Scale problem: Influence of grid spacing of digital elevation model on computed slope and shielded extra-terrestrial solar radiation

Abstract

Solar radiation is the primary energy source that drives many of Earth’s physical and biological processes and determines the patterns of climate and productivity on the surface of the Earth. A fundamental proportion of solar radiation is composed of shielded extra-terrestrial solar radiation (SESR), which can be computed using the slope and aspect derived from a digital elevation model (DEM). The objective of this paper is to determine the influence of the grid spacing of the DEM (the influence of the scale of the DEM) on the errors of slope, aspect and SESR. This paper puts forward the concepts of slope representation error, aspect representation error, and SESR representation error and then studies the relations among these errors and the grid spacing of DEMs. We find that when the grid spacing of a DEM becomes coarser, the average SESR increases; the increase in SESR is dominated by the grid cells of the DEM with a negative slope representation error, whereas SESR generally decreases in the grid cells with a positive slope representation error. Although the grid spacing varies, the distribution of the percentages of positive SESR representation errors on the slope, which is classified into 11 slope intervals, is independent of the grid spacing; this distribution is concentrated across some slope intervals. Moreover, the average absolute value and mean square error of the SESR representation error are closely related to those of the slope representation error and the aspect representation error. The findings in this study may be useful for predicting and reducing the errors in SESR measurements and may help to avoid mistakes in future research and in practical applications in which SESR is the data of interest or plays a vital role in an analysis.

Keywords

scale problem / digital elevation model / grid spacing / slope / shielded extra-terrestrial solar radiation

Introduction

“Solar radiation is the ultimate energy source of the terrestrial system” (Liu et al., 2012) because it plays important roles in meteorological (^{Pellicciotti et al., 2011}), hydrological (^{Hopkinson et al., 2010}) and biological processes (^{Häntzschel et al., 2005}; ^{Piedallu and Gégout, 2008}). “Solar radiation changes the distribution of clouds, temperature, moisture, and precipitation as well as the patterns of atmospheric circulation” (^{Qiu et al., 2004}) (i.e., solar radiation “assumes spatial heterogeneity” (^{Zhang et al., 2015})). Therefore, solar radiation has been extensively studied by scientists in many fields, such as vegetation modeling (^{Piedallu and Gégout, 2007}), solar electricity generation (^{Šúri et al., 2007}), and climate simulation (^{Yao et al., 2011}).

The solar radiation reaching a plot of land is influenced by the atmosphere and terrain relief. “When solar radiation traverses the Earth’s atmosphere, it is attenuated by various atmospheric constituents, such as ozone, oxygen, carbon dioxide, moisture, water droplets, and clouds; solar radiation reaching the top of the atmosphere is called extra-terrestrial solar radiation (ESR) or astronomical solar radiation” (^{Huang et al., 1986}). On the other hand, the ESR reaching a plot is shielded by the terrain relief around the plot; shielded ESR is called SESR in this paper.

SESR is the fundamental data through which scientists compute the direct, diffusive, and global solar radiation intensities when required meteorological parameters are known (e.g., the atmospheric transparency, cloud cover, and temperature) (^{Wang et al., 2014}; ^{Zhang et al., 2017}). In addition, “SESR is a vital astronomical parameter for evaluating solar energy resources and agricultural production potential” (^{Zuo, 1990}; ^{Qiu et al., 2005}), but solar radiation, including SESR, is difficult to measure in plots (^{Piedallu and Gégout, 2008}).

Methods for computing the SESR have gained attention. Initially, SESR has been directly computed using theoretical formulas when the latitude, solar declination, and solar hour angle are known (^{Huang et al., 1986}; ^{Li et al., 1999}) without considering the shielding caused by relief. Currently, shielding on Earth, especially on rugged terrains, is considered, and thus, theoretical formulas for computing the SESR have been developed (^{Z. Q. Li and Weng, 1988}; ^{Dozier and Frew, 1990}; Bocquet, 2010; ^{S. H. Zhang et al., 2015}). In particular, digital elevation models (DEMs) have been used to rapidly compute SESR (^{Dozier and Frew, 1990}) over large areas because they can be used to easily derive the slope and aspect, which are two variables that are required for computing the SESR (the other two required variables are latitude and solar declination).

With the development of formulas for computing the SESR, many studies have revealed that SESR or solar radiation is strongly affected by topographic factors (^{Qiu et al., 2004}; ^{Reuter et al., 2005}; ^{Ambreen et al., 2011}; ^{H. L. Zhang et al., 2013}). Of the topographic factors, slope has a positive relation with solar radiation, and the southern aspect generally receives more solar radiation than the northern aspect when slope is given (^{H. L. Zhang et al., 2010}); slope and aspect influence the distribution of SESR on rugged terrains in a seasonal manner (^{Zeng et al., 2005}). Thus, both slope and aspect are closely related to SESR.

When slope and aspect are computed by the elevations of sample points in DEMs, the elevation errors are propagated through to slope and aspect errors (^{Zhou and Liu, 2004}), respectively, which are propagated through to solar radiation errors (^{Piedallu and Gégout, 2008}). The elevation errors may be divided into two categories: “the terrain representation error” (^{Tang et al., 2001}) and the elevation sample error.

The elevation sample error has been thoroughly studied, and the methods for computing and reducing these errors are efficient, resulting in efficiently computing and reducing the slope and aspect errors (^{Zhou and Liu, 2004} and ²⁰⁰⁸; ^{Chow and Hodgson, 2009}; ^{Gao et al., 2012}; ^{Chen et al., 2014}; ^{C. Li et al., 2018a}; ^{Li et al., 2018b}). Thus, the elevation sample error is not studied in this paper.

The terrain representation error present by assuming that the elevation sample error is zero is unavoidable (^{Tang et al., 2001}). Similarly, based on the terrain representation error, we may define two concepts caused by grid spacing: the slope representation error (see Fig. 1) and the aspect representation error. The two errors cannot be avoided because grid spacings cannot be infinitely fine in practice.

