PET formulas (Heat Overlay)

The physiological equivalent temperature PET is calculated for two possible options based on the situation:

A cell is in the shade or it is night at time t: PET_{shade, night};
A cell is in the sun during the day: PET_sun;

PET Sun formula

The PET sun formula is as followed:
Failed to parse (syntax error): {\displaystyle PET_{sun} = -13.26 + 1.25 \cdot T_{a,t}+ 0.011 \cdot Q_{gl,t} - 3.37 \cdot ln(u_t) + 0.078 \cdot T_{w,t} + 0.0055 \cdot Q_{gl,t} \cdot ln(u) + 5.56 · sin(\phi_{t}) - 0.0103 \cdot Q_{gl,t} \cdot ln(u_t) \cdot sin(\phi_{t}) + 0.546 \cdot B_{b,t} + 1.94 \cdot S_{vf}} $PET_{sun}=-13.26+1.25\cdot T_{a,t}+0.011\cdot Q_{gl,t}-3.37\cdot ln(u_{t})+0.078\cdot T_{w,t}+0.0055\cdot Q_{gl,t}\cdot ln(u)+5.56\cdot sin(\phi _{t})$

where,

T_a,t is the calculated atmospheric temperature at time t;
Q_gl,t is the hourly radiation at time t;
u_t is the calculated wind speed at 1.2m above ground time t;
T_w,t is the calculated wet-bulb temperature at time t;
ϕ_t is the sun altitude at time t;
B_b,t is the bowen ratio at time t;
S_vf is the calculated sky view factor.

PET Shade and Night formula

The PET shade and night formula is as followed:
PET_{shade,night,t} = -12.14 + 1.25 · T_a,t - 1.47 · ln(u_t) + 0.060 · T_w,t + 0.015 · S_vfQ_d,t + 0.0060 · (1-S_vf) · σ · (T_a,t+273.15)⁴ where,