23 (Eq. 38) and net longwave radiation (Eq. 39, with the hourly Stefan-Boltzmann constant $\sigma =4.903\cdot {10}^{-9}/24$):

$$\begin{array}{rl}{R}_{so}& =(0.75+2\cdot {10}^{-5}z)R_a\\ R_{ns}& =(1-0.23)R_s\\ R_{nl}& =\sigma (T+273.16{)}^4(0.34-0.14\sqrt{e_a})(1.35\frac{R_s}{R_{so}}-0.35)\\ R_n& =R_{ns}-R_{nl}\end{array}$$

At night $R_{so}=0$ and the ratio $R_s/R_{so}$ is undefined. FAO-56 suggests carrying over the value from shortly before sunset; here a constant of 0.8 is used instead (parameter p_night_rs_rso). This is the point at which the previous implementation forced maximum longwave loss and clamped nighttime ET₀ to zero.

Soil heat flux and the denominator coefficient (Eq. 45/46) differ between day and night:

$$G=\{\begin{array}{ll}0.1R_n& \text{daytime}\\ 0.5R_n& \text{night}\end{array}{C}_d=\{\begin{array}{ll}0.24& \text{daytime}\\ 0.96& \text{night}\end{array}$$

Vapour pressures (Eq. 11, 54) and the slope of the saturation curve (Eq. 13):

$$e_s=0.6108\exp \left(\frac{17.27T}{T+237.3}\right),e_a=\frac{RH}{100}{e}_s,\mathrm{\Delta }=\frac{4098e_s}{(T+237.3{)}^2}$$

Wind speed at 2 m (Eq. 47). The open-field anemometers sit at 10 m while the equation is defined for 2 m — the factor is 0.75:

$$u_2=u_z\frac{4.87}{\ln (67.8z-5.42)}$$

Atmospheric pressure and the psychrometric constant (Eq. 7, 8). Measured pressure takes precedence, otherwise it is derived from elevation:

$$P=101.3{\left(\frac{293-0.0065z}{293}\right)}^{5.26},\gamma =0.000665P$$

The equation itself (Eq. 53), result in mm per hour:

$$E{T}_0=\frac{0.408\mathrm{\Delta }(R_n-G)+\gamma {\displaystyle \frac{37}{T+273}}{u}_2(e_s-e_a)}{\mathrm{\Delta }+\gamma (1+C_d{u}_2)}$$

Negative results — possible at night under strong longwave loss — are set to zero.

Limits

• ET₀ is not forest evapotranspiration. The reference surface is grass, not a stand. For the water balance of a pine stand, ET₀ is the driver, not the answer.
• The timestamps are wrong. By ICP specification mm_meh carries UTC with the start of the hour as the label; in fact the hourly labels are off by +2.0 h (2017–2023 and 2025–2026) or +1.0 h (2024) — Central European Time plus an end-of-period label. The function is called with ${\mathrm{\Delta }}_{\mathrm{UTC}}=0$ and therefore places the sun about two hours too early: $R_s/R_{so}$ comes out too small in the morning and is clipped at 1 in the afternoon, the day/night switch for $G$ and $C_d$ flips at the wrong hour, and evening hours with actual sunlight receive ET₀ = 0. The diurnal cycle suffers more than the daily total, but the daily total is not unaffected. Until the timestamps are fixed at source, calling with ${\mathrm{\Delta }}_{\mathrm{UTC}}=2$ (2024: 1) would be closer to reality — but not clean, because the offset depends on the logger's storage interval and is not a whole hour for Kienhorst (1203) and Weitzgrund.
• The anemometer height is fixed. 10 m is confirmed for all eight wind sensors by fuk.sensor_map, but it is passed as a parameter, not read per instrument. A different height requires changing the call.
• No gap filling at the hourly level. If a required variable is missing the hour is dropped — and the daily value is then a sum over fewer than 24 hours, i.e. too low. daily_completeness must therefore be read alongside ET₀: a day at 80 % completeness is not a daily value missing 20 % of its detail, it is one that is 20 % too small.
• Inputs are taken unchecked. The procedure only requires that they are not null. A stuck humidity sensor enters the computation.
• The origin code is wrong. Hourly values carry code_data_origin = 9, which in the ICP dictionary means "missing value"; 4 ("modelled data") would be correct. The daily value inherits 1 from the aggregation trigger — "measured on plot". This must be corrected before any submission to ICP Forests; in the portal and the download the rule until then is: ET is always computed, whatever the code says.