Therefore, how the slope and aspect representation errors change with the grid spacing of DEMs and how they propagate through to the SESR representation error (see Fig. 1) have remained unclear and may be an impediment in relevant academic fields. A lack of knowledge concerning the impediment may lead to incorrect conclusions in research in which the SESR is used as background data (e.g., the biological cycle and distribution of solar energy on Earth); a decision maker may make incorrect decisions when developing and using solar energy. Thus, the bottleneck has been discussed in our research. Our research lies in the scale problem (the variations in the grid spacing of DEMs, i.e., the variations in the scale of DEMs causes the slope representation error); our research is different from the existing research in that our research concentrates not on the error models but on the difference between the data computed by test DEMs and that of the 5 m DEM.

The aim of this paper is to discover how the slope, aspect and SESR representation errors of grid cells change with the grid spacing of DEMs. We find significant linear relations between the absolute average value and the mean square error of SESR representation error and those of slope representation error (or those of aspect representation error).

The results of this paper may be beneficial to research in improving the accuracy of solar radiation data. The linear relations found in this paper may help us predict and reduce the SESR error and understand the influence of topographic relief on SESR. The concepts of the slope, aspect and SESR representation errors might be used as the accuracy indexes in the research on solar radiation and terrain relief.

Methods

Materials

Introduction of the sample and test areas. We chose six sample areas that represent different landforms (see Table 1). The shade relief maps of the sample areas are illustrated in Fig. 2. The six sample areas cover latitudes from 24°39′30″ to 38°52′30″N (see the average latitudes in Table 1). The corresponding average latitude in Table 1 is used as \phi for computingU, V, and W in Eq. (1). In addition, we choose a test area representing hills and low- and medium-sized mountains to verify the equations for estimating the error in the SESR (see Eqs. (5), (6), (7) and (8), which are obtained in the sample areas). The test area covers latitudes from 27°7′30″ to 27°12′29″N and longitudes from 117°48′45″ to 117°56′15″E. The average latitude of the test area is approximately 27°10′00″N and is used as \phi in Eq. (1).

DEMs of the sample and test areas. In the sample and test areas, the 5 m grid spacing DEMs were established from 1:10000 scale topographical maps provided by the State Bureau of Surveying and Mapping from China. 5 m grid spacing DEMs can reliably describe the terrain (^{Tang et al., 2001}) and are assumed to represent actual terrains.

All of the DEMs above are used to compute the SESR and the possible sunshine duration (PSD), which is a vital parameter in computing the SESR and is discussed in Section 3.2 in detail, on the four typical days (^{Huang et al., 1986}), including the vernal equinox, summer solstice, autumnal equinox and winter solstice. The four days have corresponding D_n values (see D_n in Eq. (2)) of 80, 173, 266 and 356, respectively.

Method for computing the ESR

The ESR on a slope surface is computed by Eq. (1) (^{X. Li et al., 1999}).

(1)I_{\alpha \beta }=(\mathrm{1}/{\rho }^{\mathrm{2}})I_{\mathrm{0}}(U\sin \delta +V\cos \delta \cos \omega +W\cos \delta \sin \omega ).

whereI_{\alpha \beta } represents the ESR (in MJ·m^-2·min^-1) on the slope surface, \mathrm{1}/{\rho }^{\mathrm{2}} represents the Sun-Earth distance correction factor, I_{\mathrm{0}} represents the solar constant (0.082 MJ·m^-2·min^-1), \delta represents the solar declination and \omega represents the solar hour angle; U=\sin \phi \cos \alpha -\cos \phi \sin \alpha \cos \beta,V=\sin \phi \sin \alpha \cos \beta +\cos \phi \cos \alpha and W=\sin \alpha \sin \beta (\phi represents the latitude at the surface, \alpha represents the slope of the surface, and \beta represents the aspect of the surface). Moreover, \delta and \mathrm{1}/{\rho }^{\mathrm{2}} in Eq. (1) are computed by the following Fourier series (^{Zuo et al., 1991}).

\begin{array}{l}\delta =0\mathrm{.006894-0}\mathrm{.399512cos}\tau \mathrm{+0}\mathrm{.07207sin}\tau \mathrm{-0}\mathrm{.006799cos(2}\tau \mathrm{)+0}\mathrm{.00089sin(2}\tau )\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{-0}\mathrm{.002689cos(3}\tau \mathrm{)+0}\mathrm{.001516sin(3}\tau )\end{array}

\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{1}/{\rho }^{\mathrm{2}}=1\mathrm{.000109+0}\mathrm{.033494cos}\tau \mathrm{+0}\mathrm{.001472sin}\tau \mathrm{+0}\mathrm{.000768cos(2}\tau \mathrm{)+0}\mathrm{.000079sin(2}\tau )\hspace{0.17em},

where \tau represents the day angle (in radians) and is defined by Eq. (2).

(2)\tau \mathrm{=2}\pi (D_n-\mathrm{1})/\mathrm{365}\hspace{0.17em}(D_n=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{...},\mathrm{365}),

where D_n represents the ordinal number of the day of the year (e.g., D_n=\mathrm{2} for 2 January).

By integrating I_{\alpha \beta } in Eq. (1) for the PSD, we obtain the ESR that hits the surface, which is denoted by W_s in Eq. (3) (^{Li et al., 1999}).

(3)\begin{array}{l}{W}_s={\displaystyle \frac{T}{\mathrm{2}\pi }}\left({\displaystyle \frac{\mathrm{1}}{{\rho }^{\mathrm{2}}}}\right)I_{\mathrm{0}}{\displaystyle {\int }_{{\omega }_{sr}}^{{\omega }_{ss}}(U\sin \delta +V\cos \delta \cos \omega +W\cos \delta \sin \omega )d}\omega \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\displaystyle \frac{T}{\mathrm{2}\pi }}\left({\displaystyle \frac{\mathrm{1}}{{\rho }^{\mathrm{2}}}}\right)I_{\mathrm{0}}[U\sin \delta ({\omega }_{ss}-{\omega }_{sr})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+V\cos \delta (\sin {\omega }_{ss}-\sin {\omega }_{sr})-\sin \alpha \sin \beta \cos \delta (\cos {\omega }_{ss}-\cos {\omega }_{sr})]\hspace{0.17em},\end{array}

where{\omega }_{sr} (in radians) represents the solar hour angle at the beginning of the PSD, {\omega }_{ss} (in radians) represents the solar hour angle at the end of the PSD, and T represents the total duration of one day (i.e., 1440 min).

