外文翻譯---跨越式精確三角高程測量

1、<p><b>　　附錄A 外文翻譯</b></p><p>　　Precise Height Determination Using Leap-Frog</p><p>　　Trigonometric Leveling</p><p>　　Ayhan Ceylan1 and Orhan Baykal2</p><

2、p>　　Abstract: Precise leveling has been used for the determination of accurate heights for many years. The application of this technique is difficult, time consuming, and expensive, especially in rough terrain. These

3、difficulties have forced researchers to examine alternative methods of height determination. As a result of modern high-tech instrument developments, research has again been focused on precision trigonometric leveling. I

4、n this study, a leap-frog trigonometric leveling (LFTL) is applied </p><p>　　CE Database subject headings: Leveling; Height; Surveys.</p><p>　　Introduction</p><p>　　The development

5、of total stations has led to an investigation of precise trigonometric leveling as an alternate technique to conventional geometric leveling (Kratzsch 1978; Rueger and Brunner 1981, 1982; Kuntz and Schmitt 1986; Hirsch e

6、t al. 1990; Whalen 1984; Chrzanowski et al. 1985; Kellie and Young 1987; Young et al. 1987; Haojian 1990; Aksoy et al. 1993). Most of these papers give more practical results, rather than theoretical.</p><p>

7、;　　In this study, we treat the subject more theoretically, with current instruments. We also discuss theoretical aspects such as limits of the techniques, errors, and accuracies in leap-frog trigonometric leveling.</p

8、><p>　　Slope distances and zenith angles are measured using either a unidirectional or a reciprocal or leap-frog method of field operation in trigonometric leveling. Both of the targets in leap-frog trigonometr

9、ic leveling can always be placed at the same height above the ground. Thus, sight lengths are not limited by the inclination of the terrain, and systematic refraction errors are expected to become random because the back

10、- and foresight lines pass through the same or similar layers of air. The num</p><p>　　lengths to a few hundred meters. This reduces the accumulation of errors due to instrument settlement that is another si

11、gnificant source of systematic error.</p><p>　　1 Assistant Professor, Engineering and Architecture Faculty, Konya Selcuk Univ., 42031 Konya, Turkey. E-mail: aceylan@selcuk.edu.tr</p><p>　　2 Prof

12、essor, Civil Engineering Faculty, Istanbul Technical Univ., 80626 Istanbul, Turkey.</p><p>　　Note. Discussion open until January 1, 2007. Separate discussions must be submitted for individual papers. To exte

13、nd the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible</p><p>　　publication on August 6, 2003; ap

14、proved on August 25, 2005.</p><p>　　Principle of Unidirectional Trigonometric Leveling</p><p>　　Trigonometric leveling is the determination of height differences by means of the measured zenith

15、angles and the slope distance. Similar to geometric leveling, the height difference between two turning points (benchmarks) is computed as the sum of several single height differences obtained from each settlement.</p

16、><p>　　The measurement model of the unidirectional trigonometric leveling (UDTL) is illustrated in Fig. 1. The total station is set up at only one point and the observations are performed only in one direction.

17、</p><p>　　In Fig. 1, =geodetic (ellipsoidal) zenith angle from to ; =observed zenith angle from to ; =model error due to the refraction effect; ?ij=model error due to the deviation of the plumb line; Sij=slo

18、pe distance between Pi and Pj; hi and hj=ellipsoidal heights of Pi and Pj, respectively; Rm=mean radius of the earth (≈6,370 km)and ?hij=height difference from Pi to Pj.</p><p>　　The height difference ?hij i

19、s formulated as</p><p><b>　　(1)</b></p><p>　　where the first term is the nominal height difference, the second term is the spherical effect of the earth, and the third term is the to

20、tal effect due to the deviation of the plumb line and the vertical refraction (Coskun and Baykal 2002).</p><p>　　The coefficient of refraction, kij, is defined as the ratio between the refraction angle dZri

21、and half of the center angle (Rueger and Brunner 1982); i.e.</p><p><b>　　(2)</b></p><p><b>　　and</b></p><p><b>　　(3)</b></p><p>　　Th

22、e center angle,, can be computed as</p><p><b>　　(4)</b></p><p>　　If is introduced into Eq. (3), the model error due to the refraction effect, dZri</p><p>　　, is obtaine

