Refraction near the horizon—an empirical approach. Part 1: terrestrial refraction of the dip

Marcel E. Tschudin

doi:10.1364/AO.55.003104

Applied Optics
Vol. 55,
Issue 12,
pp. 3104-3115
(2016)
•https://doi.org/10.1364/AO.55.003104

Refraction near the horizon—an empirical approach. Part 1: terrestrial refraction of the dip

Marcel E. Tschudin

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Marcel E. Tschudin, "Refraction near the horizon—an empirical approach. Part 1: terrestrial refraction of the dip," Appl. Opt. 55, 3104-3115 (2016)

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Abstract

This study aims at providing improved closed-form refraction estimates for observations near the horizon. In this first part, over 1800 previously published direct measurements of the horizon’s depression (dip) over the sea are reanalyzed using a nonconventional robust procedure for coping with numerous real, large, and asymmetric outliers from abnormal dips. The derived 1-parameter function agrees with those proposed in modern almanacs and for land surveying. It is found that the dips of warmer and colder sea surfaces vs. air are best described with two different functions. The two proposed 3-parameter functions, also using temperature difference between air and sea and wind speed, reduce the estimated error of the 1-parameter function by $\sim ⅓$ and the number of outliers by $\sim ⅔$ .

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Corrections

Marcel E. Tschudin, "Refraction near the horizon—an empirical approach. Part 1: terrestrial refraction of the dip: erratum," Appl. Opt. 57, 3479-3480 (2018)
https://opg.optica.org/ao/abstract.cfm?uri=ao-57-13-3479

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$j$	Height- Dataset $j$ :	$〈 H 〉$ [m]	$n_{j}$ Data	Height- Weights: $w_{H j} = n_{\max} / n_{j}$	$Σ Δ_{j i}^{2}$ ${[']}^{2}$
1	Pennino 3–6 m	4.17	16	56.06	9.54
2	Meyer 6.1 m	5.29	282	2.73	106.66
2	Meyer 6.25 m	5.29	47	2.73	106.66
3	Pennino 6–14 m	8.58	16	12.46	22.38
3	Meyer 9.0 m	8.58	56	12.46	22.38
4	Koss and Thun 10 m	10.10	262	3.42	138.33
5	Koss and Thun 16 m	15.73	897	1.00	519.4
6	Pennino 30–44 m	36.08	9	99.67	6.13
7	Koss and Thun 42 m	41.60	151	5.94	35.44
8	Rouch 53 m	52.94	51	17.59	6.70
9	Rouch 71 m	70.83	58	15.47	9.25
		$Σ n_{j} = 1845$		$Σ w_{H j} Σ Δ_{j i}^{2} = 3180.4$