Note that {\omega }_{sr}=-{\omega }_{ss}. Then, we let {\omega }_{ss}={\omega }_{\mathrm{0}} and {\omega }_{sr}=-{\omega }_{\mathrm{0}} for convenience. {\omega }_{\mathrm{0}} (in radians) is computed by {\omega }_{\mathrm{0}}=\arccos (-\tan \delta \tan \phi ). [-{\omega }_{\mathrm{0}},{\omega }_{\mathrm{0}}] denotes the period of daily PSD from sunrise to sunset at a land point. If [-{\omega }_{\mathrm{0}},{\omega }_{\mathrm{0}}] is divided into several subperiods, in some subperiods, the sunshine at the point may be shielded by the relief around the point; therefore, the PSDs of the subperiods should not be considered when computing the SESR at that point. However, [-{\omega }_{\mathrm{0}},{\omega }_{\mathrm{0}}] (or [{\omega }_{sr},{\omega }_{ss}]) represents the interval in which W_s is integrated in Eq. (3) and, thus, W_s is computed without considering shielding in Eq. (3). Thus, Eq. (3) should be reconstructed. A reconstructed Eq. (3) is described in Section 2.3.

Method for computing the SESR

A distributed model (^{Qiu et al., 2005}), which is a reconstructed version of Eq. (3), is as follows.

1) Divide [-{\omega }_{\mathrm{0}},{\omega }_{\mathrm{0}}] into subperiods and determine the subperiods available for the PSD in a DEM grid cell.

The period [-{\omega }_{\mathrm{0}},{\omega }_{\mathrm{0}}] may be evenly divided into N subperiods. Each subperiod has a length of \mathrm{\Delta }\omega (\Delta \omega =\mathrm{2}{\omega }_{\mathrm{0}}/N, in radians). \mathrm{\Delta }\omega has a corresponding time step length, denoted by \mathrm{\Delta }T(in minutes), which is defined by \mathrm{\Delta }T=(\mathrm{2}{\omega }_{\mathrm{0}}/(\mathrm{2}\pi N))\times \mathrm{24}\times \mathrm{60}. \mathrm{\Delta }T should be less than 20 min (^{Li and Weng, 1988}). Thus, N can be defined by the following equation: N=\mathrm{int}({\displaystyle \frac{\mathrm{2}{\omega }_{\mathrm{0}}}{\mathrm{2}\pi }}\times {\displaystyle \frac{\mathrm{24}\times \mathrm{60}}{\mathrm{20}}})+\mathrm{1}, where int() is a function that obtains the integer part of the value in parenthesis. Therefore, Nsubperiods may be [-{\omega }_{\mathrm{0}},-{\omega }_{\mathrm{0}}+\mathrm{\Delta }\omega ],[-{\omega }_{\mathrm{0}}+\mathrm{\Delta }\omega ,-{\omega }_{\mathrm{0}}+\mathrm{2}\mathrm{\Delta }\omega ],\mathrm{...},[-{\omega }_{\mathrm{0}}+(N-\mathrm{1})\mathrm{\Delta }\omega ,{\omega }_{\mathrm{0}}]. Then, the ith subperiod is [-{\omega }_{\mathrm{0}}+i\cdot \mathrm{\Delta }\omega ,-{\omega }_{\mathrm{0}}+(i+\mathrm{1})\cdot \mathrm{\Delta }\omega ], whose beginning solar hour angle is denoted by {\omega }_i ({\omega }_i=-{\omega }_{\mathrm{0}}+i\cdot \Delta \omega), where i=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{...},N-\mathrm{1}.

The sun elevation angle corresponding to the solar hour angle {\omega }_i, denoted by h_i, is computed by the equation h_i=\arcsin (\sin \delta \sin \phi +\cos \delta \cos \phi \cos {\omega }_i). The sun elevation angles corresponding to solar hour angles {\omega }_{\mathrm{0}},{\omega }_{\mathrm{1}},{\omega }_{\mathrm{2}},\mathrm{...},{\omega }_i,\mathrm{...},{\omega }_N are denoted by h_{\mathrm{0}},h_{\mathrm{1}},h_{\mathrm{2}},\mathrm{...},h_i,\mathrm{...},h_N, respectively.

The azimuth angle corresponding to the hour angle {\omega }_i, denoted by{\mathrm{phi;}}_i, is computed by the equation {\phi }_i=\arccos [(\sin h_i\sin \varphi -\sin \delta )/(\cos h_i\cos \varphi )]. The azimuth angles corresponding to the solar hour angles {\omega }_{\mathrm{0}},{\omega }_{\mathrm{1}},{\omega }_{\mathrm{2}},\mathrm{...},{\omega }_i,\mathrm{...},{\omega }_N are denoted by {\phi }_{\mathrm{0}},{\phi }_{\mathrm{1}},{\phi }_{\mathrm{2}},\mathrm{...},{\phi }_i,\mathrm{...},{\phi }_N, respectively.

We now compute the shielding status within a DEM grid at a time corresponding to the solar hour angle {\omega }_i. The shielding status within the grid cell (a grid cell is called a grid hereafter for brevity) is determined by the sun azimuth angle {\phi }_i corresponding to {\omega }_i and the surrounding relief in the direction of {\phi }_i. The shielding status is described by the shielding coefficient, denoted by S_i, where S_i=\mathrm{0} implies that shielding occurs in the direction of {\phi }_i and S_i=\mathrm{1} implies that shielding is unavailable. The computation of the shielding coefficients is illustrated in Fig. 3. Then, the shielding coefficients of the grid corresponding to the solar hour angles {\omega }_{\mathrm{0}},{\omega }_{\mathrm{1}},{\omega }_{\mathrm{2}},\mathrm{..},{\omega }_N are denoted by S_{\mathrm{0}},S_{\mathrm{1}},S_{\mathrm{2}},\mathrm{...},S_N, respectively. Thus, the average shielding coefficients S_{\mathrm{0}},S_{\mathrm{1}},S_{\mathrm{2}},\mathrm{...},S_N range from 0 to 1, with a higher average representing more sunshine received by the grid.