23、d as follows:</p><p><b>　　(5)</b></p><p>　　The height difference between the station points Pi and Pj via unidirectional zenith angle observation is obtained from Eqs. (1) and (5)<

24、;/p><p><b>　　(6)</b></p><p>　　In practice, the effect of deviation of plumb line is very small because the zenith angles observed along the sight lengths are not longer than 500 m. Thus

25、, the second term in Eq. (6) can be ignored (Rueger and Brunner 1982). As a result, the height difference between the station points, Pi and Pj, is computed from UDTL observations as</p><p>　　Principle of Le

26、ap-Frog Trigonometric Leveling</p><p>　　Observation of leap-frog trigonometric leveling (LFTL) was performed in back and foresight reading at one setup of the total station between two turning points, the sa

27、me method used in geometric leveling. The measurement model of the LFTL is shown in Fig. 2.</p><p>　　Fig. 2. Measurement model of LFTL </p><p>　　According to Fig. 2 and Eq. （7）, the height diffe

28、rence between the station points, Pi and Pj, is obtained from LFTL observations as</p><p>　　Considering </p><p>　　where the first term is the nominal height difference, the second term is the

29、 spherical effect of the earth, the third term is the effect due to the vertical refraction, and the fourth term is the total influence of all other random errors, namely, sinking of target rods, verticality and calibrat

30、ion of rods, and uncertainties in the deviations of plumb lines. If we use the following assumptions:</p><p>　　the second term in Eq. (9) will be zero. As a result, the height difference between the station

31、points, Pi and Pj, is computed as</p><p>　　It is obvious that the height difference obtained from Eq. (11) is affected by the difference in the actual refraction coefficients and other random errors in the l

32、eap-frog trigonometric leveling (LFTL). The refraction term requires further investigation. The uncertainty in the refraction term of Eq. (11)can be minimized by making the lengths of the back- and foresights equal. Howe

33、ver, inequalities often exist between the refraction coefficients of the backsight and foresight, even if these dist</p><p>　　The accuracy of LFTL can be obtained by applying the law of variance propagation

34、to Eq. (11) under the following assumptions:</p><p>　　After propagating errors, an expression for the variance in height difference between Pi and Pj can be derived as</p><p>　　Standard deviatio

35、ns of the distances, the zenith angles, the refraction coefficients, and other random errors are denoted by, , , respectively. The variance of a 1 km level line is computed as</p><p>　　The computed standard

36、deviations of a 1 km LFTL line, based on standard deviations of ±1.0″, ±2.0″, and ±3.5 mm for zenith angles and slope distances, respectively, are summarized in Table</p><p>　　1. The uncertain

37、ty in the coefficient of refraction is taken as ±0.05 and ±0.10 for (nonsimultaneous) reciprocal zenith angle observations. The value of has been arbitrarily accepted as ±0.30 mm for total influence of al

38、l other random errors.</p><p>　　Table 1. Standard Deviations _in mm_ of a 1 km LFTL Line with Sight Distances of 100, 150, 200, and 300 m and Average Zenith Angles of 80, 85,and 90°</p><p>

39、　　Applications</p><p>　　The precise leveling (PL) and LFTL measurements were performed on a leveling network with eight points established on hilly terrain at the Campus Area of Selcuk University in Konya, T

40、urkey (Fig. 3).</p><p>　　Design and Calibration of Surveying Instruments</p><p>　　PL measurements were carried out by a measurement team of six people (one observer, one recorder, two rodmen, an

41、d two auxiliary)using a precise leveling instrument(Wild N3) equipped</p><p>　　with a parallel glass micrometer and a pair of 3 m invar rods(Wild).</p><p>　　LFTL measurements were performed by a

42、 team of four people using a pair of target rods and a total station. The accuracy of zenith angle measurement with six series is ±1″ using the total station Sokkia SET2 [telescope magnification: 30x; minimum readin

43、g: 1″; accuracy of horizontal and zenith angle measurement: ±2″; accuracy of distance measurement: ±(3 mm +2 ppm·S)]. Target rods were formed by two parts, a bottom one, which was an invar rod of 2 m, and

44、a top one, which was an iron bar 1 m in len</p><p>　　Several target plates with different patterns of various dimensions were investigated for targeting accuracy of sight distances of 200 and 300 m. A red an