				Correlation Indicators [ $\pm$ arcmin]						Outliers		Resulting Parameters:
Eq.	Analysis Type:	Estimator:	Perf. ${〈 r 〉}_{w}$	$〈 σ 〉 \pm$		$〈 ½ HGw 〉 \pm$		$〈 \frac{3}{2} MAD 〉 \pm$		$〈 N_{\| Δ \| > {1.5}^{'}} 〉 \pm$		( $F_{Da}$ )	$〈 a 〉 \pm$		$〈 b 〉$ or $〈 d 〉 \pm$			$〈 Δ c 〉 \pm$
(5)	FIT	$Δ$ -LIMITED	1.797	0.768	4.6%	0.607	3.3%	0.554	3.2%	70.0	6.9%	(1.754)	0.171
	FIT	WEIGHTED	1.808	0.785	4.6%	0.610	3.2%	0.558	3.0%	76.0	6.7%	(1.776)	0.150
	MEAN	$Δ$ -LIMITED	1.798	0.768	4.6%	0.608	3.3%	0.554	3.2%	70.0	6.9%	(1.754)	0.171	2.3%
	MEAN	WEIGHTED	1.810	0.786	4.6%	0.609	3.2%	0.559	3.1%	76.0	6.7%	(1.776)	0.149	2.8%
(9)	FIT	$Δ$ -LIMITED	1.425	0.650	3.9%	0.461	3.7%	0.449	3.5%	64.5	9.9%	(1.765)	0.161		0.427
	FIT	WEIGHTED	1.386	0.657	4.0%	0.446	2.8%	0.439	3.3%	66.0	10.5%	(1.780)	0.145		0.476
	MEAN	$Δ$ -LIMITED	1.416	0.650	3.9%	0.458	3.6%	0.445	3.5%	66.0	10.2%	(1.765)	0.160	3.0%	0.437	22%
	MEAN	WEIGHTED	1.409	0.659	4.1%	0.456	3.1%	0.443	3.5%	67.0	9.3%	(1.783)	0.142	2.0%	0.461	12%
(10)	FIT	$Δ$ -LIMITED	1.315	0.593	4.1%	0.425	2.4%	0.423	1.0%	66.6	9.7%		0.091		9.79		0.152
	FIT	WEIGHTED	1.308	0.598	4.6%	0.421	2.8%	0.420	2.6%	66.1	10.3%		0.086		9.60		0.150
	MEAN	$Δ$ -LIMITED	1.312	0.593	4.2%	0.425	2.3%	0.422	1.0%	66.6	9.7%		0.091	18%	9.85	10%	0.150	32%
	MEAN	WEIGHTED	1.305	0.595	4.4%	0.419	2.9%	0.420	2.4%	69.6	9.7%		0.089	13%	9.66	6%	0.150	22%
(11a) $Δ θ < 2 ° K$	FIT	$Δ$ -LIMITED	1.053	0.384	8.1%	0.343	1.3%	0.359	4.2%	0.5	96%		0.122		10.35		0.077
	FIT	WEIGHTED	1.001	0.395	10.1%	0.331	2.5%	0.322	2.6%	0.5	96%		0.108		9.03		0.082
	MEAN	$Δ$ -LIMITED	1.054	0.384	7.9%	0.342	1.2%	0.361	3.7%	0.5	96%		0.122	15%	10.41	10%	0.079	42%
	MEAN	WEIGHTED	1.016	0.391	9.7%	0.332	2.4%	0.332	2.7%	0.5	96%		0.116	12%	9.37	15%	0.068	43%
(11b) $Δ θ \geq 2 ° K$	FIT	$Δ$ -LIMITED	2.557	0.792	10.2%	0.885	3.5%	0.790	5.0%	25.6	12.1%		0.800		3.15		−0.308
	FIT	WEIGHTED	2.558	0.785	9.1%	0.888	5.4%	0.821	7.3%	24.6	13.7%		0.935		2.36		−0.363
	MEAN	$Δ$ -LIMITED	2.553	0.792	10.2%	0.883	3.5%	0.790	4.8%	25.6	12.1%		0.781	17%	3.14	14%	−0.299	15%
	MEAN	WEIGHTED	2.550	0.781	9.2%	0.886	5.4%	0.817	7.7%	24.1	13.0%		0.901	14%	2.36	10%	−0.346	12%
		Max =	2.558	0.792	10.2%	0.888	5.4%	0.821	7.3%	76.0	6.7%
		Min =	1.001	0.384	8.1%	0.331	2.5%	0.322	2.6%	0.5	96%