From the values of both S_i and S_{i+\mathrm{1}}, we can decide whether sunshine is available in the corresponding subperiod [{\omega }_i,{\omega }_{i+\mathrm{1}}] (^{Li and Weng, 1988}). If the subperiod [{\omega }_i,{\omega }_{i+\mathrm{1}}] has available sunshine (i.e., is available for the PSD), it is recorded by a subperiod [{\omega }_{srl},{\omega }_{ssl}], where {\omega }_{srl} is equal to {\omega }_i, and {\omega }_{ssl} is equal to {\omega }_{i+\mathrm{1}}. More generally, all of the solar angle subperiods available for the PSD are recorded by [{\omega }_{sr\mathrm{1}},{\omega }_{ss\mathrm{1}}],[{\omega }_{sr\mathrm{2}},{\omega }_{ss\mathrm{2}}],\mathrm{...},[{\omega }_{srl},{\omega }_{ssl}],\mathrm{...},[{\omega }_{srm},{\omega }_{ssm}], where m represents the number of total solar angle periods available for the PSD (see (^{Qiu et al., 2005}) for more details on computing the PSD).

Hereafter, the PSD of a grid implies the sum of the corresponding PSDs of the grid in the subperiods, including [{\omega }_{sr\mathrm{1}},{\omega }_{ss\mathrm{1}}],[{\omega }_{sr\mathrm{2}},{\omega }_{ss\mathrm{2}}],\mathrm{...},[{\omega }_{srl},{\omega }_{ssl}],\mathrm{...},[{\omega }_{srm},{\omega }_{ssm}].

2) Computing the daily SESR

The SESR in [{\omega }_{srl},{\omega }_{ssl}] can be computed by Eq. (3) if [{\omega }_{srl},{\omega }_{ssl}] is the integration interval of the integral in Eq. (3) instead of [{\omega }_{sr},{\omega }_{ss}]. In a similar manner, the SESR values of the subperiods [{\omega }_{sr\mathrm{1}},{\omega }_{ss\mathrm{1}}],[{\omega }_{sr\mathrm{2}},{\omega }_{ss\mathrm{2}}],\mathrm{...},[{\omega }_{srl},{\omega }_{ssl}],\mathrm{...},[{\omega }_{srm},{\omega }_{ssm}] can be computed. Then, the computed SESR values can be summed to yield the daily SESR, denoted by W_{\mathrm{0}\alpha \beta } (in MJ·m^-2), which takes the form of Eq. (4).

(4)\begin{array}{l}{W}_{\mathrm{0}\alpha \beta }={\displaystyle \frac{I_{\mathrm{0}}T}{\mathrm{2}\pi }}\left({\displaystyle \frac{\mathrm{1}}{{\rho }^{\mathrm{2}}}}\right)\{U\sin \delta {\displaystyle \underset{l=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}({\omega }_{ssl}-{\omega }_{srl})+V\cos \delta {\displaystyle \underset{l=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}(\sin {\omega }_{ssl}-\sin {\omega }_{srl})}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-W\cos \delta {\displaystyle \underset{l=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}(\cos {\omega }_{ssl}-\cos {\omega }_{srl})}\}.\end{array}

Method for deriving the slope and aspect from DEMs

According to Eq. (1), computing the SESR requires the slope and aspect, which are generally derived from gridded DEMs. If a terrain surface is described by the continuous function z=f(x,y), we compute the slope and aspect with the equations slope=\arctan \sqrt{{{{\displaystyle f}}_x}^{\mathrm{2}}+{{{\displaystyle f}}_y}^{\mathrm{2}}} and aspect=\pi -\arctan f_y/f_x+{\displaystyle \frac{\pi }{\mathrm{2}}}\times f_x/\left|f_x\right|, respectively. In this paper, f_x and f_y are computed using the third-order finite difference algorithm weighted by the reciprocal of the squared distance (^{Horn, 1981}) as follows: f_x=(z_{\mathrm{7}}-z_{\mathrm{1}}+\mathrm{2}(z_{\mathrm{8}}-z_{\mathrm{2}})+z_{\mathrm{9}}-z_{\mathrm{3}})/(\mathrm{8}g) and f_y=(z_{\mathrm{3}}-z_{\mathrm{1}}+\mathrm{2}(z_{\mathrm{6}}-z_{\mathrm{4}})+z_{\mathrm{9}}-z_{\mathrm{7}})/(\mathrm{8}g). In the two equations above, z_j\hspace{0.17em}(j=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{...},\mathrm{9}) denotes the elevation of the grid numbered j in Fig. 4, and g denotes the grid spacing of the DEM in Fig. 4.

Definition of error indexes

Let the slope, aspect, and SESR derived or computed using a i m grid spacing DEM be denoted by {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i, {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i and SES{R}_i (i=\mathrm{5},\mathrm{10},\mathrm{15},\mathrm{...},\mathrm{10}0\hspace{0.17em}m), respectively.

The method for deriving the slope from a 5 m (or 10 m) grid spacing DEM is illustrated in Fig. 5; the method for computing the slope representation error of grid A is illustrated in Fig. 5. We may compute the slope representation error of a grid numbered j in a 5 m grid spacing DEM by {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}\_RE={\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}-{\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{5,j}, where j denotes the number of grids (j=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{...},n), and (n denotes the total number of grids in the DEM). We compute the average absolute value of {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}\_RE by {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_AA={\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}\left|{\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}-{\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{5,j}\right|}/n; {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_AA takes its origin from the absolute error, which was used by (^{C. Li et al., 2018a}; ^{C. Li et al., 2018b}). We compute the mean square error (MSE) of {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}\_RE by {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_MSE=\sqrt{{\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{({\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{i,j}-{\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{5,j})}^{\mathrm{2}}/n}}; {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_MSE takes its origin from the MSE, which was used by (^{Zhou and Liu, 2002}, ²⁰⁰⁴, ²⁰⁰⁸). Similarly, we may define SES{R}_i\_RE, SES{R}_i\_AA and SES{R}_i\_MSE.