45、d white colored circle target was preferred for LFTL. It has been proven that the accuracy of single targeting is better than 30″ /M (M=telescope magnification) in average atmospheric conditions (Chrzanowski 1989). Conse

46、quently, the target plate in Fig. 5 is preferred.</p><p>　　Because a pair of target rods is used commonly in LFTL, the height differences between the target plates on backward and forward rods should be dete

47、rmined with the highest possible precision. The calibration of the target heights was performed using a precise leveling instrument equipped with a parallel glass micrometer. The calibration was accomplished on the slope

48、 surface. The corresponding targets on each rod were preset within an accuracy of a few millimeters in the workshop. The target rods</p><p>　　The horizontal offset (K) between the reflector and the target ro

49、d was determined by using a total station set up 10-15 m away from the rod (Fig. 4).</p><p>　　Field Work and Computations</p><p>　　The PL measurements were applied using backsight(left), foresig

50、ht(left), foresight (right), and backsight (right) methods (BFFB) on each station.</p><p>　　The LFTL measurements were performed according to Fig. 6. The measurements were repeated for the sight lengths of 5

51、0, 100, 150, and 200 m to investigate the optimum sight length. The following measurement procedure was applied on each station point:</p><p>　　1. Checking the leveling of instrument.</p><p>　　2

52、. Sighting to the reflector on the backsight and foresight rods in the direct and reverse position of the telescope, and automatic recording of the zenith angles and the distances(two readings).</p><p>　　3.

53、Sighting to Target I on the backsight and foresight rods in the direct position of the telescope and measuring the zenith angles (three readings).</p><p>　　4. Sighting to Target II on the foresight and backs

54、ight rods in the direct position of the telescope and measuring quantities as in Step 3.</p><p>　　5. Turning the telescope to the reverse position and measuring quantities as in 3 and 4.</p><p>

55、　　6. Measuring the second measurement set using the same procedures given in Steps 1–5.</p><p>　　7. Reading the temperature and the barometric pressure, and recording comments.</p><p>　　The obse

56、rved slope distance was corrected for the refractive index of the atmosphere, zero error, and scale to obtain the final value S0 of the distance between the center of the tacheometer and the reflector. Then distances SB

57、and SF between the center of the total station and the individual targets were calculated by taking into consideration the offset K (Fig. 4)</p><p>　　where and =zenith angles to the reflector and to the targ

58、et, respectively; and K=is horizontal offset of the reflector with respect to the target (Fig. 4).</p><p>　　The final height differences between the two turning points were calculated for the observations to

59、 Target I and Target II separately</p><p>　　where=calibration correction for the height differences of the corresponding targets (Chrzanowski 1989).</p><p>　　Conclusions and Recommendations</

60、p><p>　　PL and LFTL were compared using measurements on the same leveling test network, providing the following results:</p><p>　　? The optimum length of sight in the LFTL was determined as S=150 m

61、.</p><p>　　? The PL accuracy cannot be reached using LFTL. The PL accuracy of ±0.59 mm/, obtained in this study is very high, as compared to the accuracy of ±1.0 mm/ generally achieved in ordinary

62、conditions. As the LFTL achieved ±1.87 mm/, LFTL can be used instead of PL in rough terrains and for applications that do not require high accuracy.</p><p>　　? According to the cost analysis, the LFTL t

63、echnique is 39% cheaper than PL and 350% more productive than PL as well.</p><p>　　? Although PL gives high accuracy, it is slow and difficult to apply, and it is an expensive method. LFTL is fast, inexpensi

64、ve, and easy to apply in each type of field. If the results obtained using LFTL with a sight distance of 150 m are taken into consideration, LFTL can be used in applications that require lower accuracy than ±2.0 mm/

65、.</p><p><b>　　Notation</b></p><p>　　The following symbols are used in this paper:</p><p>　　dZ = model error due to refraction effect;</p><p>　　e = total of

66、influence of other random errors;</p><p>　　K = horizontal offset;</p><p>　　kij = refraction coefficient;</p><p>　　R = radius of earth;</p><p>　　S = slope distance;</

67、p><p>　　Z = zenith angle;</p><p>　　h = height difference; and</p><p>　　= model error due to deviation of plumb line.</p><p>　　跨越式精確三角高程測量</p><p><b>　　an