			KMPR Dataset (full-sized only, $N = 1845$ )				Parameters
No. Param.	Type of Terr. Refr., $R_{T}$ , estimate:	Perf. $〈 r 〉$	$\pm σ$	$\pm ½$ HGw	$\pm \frac{3}{2}$ MAD	$N_{\| Δ \| > {1.5}^{'}}$	( $F_{Da}$ )	$a \pm$		$b$ or $d \pm$		$Δ c \pm$
1	Expl. Suppl. Astr. Almanac	1.4346	0.764	0.606	0.553	137	(1.75)	0.175
	UKHO/USNO Naut. Almanac	1.4381	0.770	0.611	0.548	141	(1.763)	0.162
	Eq. (5): FIT & $Δ$ -LIMITED	1.4383	0.765	0.607	0.556	140	(1.754)	0.171	2.3%
	Thom ( $K_{Mean} = 7.60$ )	1.440	0.768	0.608	0.554	138	(1.753)	0.173
2	Eq. (9): FIT & WEIGHTED	1.140	0.653	0.445	0.439	132	(1.780)	0.145	2.0%	0.48	12%
	Freiesleben, Eq. (4) in [11]	1.257	0.865	0.439	0.422	170	(1.80)	0.125		1.10
	Albert U.S.S. (Peters [8])	1.333	0.962	0.435	0.443	181	(1.757)	0.168		1.23
	Galilee (Peters [8])	1.509	0.917	0.582	0.548	164	(1.624)	0.301		0.24
	Fus (Kazanskii [14])	1.601	1.235	0.496	0.495	169			see text
	Carnegie (Peters [8])	1.763	1.070	0.679	0.642	245	(1.588)	0.337		0.05
3	Eq. (10): MEAN & WEIGHTED	1.063	0.591	0.419	0.420	139		0.089	13%	9.66	5.5%	0.150	22%
3	Hasse [13], Table 1	1.326	0.934	0.449	0.442	174		0.145	Table values (see text)
$2 \times 3$	Eq. (11a): FIT & WEIGHTED	1.000	0.519	0.413	0.406	49	$Δ θ < 2 ° K :$	0.108	12%	9.033	15%	0.082	43%
	Eq. (11b): MEAN & WEIGHTED	1.000	0.519	0.413	0.406	49	$Δ θ \geq 2 ° K :$	0.901	14%	2.364	10%	−0.346	12%
Theory	Std. Atm., US 1976	1.4346	0.766	0.607	0.551	140			$T_{std}^{'} = - 0.0065 [° K / m]$
	$''$	11.37	6.279	6.204	2.826	992			$T_{air-sea}^{'} = (T_{air} - T_{sea}) / HoE$
	Freiesl., Eq. (3) in [11], $Δ T_{air-sea}$	9.40	6.312	3.765	2.807	1073	(1.69)	0.235		7.50
		Max =	6.312	6.204	2.826	1073
		Min =	0.519	0.413	0.406	49

$j$	Height- Dataset $j$ :	$〈 H 〉$ [m]	$n_{j}$ Data	Height- Weights: $w_{H j} = n_{\max} / n_{j}$	$Σ Δ_{j i}^{2}$ ${[']}^{2}$
1	Pennino 3–6 m	4.17	16	56.06	9.54
2	Meyer 6.1 m	5.29	282	2.73	106.66
2	Meyer 6.25 m	5.29	47	2.73	106.66
3	Pennino 6–14 m	8.58	16	12.46	22.38
3	Meyer 9.0 m	8.58	56	12.46	22.38
4	Koss and Thun 10 m	10.10	262	3.42	138.33
5	Koss and Thun 16 m	15.73	897	1.00	519.4
6	Pennino 30–44 m	36.08	9	99.67	6.13
7	Koss and Thun 42 m	41.60	151	5.94	35.44
8	Rouch 53 m	52.94	51	17.59	6.70
9	Rouch 71 m	70.83	58	15.47	9.25
		$Σ n_{j} = 1845$		$Σ w_{H j} Σ Δ_{j i}^{2} = 3180.4$