Because “the aspect is a circular variable” (^{Ruiz‐Arias et al., 2009}) (e.g., the aspects of \mathrm{2}\pi and 0 are the same as the northern aspect), we have to compute the aspect representation error of {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i by {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{i,j}\_RE=\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{i,j})-\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{5,j}). Then, we compute the average absolute value and the MSE of {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i\_RE by {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i\_AA={\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}\left|\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{i,j})-\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{5,j})\right|}/n and {\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i\_MSE=\sqrt{{\displaystyle \underset{j=\mathrm{1}}{\overset{n}{\mathrm{\Sigma }}}{[\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{i,j})-\cos ({\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_{5,j})]}^{\mathrm{2}}/n}}, respectively.

Results and discussion

Change in positive, negative, or zero \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE with grid spacing of the DEM

For convenience of discussion, \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is divided into three classes by their signs: positive, negative, and zero. We find Result 1 and Result 2.

Result 1. When the grid spacing of a DEM becomes coarser, the percentage of grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is greater than that of grids with a positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE; the ratio of the first percentage to the second percentage increases with the grid spacing.

Result 2. When the grid spacing of a DEM becomes coarser, the average decrease in the slope of grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is greater than the average increase in the slope of grids with a positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE; the ratio of the first average to the second average increases with the grid spacing.

Result 1 and Result 2 are explained by Fig. 6. Result 1 reveals that the grids with a positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, whose percentage cannot be ignored compared with those of the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, may exert an influence on the average slope when the grid spacing changes. The influence could be explained by Result 2.

By combining Result 1 and Result 2, we may more deeply understand the fact that the average slope decreases with grid spacing (^{Gao, 1997}; ^{Chow and Hodgson, 2009}). Result 1 states that the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE comprise the majority of grids and increase with grid spacing; Result 2 states that these grids dominate the decrease in slope and contribute a greater decrease in slope when the grid spacing becomes coarser. Together, the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE may dominate the change in slope when the grid spacing becomes coarser. Then, the slope generally decreases (i.e., the represented relief reduces) and, thus, the average slope decreases when the grid spacing becomes coarser.

Change in positive, negative, or zero SESR\_RE with grid spacing of the DEM

In Section 3.1, the grids are divided into three classes when the grid spacing changes. In this section, we discuss which classes of grids dominate the change in SESR.

Result 3. When the grid spacing of a DEM becomes coarser, the SESR generally increases on the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE.

We now explain Result 3 using the following data: We find that 2732914 grids have a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE when the grid spacing is 10 m in Nanping on the vernal equinox. Of the 2732914 grids, 2135772 grids (a percentage of 2135772/2732914≈78.15%) contribute to increases in the SESR (i.e., 2135772 grids have a positive SESR\_RE). More generally, the corresponding percentages range from 50.15% to 95.10% in all the sample areas on the four selected days.

We now explain the reason for Result 3: First, we assume that when a grid has a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, it may mainly be surrounded by grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE. This assumption could be explained by an example related to grid A (see grid A in Fig. 5). Suppose that A has a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE. Then, by this assumption, the surrounding grids around A (see the 24 grids surrounding A in Fig. 5) may mainly have a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE (i.e., the slope decreases in most surrounding grids). This assumption has been confirmed (see Appendix A).

The reason for Result 3 may be as follows. When the slope in most surrounding grids decreases (see the assumption just mentioned), the represented relief decreases in these grids, reducing the shielding on A (see Fig. 5) caused by most of the surrounding grids (i.e., A may receive more sunshine and, thus, the PSD in A may increase, resulting in an increase in the SESR on A).

Then, we confirm the reason for Result 3. We find that in the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, the shielding coefficients, PSD and SESR generally increase, with their average increases ranging from 0.02 to 0.36, 0.02 to 1.58 h and 0.05 to 6.61 MJ/m², respectively, in all sample areas on the four selected days. Then, the reason for Result 3 has been confirmed. In addition, we find Result 4.

Result 4. When the grid spacing of a DEM becomes coarser, the SESR generally decreases on the grids with a positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE.

Specifically, the percentage of grids where the SESR decreases (SESR\_RE is negative) and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is positive compared to grids where \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is positive is from 51.67% to 89.14% in all sample areas on the four selected days. We may explain the reason for Result 4 in a similar way that we do for Result 3.

Result 5. When the grid spacing of a DEM becomes coarser, the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE dominate the increase in SESR (or average SESR).

Result 5 is explained by Fig. 7. By combining Results 1 through 4, we can explain the reason for Result 5. In addition, by combining Results 3 through 5, we may explain the reason for Result 6.

Result 6. The average SESR increases when the grid spacing of a DEM becomes coarser.

The significance of Results 3 through 6 may be as follows. The slope derived from the DEM changes with the grid spacing, resulting in a significant change in the SESR (^{Ruiz-Arias et al., 2009}; ^{Zhang et al., 2013}), can be thoroughly explained on the grid level by using Results 3 through 5. By Result 6, we could expect a positive relation between the average SESR and the grid spacing and then give a more thorough understanding to the fact that the grid spacing positively influences the SESR (^{Zhang et al., 2010}). We, however, fail to find the relation between SESR\_RE and \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_RE.

A discussion about Result 6 is as follows. Suppose that theoretically, we have a DEM (named DEM A) with an infinitely fine grid spacing (although an infinitely fine grid spacing does not exist in practice) that perfectly represents a land surface. DEM A may yield the real SESR on the land surface. Suppose that a DEM in practice (named DEM B) also represents the land surface. Because the grid spacing of DEM B is always more than that of DEM A, the average SESR computed from DEM B is always more than that computed from DEM A, according to Result 6. Then, broadly speaking, any DEM in practice could yield the SESR whose average is always greater than the average of the corresponding real SESR. The discussion may be meaningful in practice.

