fanlili_journal of atmospheric and oceanic technology

1、606J OURNAL O F A TM OSPHERIC AND O CEA NI C T EC HNOLOGYVOLUME 28Cotidal Charts near Hawaii Derived from TOPEX/Poseidon Altimetry DataLI-LI FAN, BIN WANG, AND XIAN-QING LVPhysical Oceanography Laboratory, Ocean University of China, Qingdao, Shandong, China(Manuscript received 1 July 2010, in final

2、form 17 October 2010)ABSTRACTHarmonic analysis of 10 yr of Ocean Topography Experiment (TOPEX)/Poseidon (T/P) along-track al- timetry is performed toderive the semidiurnal and diurnal tides(M2, S2, N2, K2, K1, O1, P1, and Q1) near Hawaii. The T/P solutions are evaluated through intercomparison for c

3、rossover points of the ascending and descending tracks and comparison with the data of tidal stations, which show that the T/P solutions in the study area are reliable. By using a suitable order polynomial to fit the T/P solutions along every track, the harmonic constants of any point on T/P tracks

4、are acquired. A new fitting method, which is characterized by applying the harmonics from T/P tracks to produce directly empirical cotidal charts, is developed. The har- monic constants derived by this fitting method show good agreement with the data of tidal stations, the results of National Astron

5、omical Observatory 99b (NAO.99b), TOPEX/Poseidon 7.2 (TPXO7.2), and Finite Ele- ment Solutions 2004 (FES2004) models, which suggests that the fitting method is reasonable, and the highly accurate cotidal chart could be directly acquired from T/P altimetry data by this fitting method.1. IntroductionT

6、he studies of tides and cotidal charts have a long history. At the early stage, the cotidal charts of certain sea areas mainly depended on observations at coastal and island tidal gauge stations (e.g., Ogura 1933; Nishida 1980; Fang 1986), but the information of tides obtained by these direct observ

7、ation methods is very limited. Since the launch of satellite altimetry Ocean Topography Ex- periment (TOPEX)/Poseidon (T/P) in August 1992, which provides unprecedented precise sea surface height data, the study of ocean tides has progressed dramatically. Especially in the past 10 yr, T/P altimeter

8、data have been widely used to study the distribution and charac- teristics of ocean tides and build either a global or re- gional tidal model. The first application of T/P altimeter datacovering about 7 months was carried outtoderive ocean tides in the Asian semi-enclosed sea by Mazzega and Berge (1

9、994). By interpolating the tidal variations of the sea surface, Yanagi et al. (1997) used T/P altim- eter data covering about 3 yr to construct cotidal charts of eight tidalconstituents in the Yellow and East China Seas,showingsignificantimprovementoverMazzegaandBergeswork. Teagueetal.(2000) evaluat

10、edthetidesin the Bohai and Yellow Seas derived from 5 yrof T/P data through a comparison with data from in situ pressure gauges and coastal tide stations, and their fairly good agreement suggested that T/P could provide accurate tide data in specific regions. Fang et al. (2004) applied the tidal har

11、monics from T/P altimetry data covering 10 yr at coastal and island stations to give empirical cotidal charts of five principal constituents in the Bohai, Yellow, and East China Seas, showing higher accuracy than the previous charts for the offshore area. They acquired the harmonics at every grid po

12、int by using quadratic polynomials to fit its surrounding data points within a certain influence distance in the T/P track co- ordinates. In short, the new methods have been emerging in an endless stream and the study of ocean tides has also significantly improved with the satellite altimeter data i

13、ncreasing.Inthis paper, we propose anew fitting method that is quite different from other existing methods and models. Itdoesnot depend onthe complex dynamic system and has nothing to do with models, so that terrible calcula- tions are avoided. The method is characterized by ap- plying the harmonics

14、 from T/P tracks to produce directly empirical cotidal charts. Based on the continuity of har- monic constants in space, we apply the harmonics on T/P tracks to fit those between T/P tracks so that the har- monics inthewholestudy areas can beestimated.Corresponding author address: Xian-Qing Lv, Labo

