Multiangular spectra (Section 2.2.1) were processed into estimates of directional scattering coefficients (DSCtree(Ω), [sr-1]). Note that due to the biconical geometry of the measurements (Fig. 1), our processing results in an approximation of true DSCtree(Ω) that would be observed in an infinitesimally narrow solid angle. The DSCtree(Ω) gives the probability density of scattered photons (per steradian) in a given view direction Ω, or, in other words, the fraction of intercepted photons scattered into a unit solid angle around Ω. Multiplication of DSCtree(Ω) by π gives the ratio of signal measured from the tree to that measured in nadir view from an ideal (non-absorbing) Lambertian surface of same surface area. This is conceptually similar to the bidirectional reflectance factor (BRF) commonly used for quantifying the scattering by surfaces in remote sensing [7]. Alternatively, multiplication of DSCtree(Ω) by 4π gives the ratio of signal measured from the tree to that measured from an ideal isotropic scatterer that scatters in all spherical directions. Note that for simplicity, we have omitted the wavelength sign of the spectral quantities in the equations presented in Sections 2.3.2 and 2.3.3.

The computation of DSCtree(Ω) has been reported in Hovi et al. [2] and in Forsström et al. [1]. For completeness, we provide the basic computation steps also here. For derivation of the measurement equations and estimation of uncertainties in DSCtree(Ω), see [2]. The equation for DSCtree(Ω) is

where DNtree(Ω) and DNWR_tree are the measured signals [digital number] from the tree and white reference (white reference was always measured in nadir view), Streei) is the silhouette area of the tree in the direction of illumination [m2], SWR_tree is the surface area of the white reference panel [m2], RWR_tree [fraction] is the reflectance of the white reference panel, and fWR_tree and ftree(Ω) [fraction] are estimates of the ratio of the measured signal to that measured by an isotropic detector, for the white reference and the tree, respectively. To explain shortly, Eq. (1) calculates the ratio of signals from the tree and white reference (first term on the right-hand side), multiplies it with the ratio of the radiation intercepted by the white reference and the tree (second term), multiplies the result with the DSC of the white reference panel at nadir (third term), and finally multiplies the result with a correction factor (fourth term) that takes into account the spectrometer's sensitivity within its FOV (i.e., detectors of the spectrometer had approximately Gaussian point-spread-functions with sensitivity falling off away from the center towards the edges). The derivation of the correction factors fWR_tree/ftree(Ω), one for each of the three detectors, has been explained in Section 3.6.2 of [2] and in Section of [1].

Eq. (1) assumes that DNtree(Ω) and DNWR_tree are free of stray light. Stray light fraction in each view angle was known from the measurements of an empty goniometer, i.e., the background canvas in place but without the tree. Thus, the stray light could be computed for each tree based on the white reference measurement. However, the tree (or white reference panel) and its shadow partly covered the illuminated background, and thus obscured a fraction of stray light. For an accurate stray light removal, we used the formulae

where DNtree(Ω) and DNWR_tree are the signals from the tree and white reference free from stray light, DNtotal,tree(Ω) and DNtotal,WR_tree are the signals from the tree and white reference that contain stray light, DNstray(Ω) and DNstray,WR_tree are stray light that would be measured in an empty goniometer (calculated based on the white reference measurements and stray light fraction (DNstray(Ω)), or measured separately for each tree (DNstray,WR_tree)), and btree(Ω) and bWR_tree are the fractions of stray light not obscured by the tree or white reference panel. Calculations of btree(Ω) and bWR_tree were performed for each of the detectors of the spectrometer separately, using the multiangular silhouette photographs, and additionally photographs taken of the light beam. Using photogrammetric techniques, it was possible to calculate the fraction of illuminated background that the spectrometer's detector could ‘see’ in the presence of the tree. The process has been described and illustrated in Section 3.6.3 of [2] and in Section of [1].

Despite the corrections, there remained jumps between the detectors of the spectrometer. In addition, there was high-frequency noise present close to 350 nm and close to 2500 nm. To remove the noise, the spectra were smoothed with a Savitzky-Golay filter [8]. Finally, the sensor jumps were removed by multiplying the spectra obtained by the SWIR1 and SWIR2 detectors by correction factors, which were obtained by comparing the difference of DSC between SWIR1 (1001 nm) and VNIR (1000 nm), and then by comparing the remaining difference between SWIR2 (1801 nm) and SWIR1 (1800 nm). We provide both original (DSCtree,raw), as well as jump-corrected and filtered (DSCtree,filt) spectra. The former were used by Forsström et al. [1], and the latter by Hovi et al. [2]. Uncertainty of DSCtree(Ω) is estimated to be 15–30% in relative terms (see Section 3.6.5 of [2] for details). The uncertainty is the highest in the regions were the signal from the tree is at its lowest and thus the contribution of stray light the highest, i.e., in the blue and red wavelengths, and in the water absorption regions in the shortwave-infrared.

An estimate of the tree's hemispherical reflectance (Rtree), i.e., the fraction of intercepted radiation scattered into hemisphere, can be obtained from the DSCtree(Ω) values by numerical Gauss-Legendre integration. To ensure systematic distribution of observations over the hemisphere, we did not use principal and cross-planes here. The nadir observations were also dropped out because nadir does not belong to the Gauss-Legendre nodes. Thus, Rtree was calculated as

where i are the view azimuth angles, j are the view zenith angles, wj are the Gauss-Legendre weights for each view zenith angle. Note that here we have separated positive and negative zenith angles into separate azimuths. Thus, there are 12 instead of 6 azimuth angles. Multiplication with 2π is required because DSCtree(Ω) is per one steradian, and hemisphere has 2π steradians. The calculations of both DSCtree(Ω) and Rtree from the raw data are demonstrated in the Python code provided with the data.

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