The result that any DEM in practice could yield the SESR whose average is always greater than the average of the corresponding real SESR was tested in an indirect way in a test area which covers the latitudes from 27°14′10″ N to 27°14′32″ N and longitudes from 119°54′56″ E to 119°56′10″ E and with an approximate area of 2.288 km². We obtained the 0.5 m grid spacing DEM of the test area which was regarded to represent the real land surface of the test area (the DEM with infinite fine grid spacing does not exist in practice) and 30 m grid spacing DEM of the test area. Then, we computed the SESRs on the four selected days by using the two DEMs. The corresponding average values of the SESRs computed by the 0.5 m (30 m) grid spacing DEM were 33.35 (34.74) MJ/m² on the vernal equinox, 33.54 (34.81) MJ/m² on the summer solstice, 32.87 (34.26) MJ/m² on the autumnal equinox and 26.47 (27.72) MJ/m² on the winter solstice, respectively. The average value of SESR increased with the grid spacing of DEM. Thus, we have verified the result in an indirect way.

Relations between the error indexes of SESR\_RE and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE or \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_RE

Result 7. There are positive relations of \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or SESR\_MSE with \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE, respectively; there are positive relations of SESR\_AA or SESR\_MSE with \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_AA or \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_MSE, respectively.

The positive relations stated in Result 7 are found in each sample area on each of the four days and are described by Eqs. (5), (6), (7) and (8).

(5)SES{R}_i\_AA=a_{\mathrm{1}}\times {\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_AA+b_{\mathrm{1}},

(6)SES{R}_i\_MSE=a_{\mathrm{2}}\times {\mathrm{s}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_MSE+b_{\mathrm{2}},

(7)SES{R}_i\_AA=a_{\mathrm{3}}\times {\mathrm{a}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i\_AA+b_{\mathrm{3}},

(8)SES{R}_i\_MSE=a_{\mathrm{4}}\times {\mathrm{a}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}}_i\_MSE+b_{\mathrm{4}},

In Eqs. (5), (6), (7) and (8), a_k and b_k\hspace{0.17em}(k=\mathrm{1},\hspace{0.17em}\mathrm{2},\hspace{0.17em}\mathrm{3},\hspace{0.17em}\mathrm{4}) are parameters. The four equations are confirmed by a significance test with a confidence level of 0.05. a_k and b_k are given in Table 2. Note that in the four equations, a_k is positive.

We explain the reason for the positive relations described by Eqs. (5) and (6). When the grid spacing of the DEM becomes coarser, the difference between the surface represented by the DEM and the real surface may increase, resulting in a higher \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE (i.e., \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE may increase, which means a positive relation between the grid spacing and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE). The positive relation just mentioned can be confirmed by the coefficients between the grid spacing and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA (\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE), which range from 0.922 to 0.985 (0.877 to 0.977) in all the sample areas.

Similarly, we may explain the reason for the positive relations described by Eqs. (7) and (8). When the grid spacing of the DEM becomes coarser, the difference between the surface represented by the DEM and the real surface may increase, resulting in a higher \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_RE (i.e., \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_AA or \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_MSE may increase, which means a positive relation between the grid spacing and \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_AA or \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_MSE). The positive relation just mentioned could be confirmed by the coefficients between the grid spacing and \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_AA (\mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_MSE), which range from 0.949 to 0.993 (0.928 to 0.990) in all the sample areas.

Based on Result 6, the increase in the average SESR with the grid spacing of a DEM may mean a larger SESR representation error. Then, we may expect a positive relation between the grid spacing and SESR\_AA (SESR\_MSE). The positive relation just mentioned could be confirmed by the corresponding coefficients, which range from 0.870 to 0.984 (0.835 to 0.977) in all the sample areas on the four selected days.

Generalizing the discussion above, we may explain the reason for the positive relations described by Eqs. (5) and (6), (7) and (8).

To verify the reliability of Eqs. (5), (6), (7) and (8), we conduct an experiment in the test area. Considering that of the six sample areas, Nanping is the one located nearest to the test area (the distance between Nanping and the test area is approximately 104 km), we may compute the SESR\_AA in the test area by substituting the a_k and b_k values obtained in Nanping (given in Table 2) and the \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA value computed in the test area into Eq. (5). The technical framework for verifying Eq. (5) is given in Fig. 8.

The average relative error (ARE) of the SESR\_AA computed by Eq. (5) is defined in Fig. 8 and ranges from 6.36% to 12.55% (see Table 3). Similarly, we verify Eqs. (6), (7) and (8) in the test area and compute the corresponding AREs, which range from 6.36% to 9.24%, 9.51% to 12.55% and 7.07% to 9.74%, respectively (see Table 3). Then, Eqs. (5), (6), (7) and (8) may be verified if the upper limit of the ARE is set to 12.55%.

The significance of Eq. (5) may be as follows. By Eq. (5), a_{\mathrm{1}} and b_{\mathrm{1}} in an area can be computed by the SES{R}_i\_AA and {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_AA, which are computed in a few grids of the DEM representing the area. Then, based on the computed values for a_{\mathrm{1}} and b_{\mathrm{1}}, researchers and users are able to compute the SES{R}_i\_AA on any other grids of the DEM via {\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_i\_AA, which can be easily computed, without the complicated computation of the SESR. In addition, Eqs. (6), (7) and (8) have the same significance.

Change in positive or negative SESR\_RE or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE with grid spacing of the DEM when the slope is classified

Considering that the slope is usually classified in practice, we study the change in positive or negative SESR\_RE with the grid spacing of the DEM when the slope derived from a 5 m grid spacing DEM was classified into 11 intervals ([\mathrm{0}°,\hspace{0.17em}\mathrm{5}°),\hspace{0.17em}[\mathrm{5}°,\hspace{0.17em}\mathrm{10}°),\hspace{0.17em}[\mathrm{10}°,\hspace{0.17em}\mathrm{15}°),\hspace{0.17em}\mathrm{...},\hspace{0.17em}[\mathrm{45}°,\hspace{0.17em}\mathrm{50}°), [\mathrm{50}°,\hspace{0.17em}\mathrm{9}0°]) numbered from 1 to 11. Because the percentage of grids with a slope interval of [\mathrm{50}°,\hspace{0.17em}\mathrm{9}0°] was less than 0.3% in all sample areas, we do not divide the interval [\mathrm{50}°,\hspace{0.17em}\mathrm{9}0°] into subintervals.

We compute the percentages of the grids with a positive SESR\_RE on every classified slope when the grid spacings are 10 m, 15 m, …, 100 m (see Fig. 9).