				Correlation Indicators [ $\pm$ arcmin]						Outliers		Resulting Parameters:
Eq.	Analysis Type:	Estimator:	Perf. ${〈 r 〉}_{w}$	$〈 σ 〉 \pm$		$〈 ½ HGw 〉 \pm$		$〈 \frac{3}{2} MAD 〉 \pm$		$〈 N_{\| Δ \| > {1.5}^{'}} 〉 \pm$		( $F_{Da}$ )	$〈 a 〉 \pm$		$〈 b 〉$ or $〈 d 〉 \pm$			$〈 Δ c 〉 \pm$
(5)	FIT	$Δ$ -LIMITED	1.797	0.768	4.6%	0.607	3.3%	0.554	3.2%	70.0	6.9%	(1.754)	0.171
	FIT	WEIGHTED	1.808	0.785	4.6%	0.610	3.2%	0.558	3.0%	76.0	6.7%	(1.776)	0.150
	MEAN	$Δ$ -LIMITED	1.798	0.768	4.6%	0.608	3.3%	0.554	3.2%	70.0	6.9%	(1.754)	0.171	2.3%
	MEAN	WEIGHTED	1.810	0.786	4.6%	0.609	3.2%	0.559	3.1%	76.0	6.7%	(1.776)	0.149	2.8%
(9)	FIT	$Δ$ -LIMITED	1.425	0.650	3.9%	0.461	3.7%	0.449	3.5%	64.5	9.9%	(1.765)	0.161		0.427
	FIT	WEIGHTED	1.386	0.657	4.0%	0.446	2.8%	0.439	3.3%	66.0	10.5%	(1.780)	0.145		0.476
	MEAN	$Δ$ -LIMITED	1.416	0.650	3.9%	0.458	3.6%	0.445	3.5%	66.0	10.2%	(1.765)	0.160	3.0%	0.437	22%
	MEAN	WEIGHTED	1.409	0.659	4.1%	0.456	3.1%	0.443	3.5%	67.0	9.3%	(1.783)	0.142	2.0%	0.461	12%
(10)	FIT	$Δ$ -LIMITED	1.315	0.593	4.1%	0.425	2.4%	0.423	1.0%	66.6	9.7%		0.091		9.79		0.152
	FIT	WEIGHTED	1.308	0.598	4.6%	0.421	2.8%	0.420	2.6%	66.1	10.3%		0.086		9.60		0.150
	MEAN	$Δ$ -LIMITED	1.312	0.593	4.2%	0.425	2.3%	0.422	1.0%	66.6	9.7%		0.091	18%	9.85	10%	0.150	32%
	MEAN	WEIGHTED	1.305	0.595	4.4%	0.419	2.9%	0.420	2.4%	69.6	9.7%		0.089	13%	9.66	6%	0.150	22%
(11a) $Δ θ < 2 ° K$	FIT	$Δ$ -LIMITED	1.053	0.384	8.1%	0.343	1.3%	0.359	4.2%	0.5	96%		0.122		10.35		0.077
	FIT	WEIGHTED	1.001	0.395	10.1%	0.331	2.5%	0.322	2.6%	0.5	96%		0.108		9.03		0.082
	MEAN	$Δ$ -LIMITED	1.054	0.384	7.9%	0.342	1.2%	0.361	3.7%	0.5	96%		0.122	15%	10.41	10%	0.079	42%
	MEAN	WEIGHTED	1.016	0.391	9.7%	0.332	2.4%	0.332	2.7%	0.5	96%		0.116	12%	9.37	15%	0.068	43%
(11b) $Δ θ \geq 2 ° K$	FIT	$Δ$ -LIMITED	2.557	0.792	10.2%	0.885	3.5%	0.790	5.0%	25.6	12.1%		0.800		3.15		−0.308
	FIT	WEIGHTED	2.558	0.785	9.1%	0.888	5.4%	0.821	7.3%	24.6	13.7%		0.935		2.36		−0.363
	MEAN	$Δ$ -LIMITED	2.553	0.792	10.2%	0.883	3.5%	0.790	4.8%	25.6	12.1%		0.781	17%	3.14	14%	−0.299	15%
	MEAN	WEIGHTED	2.550	0.781	9.2%	0.886	5.4%	0.817	7.7%	24.1	13.0%		0.901	14%	2.36	10%	−0.346	12%
		Max =	2.558	0.792	10.2%	0.888	5.4%	0.821	7.3%	76.0	6.7%
		Min =	1.001	0.384	8.1%	0.331	2.5%	0.322	2.6%	0.5	96%

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