68、d </b></p><p>　　摘要：精密水準測量已用于精確高程測定許多年。這項技術(shù)的應用不僅困難，費時，而且價格昂貴，特別是在惡劣的地形。這些困難迫使研究人員尋找替代高原來高程測定的方法。隨著現(xiàn)代科學技術(shù)手段的發(fā)展，研究又集中在精密三角高程測量上。在這項研究中，跨越式三角高程測量（LFTL）適用在土耳其Konya的一個大學校園——Selcuk University Campus里，通過在同一個樣品

69、測試網(wǎng)絡中的不同視距的比較，以確定最佳視距。實驗結(jié)果同幾何水準在精度，成本和可行性等方面的比較可以知道，跨越式測量在視線距離S = 150米的三角高程測量中的標準差為± 1.87毫米/√公里，效率為5.6公里/天。</p><p>　　CE數(shù)據(jù)庫關(guān)鍵詞：水準;高度;調(diào)查。</p><p><b>　　簡介</b></p><p>　　

70、全站儀的發(fā)展，導致了精密三角高程測量代替?zhèn)鹘y(tǒng)的幾何水準測量的技術(shù)研究(Kratzsch 1978; Rueger and Brunner 1981, 1982; Kuntz and Schmitt 1986; Hirsch et al. 1990; Whalen 1984; Chrzanowski et al. 1985; Kellie and Young 1987; Young et al. 1987; Haojian 1990; A

71、ksoy et al. 1993)。這些論文大多是給出了實際的成果，而不是理論上的成果。</p><p>　　在這項研究中，就目前所擁有的儀器來說，我們更加注重理論上的研究。我們還討論了諸如技術(shù)，誤差范圍和跨越式精密三角高程測量的理論方面。</p><p>　　斜距和天頂角的測量是用一個單向或?qū)ο蛴^測，或著跨越式的三角高程測量方法來測量的。在跨越式三角高程測量中可以把兩個目標都始終放置在離

72、地面高度相同的位置。因此，視覺長度不受限于地形傾斜度，折射誤差和系統(tǒng)誤差可以成為隨機誤差，因為后視和前視通過的空氣層是相同或者相似的。延長了視線到幾百米的長度可以減少每公里的換站次數(shù)。這就減少了另一種重要的系統(tǒng)誤差來源——儀器誤差的累積差。</p><p>　　1助理教授，建筑與環(huán)境工程學院，科尼亞塞爾丘克大學。42031科尼亞，土耳其。電子郵箱：aceylan@selcuk.edu.tr</p>

73、<p>　　2教授，土木工程學院，伊斯坦布爾技術(shù)大學。80626伊斯坦布爾，土耳其。</p><p>　　注。公開討論，直到2007年1月1日。單獨討論,個人必須提交論文。為了延長一個月的截止日期，以書面請求必須提交給ASCE的執(zhí)行主編。此稿已提交文件審查并且可能于2003年8月6日批準，2005年8月25日批準。</p><p>　　單向三角高程測量原理</p>

74、<p>　　三角高程測量是通過測天頂角和斜距的來測定高程的差異的。同幾何水準相類似，??兩個轉(zhuǎn)折點（基準）的高差計算是獲得的每個結(jié)算點高度差之和。該單向三角高程測量（UDTL）測量模型如圖1所示，全站儀只設置在一個點上，并且只在一個方向進行觀測。</p><p>　　在圖1中， =大地測量（橢球）從PI到pj的天頂角; Zij =從PI到pj的觀測的天頂角; dZr =由于折射作用而引起的模型誤差;

75、?ij =由于鉛垂線的偏差而引起的模型誤差; Sij =PI到pj的斜距；hi 和hj =PI到pj的橢球高，圓周率; Rm =地球平均半徑（≈6,370 km）；Δhij=pi到pj的高差。高差Δhij的公式為</p><p><b>　　(1)</b></p><p>　　其中第一項是名義上的高度差，第二項是地球的球形偏差，第三項是由于垂線偏差和垂直折光而引起的總

76、的影響，（Coskun and Baykal 2002）。</p><p>　　折射系數(shù)，kij，是指折射角dZri和圓心角（Rueger and Brunner 1982）之間的半數(shù)的比例；i.e。</p><p><b>　　(2)</b></p><p><b>　　和</b></p><p>