Figure 9 shows that although the grid spacing changes, the distribution of the percentages on the classified slope fail to change in terms of statistics—we perform a one-way analysis of variance with a null hypothesis that the distributions of the percentages of different grid spacings are from the same distribution; we find that all corresponding p-values are 1.000, and the null hypothesis cannot be rejected. Similarly, we find that the distribution of the percentages of grids with a negative SESR\_RE on every classified slope fail to change in terms of statistics though the grid spacing changes—all the corresponding p-values are 1.000. These findings can be expressed in Result 8.

Result 8. Based on statistics, the distribution of the percentage of grids with a positive SESR\_RE or a negative SESR\_RE on the classified slope is independent of the grid spacing of the DEM.

We explain the reason for Result 8. First, the distribution of the percentage of grids with a positive (or negative) \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE on the classified slope is independent of the grid spacing of the DEM (^{Chen, 2014}). Second, based on Result 3 and Result 4, a positive SESR\_RE generally appears on grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, and vice versa. Together, the independence of the distribution of a positive or negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE on the classified slope may be extended to the distribution of a negative or positive SESR\_RE.

In addition, Fig. 9 shows that the distribution of the percentages of positive SESR\_RE values are concentrated on the slope intervals numbered from 5 to 9 (i.e., the majority is found in this concentration). The corresponding concentrations of positive or negative SESR\_RE values on the corresponding slope intervals are found in all sample areas on the four selected days. The findings mentioned above may be caused by the following reasons:

When the grid spacing of a DEM becomes coarser, whether the slope of a land surface derived from the DEM increases or decreases is determined by the first-order and third-order derivative (or partial derivative) of the function that describes the land surface (^{Chen, 2013}). In other words, the function describing the geometrical feature of the land surface determines whether the slope increases or decreases (i.e., whether \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is positive or negative). Whether \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is positive or negative generally determines whether SESR\_RE on the land surface is negative or positive (based on Result 3 and Result 4, respectively). Then, whether SESR\_RE is negative or positive may be generally determined by the geometrical feature of the land surface whose slope derived from the 5 m grid spacing DEM is a certain value, resulting in a relation between the slope and the sign of SESR\_RE. Based on the relation, we may explain the concentration of the positive or negative SESR\_RE values within the corresponding slope intervals.

Distribution of the values of SESR\_AA (or SESR\_MSE) on the classified slope

We analyze the SESR\_AA (or SESR\_MSE) for the 11 intervals and find Result 9.

Result 9. The SESR\_AA (or SESR\_MSE) generally increases when the number of slope intervals increases from 1 to 11.

As shown in Fig. 10, SESR\_AA generally increases when the number of slope intervals increases from 1 to 11 (i.e., the slope increases in all the sample areas on the four selected days). In addition, as shown in Fig. 11, the SESR\_MSE generally increases when the number of slope intervals increases from 1 to 11 in all sample areas on the four selected days.

We explain the reason for Result 9 by using the following two results.

\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA (or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE) generally increases when the number of slope intervals increases in all the sample areas on the four selected days, as shown in Fig. 12 (Fig. 12 shows \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA on the 11 intervals only in Nanping for brevity).

A positive relation between \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA (\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE) and SESR\_AA (SESR\_MSE) in Result 7 still exists when the slope is classified into 11 intervals (when the slope is classified, the correlation coefficients between the \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA and SESR\_AA range from 0.805 to 1.000 in all sample areas on the four selected days).

By combining the two results, we conclude that SESR\_AA or SESR\_MSE generally increase when the number of slope intervals increases.

Based on Result 9, we may more deeply understand that “the slope has a positive relation with solar radiation” (Hang et al., 2010).

Discussion on the influence of the locations of the sample areas on Results 3, 4, 6, 7, 8 and 9

The computation of SESR reveals that the locations of the sample areas (in the Northern or Southern Hemisphere) and the landforms of the sample areas could influence Results 3, 4, 6, 7, 8, and 9.

We have compared the results of the sample areas that are classified by landforms and fail to find any obvious pattern related to the landforms, though we could study the SESR under the diversified possibilities based on whether the grids in the DEMs of the sample areas face into or away from the sun during a day (see Fig. 13). This failure may be caused by two reasons. First, the number of sample areas may be insufficient; second, we do not have sample areas at the same latitude representing various landforms. This assessment should be performed in future work. In addition, this failure means that we may study the influence of the locations of the sample areas, avoiding the influence of the landforms of the sample areas.

We now discuss how the locations of the sample areas influence Results 3, 4, 6, 7, 8, and 9. Suppose that the six sample areas are moved to the equator (0°), to the latitude of 20°S and to the latitude of 40°S (we do not discuss the change in the SESR during the polar day and the polar night, so the sample areas are not moved to 80°S where the polar day and polar night occur; considering that the south-east point of the continent of South American does not touch 60°S or a higher latitude than 60°S, we do not move the sample areas to 60°S). Then, we compute the corresponding SESRs in the sample areas at the given latitudes. When analyzing the SESRs, we could decide how the locations of the sample areas influence Results 3, 4, 6, 7, 8, and 9 without considering the influence of the landforms of the sample areas.

Discussion on Result 3. Let Nanping be moved to the equator. When the grid spacing is 10 m (50 m), approximately 91.13% (92.23%) of the grid with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE contribute to the increase in the SESR in Nanping on the vernal equinox. More generally, the corresponding percentage ranges from 51.03% to 97.06% in all the sample areas on the four selected days when the sample areas are moved to latitudes of 0°, 20°S and 40°S. Then, Result 3 is confirmed at latitudes of 0°, 20°S and 40°S.

Discussion on Result 4. Let Nanping be moved to the equator. The percentage of grids where the SESR decreases (SESR\_RE is negative) and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is positive compared to grids where \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE is negative ranges from 75.66% to 85.54% in Nanping on the vernal equinox. The corresponding percentages in all the sample areas on the four selected days are more than 50.00% (when the sample areas are moved to latitudes of 0°, 20°S and 40°S) but under a few conditions, e.g., when Zhangzhou is moved to 40°S and the grid spacing is more than 25 m, the corresponding percentages range from 48.14% to 49.67% on the summer solstice (however, the number of the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE and positive SESR\_RE is more than that of the grids with a positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE and negative SESR\_RE under this condition. The ratio of the first number to the latter number is 1.45 to 3.22; the ratio of the corresponding mean increase in SESR to the corresponding mean decrease in SESR is 1.61 to 2.76 under this condition. Then, the SESR has to also increase with the grid spacing under this condition.). Together, we may state that generally, SESR decreases in the grids with positive \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE.

