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

doi:10.1007/s11707-019-0770-z

Front. Earth Sci.

2020, Vol. 14

Issue (1) : 171-187 https://doi.org/10.1007/s11707-019-0770-z

RESEARCH ARTICLE

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

Nan CHEN^1,²(

)

¹. Key Laboratory of Spatial Data Mining & Information Sharing (Ministry of Education), Fuzhou University, Fuzhou 350108 China
². Spatial Information Research Center of Fujian Province, Fuzhou University, Fuzhou 350108 China

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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

Corresponding Author(s): Nan CHEN

Online First Date: 27 December 2019 Issue Date: 24 March 2020

Cite this article:

Nan CHEN. Scale problem: Influence of grid spacing of digital elevation model on computed slope and shielded extra-terrestrial solar radiation[J]. Front. Earth Sci., 2020, 14(1): 171-187.

URL:

https://academic.hep.com.cn/fesci/EN/10.1007/s11707-019-0770-z
https://academic.hep.com.cn/fesci/EN/Y2020/V14/I1/171

Fig.1 Slope change caused by the change in the grid spacing of the digital elevation model. For convenience of explanation, we give a profile of the land surface,

A B C^

, in the figure.

Fig.2 Shade relief maps of the six sample areas

Tab.1 The locations and landforms of the sample areas

Fig.3 Computation of the shielded coefficients. For convenience of explanation, we give a profile of the land surface (the blue curve in the figure). The red lines with arrows represent the directions of sunbeams, whose elevation angle is denoted by

h

(in the figure on the left) and whose azimuth angle is denoted by

φ

(in the figure on the right figure).

Fig.4 A 3 × 3 gridded digital elevation model. The elevations of grid cells 1 through 9 are necessary for computing the slope (or the aspect) of grid cell 5. In this figure,

g

denotes the grid spacing of the digital elevation model.

Fig.5 Sketch map of the derivation of the slope of grid cell A from 5 m grid spacing or 10 m grid spacing digital elevation model (DEM) using the third-order finite difference algorithm weighted by the reciprocal of the squared distance. When we derive the slope of A using a 5 m grid spacing DEM, the elevations of the green grid cells are required—the derived slope is denoted by

S l o p e 5, A

; when we derive the slope of A using a 10 m grid spacing DEM, the elevations of the yellow grid cells and grid cell A are required—the derived slope is denoted by

S l o p e 10, A

. The difference,

S l o p e 10, A − S l o p e 5, A

, is the slope representation error of A.

Fig.6 Change in the slopes of grid cells with a positive or negative slope representation error in Nanping. The slope representation error is denoted by

S l o p e_R E

Fig.7 Change in the

S E S R

(shielded extra-terrestrial solar radiation) in grid cells with a positive or negative

S l o p e_R E

(slope representation error) in Nanping on the vernal equinox. The

S E S R

representation error is denoted by

S l o p e_R E

in this figure.

Sample area	Day	$a 1$	$a 2$	$a 3$	$a 4$	$b 1$	$b 2$	$b 3$	$b 4$
Nanping	Vernal equinox	21.61	23.90	9.94	10.30	0.29	0.48	0.63	0.74
	Summer solstice	20.48	22.66	9.42	9.76	0.32	0.48	0.65	0.72
	Autumnal equinox	21.23	23.50	9.76	10.13	0.28	0.48	0.62	0.73
	Winter solstice	29.98	30.91	13.80	13.32	0.18	0.37	0.66	0.70
Shenmu	Vernal equinox	21.75	22.19	3.96	5.00	0.03	0.18	0.33	0.52
	Summer solstice	9.84	13.85	1.78	3.09	0.15	0.41	0.30	0.64
	Autumnal equinox	21.32	21.79	3.89	4.91	0.03	0.18	0.32	0.52
	Winter solstice	29.63	27.35	5.41	6.17	-0.01	0.13	0.39	0.55
Sanming	Vernal equinox	18.15	21.54	8.34	9.47	0.42	0.53	1.03	1.03
	Summer solstice	17.08	19.20	7.84	8.41	0.54	0.79	1.12	1.25
	Autumnal equinox	17.82	21.17	8.19	9.30	0.42	0.53	1.01	1.02
	Winter solstice	26.10	28.46	12.03	12.56	0.21	0.29	1.07	0.92
Yanchuan	Vernal equinox	21.89	24.64	13.48	14.64	1.03	1.67	1.12	1.42
	Summer solstice	19.87	22.94	12.23	13.63	0.77	1.16	0.86	0.93
	Autumnal equinox	21.53	24.25	13.26	14.41	1.01	1.64	1.10	1.40
	Winter solstice	22.83	23.39	14.03	13.88	0.27	0.67	0.37	0.45
Yijun	Vernal equinox	21.80	23.57	10.16	12.53	0.48	1.22	0.73	1.26
	Summer solstice	17.64	21.65	8.22	11.51	0.49	1.04	0.69	1.07
	Autumnal equinox	21.43	23.20	9.99	12.33	0.48	1.21	0.72	1.24
	Winter solstice	22.08	21.09	10.31	11.22	0.06	0.53	0.31	0.56
Zhangzhou	Vernal equinox	12.03	12.68	1.38	2.27	0.02	0.09	-0.02	-0.36
	Summer solstice	8.12	10.65	0.93	1.90	0.04	0.11	0.01	-0.27
	Autumnal equinox	11.73	12.39	1.34	2.22	0.02	0.09	-0.02	-0.35
	Winter solstice	24.33	23.52	2.79	4.23	0.01	0.07	-0.08	-0.78

Tab.2 The parameters in Eqs. (5) through (8)

Fig.8 Technical framework for verifying Eq. (5) in the test area.