77、;<b>　　(3)</b></p><p><b>　　圓心角的計算公式為</b></p><p><b>　　(4)</b></p><p>　　如果將代入公式（3），由于折射而引起的模型誤差dZri，計算公式如下</p><p><b>　　(5)</b&

78、gt;</p><p>　　根據(jù)公式（1）及（5）來測定單向天頂角觀測站點Pi 和Pj之間的高差</p><p><b>　　(6)</b></p><p>　　在實踐中，垂線偏差的影響非常小，因為觀察天頂角的視線長度不超過500米。因此公式(6)可以忽略(Rueger and Brunner 1982)。所以，分站點Pi和pj之間的高差要從U

79、DTL計算，計算公式為</p><p>　　跨越式三角高程測量原理</p><p>　　跨越式三角高程觀測（LFTL）是為了全站儀在兩個轉(zhuǎn)折點之間進行前后讀數(shù)，同樣的方法也用在幾何水準測量中。該LFTL測量模型如圖2所示。</p><p>　　根據(jù)圖2和公式（7），站點Pi和Pj之間的高差可以從LFTL觀測中計算得到，公式為</p><p>

80、<b>　　考慮到</b></p><p>　　其中第一項是標準高差，第二項是地球的球形誤差影響，第三項是由于垂直折射的影響，第四項是所有隨機誤差的總的影響，即目標下沉，垂直度和目標校準，鉛垂線偏差的不確定性。如果我們使用下面的假設</p><p>　　公式（10）中的第二項代入到公式（9）將為0。因此，站點Pi和Pj之間的高差計算如下</p><

81、p>　　很明顯，方程（11）中得到的高差，是受實際中的折射系數(shù)和跨越式三角高程測量（LFTL）其他隨機誤差的影響。折射系數(shù)需要進一步的長期的調(diào)查。方程（11）中的折射系數(shù)的不確定性可以通過是前后視距相等來最小化。不過，經(jīng)常存在前后視距的折射系數(shù)不相等的情況，即使他們的前后距離是相等的。在任何情況下，</p><p>　　LFTL的測量方法將使得折射系數(shù)的誤差降到最小。對于特殊情況下，平均長度為k的一個折射系

82、數(shù)可以從相互頂角觀測中計算。公式為</p><p>　　該LFTL準確性可以通過公式（11）運用方差傳播規(guī)律來處理。根據(jù)下面的假設：</p><p>　　經(jīng)過誤差傳遞，通過Pi 和 Pj之間的高差，可以推導出</p><p>　　距離，天頂角，折射系數(shù)，以及其他隨機誤差的標準偏差分別記為, , 。每公里的水平線上方差的計算公式為</p><p&g

83、t;　　在每公里LFTL線的基礎(chǔ)上計算標準差，由表1可知，天頂角的標準差為± 1.0”，± 2.0”，斜距的標準差為± 3.5mm。為了方便天頂角的觀測具有不確定性的折光系數(shù)取為± 0.05和± 0.10。所有其他的隨機誤差的影響可以設為±0.30 mm</p><p>　　表1為每1千米的標準差（毫米），視距為100，150，200，300米，平均天頂

84、角為80，85和90 ° </p><p><b>　　應用</b></p><p>　　在科尼亞，土耳其塞爾丘克大學校園??區(qū)（圖3）的丘陵地形上，建立了一個8個站點的水準網(wǎng)來進行精密水準測量（PL）和LFTL。</p><p>　　設計與測量儀器的校準</p><p>　　PL測量是由一個六個人的測量隊伍組

85、成（一個觀察員，一個記錄員，兩個rodmen，和兩個輔助人員）測量使用一臺精密水準測量儀器，配有一個平行玻璃千分尺，一對3m的千分尺。</p><p>　　LFTL測量是由一個4人的隊伍配有一對目標桿和一臺全站儀來進行的。使用全站儀索佳SET2[望遠鏡放大倍數(shù)：30倍，最小讀數(shù)：1”;水平角和天頂角度測量精度：± 2“;距離的測量精度：±(3 mm +2 ppm?S)]測量天頂角的精度為