Discussion on Result 6. Let Nanping be moved to the equator. The average SESR in Nanping on the vernal equinox increases with the grid spacing, e.g., the average SESR is 31.37 MJ·m⁻², 34.19 MJ·m⁻² and 35.39 MJ·m⁻² when the grid spacing is 5 m, 50 m and 100 m, respectively (see Fig. 14). More generally, the average SESR increases with the grid spacing in all the sample areas on the four selected days when the sample areas are moved to latitudes of 0°, 20°S and 40°S.

Discussion on Result 7. We find the positive relations described by Eqs. (5), (6), (7) and (8) in all the sample areas on the four selected days when they are moved to 0° or 20°S or 40°S. The four equations are confirmed by a significance test with a confidence level of 0.05. Note that in the four equations, a_k is positive (the minimum a_k is approximately 0.54). The a_k and b_k are not given here for brevity; we obtain 72 sets of a_{\mathrm{1}} and b_{\mathrm{1}} (72= 6 × 4 × 3, with 6 representing the six sample areas, 4 representing the four selected days and 3 representing the three latitudes) and 72 sets of a_{\mathrm{2}} and b_{\mathrm{2}}, 72 sets of a_{\mathrm{3}} and b_{\mathrm{3}} and 72 sets of a_{\mathrm{4}} and b_{\mathrm{4}}.

Discussion on Result 8. We perform a one-way analysis of variance with a null hypothesis that the distributions of the percentages of grids with a positive SESR\_RE on every classified slope are from the same distribution; we find that all corresponding p-values are 1.000, and the null hypothesis cannot be rejected. Similarly, we find that the distribution of the percentages of grids with a negative SESR\_RE on every classified slope fails to change in terms of statistics though the grid spacing changes—all the corresponding p-values are 1.000.

Discussion on Result 9. When the number of slope intervals increases from 1 to 11, the SESR\_AA (SESR\_MSE) increases or has an upward trend under a few conditions if all sample areas are moved to 0° or 20°S or 40°S.

We fail to find any fundamental changes to the Results 3, 4, 6, 7, 8 and 9 that are caused by the locations of the sample areas.

Limitations and applications of the findings in this study

Limitations of the findings in this study

Although aspect influences SESR (^{Zeng et al., 2005}) (or solar radiation (^{Wang et al., 2014})), we find that the sign of \mathrm{A}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\_RE does not obviously relate to that of \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE; we fail to find some obvious pattern in the distribution of the percentages of the \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE on aspect which is classified into 9 classes (north, north-east, east, south-east, south, south-west, west, north-west and flat). We, however, fail to discover the reason explaining these results.

In this study, we do not consider how the errors in latitude and solar decline affect the SESR representation error, although latitude and solar decline are required variables for computing the SESR (see Eq. (1)). The corresponding affect could be studied in future work.

We do not discuss the change in the SESR during the polar day and polar night because the change may be quite complicated and could not be discussed thoroughly in this paper for lack of room. The corresponding discussion may be interesting and worth doing in the future.

Applications of the findings in this study

Applications of Eqs. (5), (6), (7), and (8). These four equations can be used to compute \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE, both of which are not easily computed in practice because of the difficulty in obtaining the actual solar radiation or the SESR over a large region. This difficulty is generally caused by factors such as “the inappropriate density of stations” that survey solar radiation (“particularly in complex peripheral national territories”) (^{Ambreen et al., 2011}) and the lack of satisfactory sunshine hour data. We, however, can easily survey the actual slope or aspect and derive the slope or aspect from the DEM, by which we can compute \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE (or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE). Then, we can compute \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE using the corresponding equation from the four equations above.

Applications of the results (Results 1 through 9). These results may add to the current knowledge of how \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE propagates toward \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE and how to estimate \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE. Then, when researchers conduct studies in which the SESR is the data of interest, they may avoid incorrect results caused by the SESR representation error. Moreover, users who make important decisions using the SESR can avoid mistakes. Furthermore, this study might be beneficial for research in the fields of spatial data errors and digital terrain analysis. In addition, based on Result 9, the lower the slope on a grid is, the more reliable the SESR computed on the grid is. Then, the method for computing the SESR may be applied in regions with low slopes.

Conclusions

In this paper, we have defined a set of concepts and developed the corresponding methods to describe the representation errors in slope, aspect and SESR; the representation errors are unavoidable and useful for understanding how the SESR varies with the grid spacing of a DEM. The scale of DEM exerts an obvious influence on the computed slope and the SESR. We draw the following conclusions:

Let a surface be represented by a gridded DEM.

1) When the grid spacing of the DEM becomes coarser, the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE dominate the change in slope, resulting in a decrease in the average slope and an increase in the PSD and the SESR. When the grid spacing becomes coarser, the SESR generally increases in the grids with a negative \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_RE, and vice versa. The DEM could yield the SESR whose average is generally greater than the average of the corresponding real SESR.

2) We may compute \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE using \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE (\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE), respectively, in practice because the corresponding relations are sufficiently linear.

3) When the slope is classified into 11 intervals, \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA or \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE increase with the number of intervals. The distribution of the percentages of positive SESR\_RE values on the classified slope is independent of the grid spacings of the DEM and concentrates on some slope intervals.

4) The locations of the sample areas (in the Northern or Southern Hemisphere) fail to cause any essential changes to Results 3, 4, 6, 7, 8 and 9.

The results of this study may lead to fast forecasting of SESR\_RE at a global level and may be beneficial to the development of the accuracy theory of spatial data, extending its application into studies on solar radiation and energy. In future work, we will study how the errors in latitude and solar declination propagate toward the SESR and discuss how seasons influence the relation between \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA (\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE) and \mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_AA (\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}\_MSE).

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