S E S R_A A

denotes the average absolute value of the slope representation error, and

S E S R_A A

denotes the average absolute value of the shielded extra-terrestrial solar radiation. Equation (5) is given in Section 3.3. We compute the difference between the

S E S R_A A

computed by Eq. (5) and the actual

S E S R_A A

. Then, we define the ratio of the absolute value of the difference to the actual

S E S R_A A

as the relative error of

S E S R_A A

computed by Eq. (5). The average value of the relative error is defined as the average relative error of

S E S R_A A

computed by Eq. (5).

ARE	Vernal equinox	Summer solstice	Autumnal equinox	Winter solstice
ARE of $S E S R_A A$ computed by Eq. (5)	10.70%	11.56%	10.72%	8.31%
ARE of $S E S R_M S E$ computed by Eq. (6)	8.85%	9.24%	8.86%	6.36%
ARE of $S E S R_A A$ computed by Eq. (7)	11.75%	12.55%	11.76%	9.51%
ARE of $S E S R_M S E$ computed by Eq. (8)	9.26%	9.74%	9.27%	7.07%

Tab.3 The average relative errors of

S E S R_A A

and

S E S R_M S E

in the test area*

Fig.9 Percentage of grid cells with a positive representation error of the shield extra-terrestrial solar radiation, denoted by

S E S R_R E

, in Nanping on every classified slope on the vernal equinox. The shielded extra-terrestrial solar radiation is computed from digital elevation models (DEMs) with grid spacings of 10 m, 15 m, …, 100 m. The slope derived from the DEM with 5 m grid spacing is classified into 11 intervals,

[0 °, ? 5 °), ? [5 °, ? 10 °), ? [10 °, ? 15 °), ? ..., ? [45 °, ? 50 °), ? [50 °, ? 9 0 °]

, numbered from 1 to 11. The percentages of grid cells with a positive or negative

S E S R_R E

in other sample areas are not given here because of lack of space.

Fig.10 Average absolute value of the shielded extra-terrestrial solar radiation representation error (i.e., the AA of SESR_RE) distributed across the 11 slope intervals of

[0 °, ? 5 °), ? [5 °, ? 10 °), ? [10 °, ? 15 °), ? ..., ? [45 °, ? 50 °), ? [50 °, ? 9 0 °]

. The shielded extra-terrestrial solar radiation is computed from digital elevation models with grid spacings of 10 m, 15 m, …, 100 m in Nanping on the vernal equinox.

Fig.11 Mean square errors of shielded extra-terrestrial solar radiation representation error (i.e., the MSEs of SESR_RE) distributed across the 11 slope intervals of

[0 °, ? 5 °), ? [5 °, ? 10 °), ? [10 °, ? 15 °), ? ..., ? [45 °, ? 50 °), ? [50 °, ? 9 0 °]

. The shielded extra-terrestrial solar radiation is computed from digital elevation models with grid spacings of 10 m, 15 m, …, 100 m in Nanping on the vernal equinox.

Fig.12 Average absolute value of the slope representation error (i.e., the AA of Slope_RE) (in radians) for the 11 slope intervals of

[0 °, ? 5 °), ? [5 °, ? 10 °), ? [10 °, ? 15 °), ? ..., ? [45 °, ? 50 °), ? [50 °, ? 9 0 °]

. The slope representation error is computed from digital elevation models with grid spacings of 10 m, 15 m, …, 100 m in Nanping.

Fig.13 Side facing into or away from the sun. In the figure on the left, (a) Sunrise, face ACD of pyramid ABCD faces into the sun (represented by the red disk), at sunrise, while face ABC faces away the sun. In the figure on the right, (b) Sunset, face ACD faces away the sun, while face ABC faces into the sun. Then, although the aspect of ACD or ABC is certain, whether ACD or ABC faces into or away the sun varies with time. Furthermore, because all the grid cells in the DEMs of the sample areas possess diversified slopes and aspects, the grid cells may have diversified possibilities on whether they face into or away from the sun during the day, making it possible for us to study the shielded extra-terrestrial solar radiation under diversified possibilities.

Fig.14 The average shielded extra-terrestrial solar radiation (SESR) in Nanping moved to the equator on the vernal equinox, summer solstice, autumnal equinox and winter solstice when the grid spacings of the digital elevation model are 5, 10, ..., 100 m.

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