86、77;1″。目標桿由兩部分組成，底部是一個2m的銦瓦尺，頂部是一個長1m，直徑2cm的鐵棒。這個部分被裝在一起。反射器是安裝在距離底部1.70米的高度，以實施遠程測量，兩個垂直角度觀察目標分別在2.20和3.00米。一個圓形氣泡（10'精度）和被用來探測目標桿的三腳架（圖4）。 </p><p>　　針對200米和300米視距的測量精度，在不同層面的幾個不同模式下進行了調(diào)查，LFTL測量的首選的圓形目標是

87、紅色和白色。它已經(jīng)證明，在平均大氣壓條件下（Chrzanowski 1989），單一定位精度優(yōu)于30“/ M（M=望遠鏡放大倍數(shù)）。因此，圖5的目標板是首選。</p><p>　　因為，通常用于LFTL測量的前后目標，應該盡可能的提高他們之間高差的精度。用精密水準測量儀器與平行玻璃微米裝備對該目標的高度進行校準。校準是在一個斜面上進行的。相應的每個桿的目標都提前做一個幾毫米的精度的設置。目標桿可以放在測量同一個轉(zhuǎn)

88、折點目標高度的水準儀大概幾米遠的地方。對于每一個目標板，高差可以通過平行玻璃千分尺來讀取的差異來確定。這些更正被添加到兩者之間的轉(zhuǎn)折點的高差計算中。這些更正過的標準偏差通過重復量測獲得，精度大約為±0.1 mm (Chrzanowski 1989)。</p><p>　　反射鏡和目標桿之間的水平位移是用設置在距離目標10-15 m遠的全站儀來測定（圖4）。</p><p><

89、;b>　　野外工作和計算 </b></p><p>　　PL測量在每個測站上應使用后視（左），前見（左），前見（右）和后視（右）的方法（BFFB）。</p><p>　　LFTL的測量根據(jù)圖6。測量的視距在50，100，150和200米之間重復，確定最佳視線長度。以下測量程序應用于每站點：</p><p>　　1。檢查儀器，整平。 </p&g

90、t;<p>　　2。直接瞄準前視和后視的反射棱鏡，扭轉(zhuǎn)望遠鏡的位置，自動記錄天頂角和距離（讀兩次） </p><p>　　3。望遠鏡在能夠直接看到前后目標的位置，瞄準目標I，測量天頂角（讀三次）。 </p><p>　　4。望遠鏡在能夠直接看到前后目標的位置，瞄準目標II，按照步驟3測量距離。 </p><p>　　5。轉(zhuǎn)動望遠鏡到反向的位置，繼續(xù)3，

91、4步驟。 </p><p>　　6。第二次測量的步驟同1-5相同。 </p><p>　　7。讀溫度和氣壓，并記錄。</p><p>　　其中和 =分別是反射鏡和目標的天頂角；K=是反射目標的水平位移（圖4）</p><p>　　分別計算目標I和目標II，兩個轉(zhuǎn)折點的最終高差。</p><p>　　其中=校準了相應的

92、修正目標的高度差(Chrzanowski 1989)。</p><p><b>　　結(jié)論與建議</b></p><p>　　PL和LFTL在使用同一水準測試網(wǎng)測量的情況下相比較，提供了以下結(jié)果：</p><p>　　?在LFTL測量中視線S = 150米被認定為最佳長度 </p><p>　　?PL的精度無法達到LFTL

93、的精度。同普通條件下的精度~±1.0 mm/√km相比，在這項研究中PL獲得了非常高的精度，達到了±0.59 mm/√km。由于LFTL的精度達到±1.87 mm/√km，LFTL可以用來代替在地勢崎嶇地區(qū)的低精度的PL測量。</p><p>　　?根據(jù)成本分析，LFTL技術(shù)比PL便宜39％以上，并且效率是PL的350％以上</p><p>　　?盡管PL測量

94、能夠提供高精確度，但是很緩慢，難以適用，并且花費巨大。 LFTL快速，價格低廉，易于應用在各種環(huán)境。如果所得的結(jié)果用LFTL測量，150米的視距，那么LFTL可用于精度要求低于?±2.0 mm/√km的應用中。</p><p><b>　　符號 </b></p><p>　　以下符號用于本文件中：</p><p>　　dz=由于折射影

95、響而引起的模型誤差; </p><p>　　é =總的其他隨機誤差的影響; </p><p><b>　　K=水平位移; </b></p><p>　　kij =折射系數(shù); </p><p><b>　　R =地球半徑; </b></p><p><b>