Supplementary Material 3. Assumptions to model variations in solar insolation reaching densely-forested slopes in mountainous terrain

It may be assumed that variations in solar insolation are dependent upon the integrated influence of the obstruction effect. The sky view factor (SVF) may be recognise as a reliable numerical value which can express the integrated influence of the topographic relief of adjoining mountain ranges and the contour of adjacent tree stands on the shading. Thus, insolation on a given site should be dependent upon insolation on an unobstructed site, Hopen, and SVF: Hi=f(SVFi,Hopen), where f⋅ is an unknown function. Based on an algorithm to calculate theoretical daily mean total insolation on a horizontal surface β=0 (Online Resources 2) some computer simulations were performed taking the following assumptions into account:

a)  the reflected part of solar was omitted,

b)  elevation angle (ε) was constant in every direction α,

c)  clearness index (KT) was constant through a whole year.

The results calculated for research sites M1, M2 and S5 are presented on Fig. ESM3-1 for different sky conditions based on Iqbal’s (1983) classification (KT=0.7 clear sky day, KT=0.4 partially cloudy sky day, KT=0.1 cloudy sky day).

Fig. ESM3-1. Results of computer simulations of relationship between daily insolation on horizontal surfaces relative to a theoretical unobstructed site, Hopen, for different sky conditions based on Iqbal’s (1983) classification
(KT=0.7 clear sky day, KT=0.4 partially cloudy sky day, KT=0.1 cloudy sky day).

The obtained results show that the relationship is mainly linear with minor distinguishable non-linearity. If one omits such noticeable but small non-linearity, which occurs usually during sunny days in winter (Hopen3kWh/m2), it may be generally stated that a slope coefficient is equal to approximately 1 (for KT=0.7) and its value is proportional to the clearness index; the lower KT is, the lower value of a slope coefficient is observed. Furthermore, the influence of SVF on a slope coefficient also depends on KT; during sunny days SVF is proportional to an absolute term in this linear relationship; for cloudy and overcast days SVF reduces the value of a slope coefficient. Additionally, it may be concluded that noticeably non-linearity occurs only when the sun’s disk is getting hidden behind obstructions (sites M2 and S5 at a height of 0.2m). In the presented research an unobstructed site was missing and recorded data were compared to the least shaded site M1. Such comparison based on numerical simulations is presented in Fig. ESM3-2. It may be reckoned that the expected relationship is linear, i.e. Hi=fSVFi⋅Hopen and HM1=fSVFM1⋅Hopen, thus Hi=fSVFifSVFM1⋅HM1.

Fig. ESM3-2. Results of computer simulations of relationship between daily insolation on horizontal surfaces relative to the least obstructed site M1 for different sky conditions based on Iqbal’s (1983) classification (KT=0.7 clear sky day, KT=0.4 partially cloudy sky day, KT=0.1 cloudy sky day)

It should be emphasised that the presented above simulations presupposed a constant value of the clearness index (KT). It means that weather conditions were constant throughout a whole year. Random weather conditions may also be simulated by substituting a random number for KT. Two additional simulation were performed and the results are shown in Fig. ESM3-3. At first, uniformly distributed pseudorandom values ranged from 0.1 to 0.9 (i.e. all weather conditions had the same probability) were substituted for KT (Fig. ESM3-3, left). Secondly, normally distributed pseudorandom values (μ=0.5, σ=0.1, i.e. partially cloudy sky day, KT=0.5, was the most probable) were substituted for KT (0.05≤KT≤0.95, Fig. ESM3-3, right). Furthermore, the recorded data differed significantly even if logged at the same time, due to the distances between sites and local meteorological conditions. Therefore, in both cases the drawn random values of KT could also vary randomly according to site by ±20%, i.e. assuming KT=0.6 (random sampling) for a given day thus KT for research sites were in range from 0.48 to 0.72 (uniformly distributed pseudorandom values). It may be observed that the results of both simulations are very similar to the measured data (Fig. 5). It should be remembered that simulations did not take into considerations the reflected component of solar radiation, possibly measurement errors, inconstant values of elevation angles and above all, the form and accuracy of the equations ESM1-9 and ESM1-13. Therefore, the suggested linear model, Eq. (9), can explain the observed variations in solar insolation with very good accuracy.

Fig. ESM3-3. Results of computer simulations of relationship between daily insolation on horizontal surfaces relative to the least obstructed site M1 with uniformly distributed pseudorandom KT, 0.1<KT<0.9 (left) and normally distributed (μ=0.5, σ=0.1) pseudorandom KT (0.05≤KT≤0.95, right)

In the presented research the range of the explaining variable SVF was relatively small (0.79 – 0.65). In general, SVF varies in a range from 1 (unobstructed area) to very low values (deep canyons or small openings in a stand). An universal relationship between daily insolation on horizontal surfaces relative to a theoretical unobstructed site based on computer simulations is presented in Fig. ESM3-4. In such a general case rather a segment piecewise regressions should be used. It may be expected that a two-segment piecewise linear regression could be sufficient to solve this task and an intersection point should be dependent upon the moment (equivalent to insolation on an unobstructed site) when the sun’s disk is completely hidden behind obstructions. However, such a segment piecewise relationship is not clearly visible in the results of computer simulations with random values of KT (Fig. ESM3-5); only for SVF ranged from 0.4 to 0.2 an intersection point may be distinguished.

Fig. ESM3-4. Simulations of a general relationship between daily insolation on horizontal surfaces relative to a theoretical unobstructed site for different sky conditions based on Iqbal’s (1983) classification (KT=0.7 clear sky day, KT=0.4 partially cloudy sky day, KT=0.1 cloudy sky day)

Fig. ESM3-5. Results of computer simulations of relationship between daily insolation on horizontal surfaces relative to a hypothetical unobstructed site with uniformly distributed pseudorandom KT, 0.1<KT<0.9 (left) and normally distributed (μ=0.5, σ=0.1) pseudorandom KT (0.05≤KT≤0.95, right)

Supplementary References

Iqbal M., 1983. An introduction to solar radiation. Academic Press, Toronto

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