15、ratory of Physical Oceanography, Ocean University of China, Qingdao 266100, China.E-mail: DOI: 10.1175/2010JTECHO809.12011 American Meteorological Society607APRIL 2011F A N E T A L .TABLE 1. RMS of amplitude difference and vector difference at 339 crossover points between harmonics

16、derived from ascending and descending tracks for eight principal constituents. Respec- tively, DH and D represent amplitude difference and vector dif- ference between them (cm).M2S2N2K2K1O1P1Q1DH D0.761.060.960.990.691.031.761.201.672.510.831.161.051.650.911.44tracks) within the study area, which ar

17、e indicated by red dots in Fig. 1. We carry out comparisons of harmonics at 339 crossover points for the eight principal constituents(M2, S2, N2, K2, K1, O1, P1,andQ1).Because of thelimited space, we only give the RMS of amplitude dif-ference and vector difference for 339 crossover points in Table 1

18、. The vector difference D between the ascending and descending solutions (Fang et al. 2004) is defined byFIG. 1. Distributions of TOPEX/Poseidon ground tracks and tidal stations near Hawaii. The T/P ground tracks (lines), T/P crossover points (dots), islands (small dark areas), and (S1S7) tidal stat

19、ions (triangles) are shown.)2D5 (H cosG H cosGaadd)21/2,1 (H sinG H sinG(1)The paper is organized as follows. Harmonic analysis of 10 yr of T/P along-track altimetry is performed to derive eight principal constituents near Hawaii, and the T/P solu- tions are evaluated through an intercomparison for

20、cross- over points and a comparison with the data of tidal stations in section 2. In section 3, harmonic constants of any point on T/P tracks are acquired by using suitable-order polynomials to fit T/P solutions along every track. In section 4, a new fitting method, which is characterized by applyin

21、g the har- monics from T/P tracks to produce empirical cotidal charts, is developed and evaluated through comparison with the data from tidal stations, the results of the National Astro- nomical Observatory 99b (NAO.99b), TOPEX/Poseidon 7.2 (TPXO7.2), and Finite Element Solutions 2004 (FES2004) mode

22、ls. The paperends with conclusions in section 5.ddddwhere the subscripts a and d represent ascending and descending solutions, respectively.From Table 1, we can see that the RMS of the am- plitude difference for eight principal constituents is less than 2 cm, and the maximum and minimum are 1.76 (K2

23、 tide) and 0.69 (N2 tide) cm, respectively; the RMS of vector difference is about 2 cm, and the maximum and minimum are 2.51 (K1 tide) and 0.99 (S2 tide) cm, re- spectively. These small differences indicate that the T/P solutions we obtained above are reliable.b. Comparison between T/P-derived harmo

24、nics and the data of tidal stationsWedownload the sealevel data ofseventidal stations (online at /UHSLC/), which are indicated by green triangles in Fig. 1, and we perform harmonic analyses for them. The amplitude difference and vectorial differencebetween the T/Psolutions

25、and the data from the tidal stations for eight principal con- stituents are listed in Tables 2 and 3.From Tables 2 and 3, we can see that the RMS of the amplitude difference for the eight principal constitu- ents is less than 2 cm, for example, the maximum and minimum are 1.73 (M2 tide) and 0.47 (N2

26、 tide) cm, respectively; the RMS of the vector difference for eight principal constituents is less than 3; and the maximum and minimum are 2.95 (M2 tide) and 0.99 (N2 tide) cm, re- spectively. These comparisons validate the T/P solutions.2. Harmonics derived from T/P altimeter dataThe T/P altimeter

27、data from October 1992 to June 2002 (2 ; 363 cycles), which we use in this paper, is pro- vided by the National Aeronautics and Space Adminis- tration (NASA). First, we process the approximately 10 yr of T/P altimeter data to gain the time series of tidal elevations of each observation point on the

28、tracks. Then, we make use of this information to perform harmonic analyses for deriving the harmonics. Considering that too few observation times may lower the reliability of harmonic analysis, we only select the points that have more than 300 T/P observations.a. Intercomparison of T/P-derived harmo

29、nics at crossover pointsThere are in total 339 crossover points on the 54 ground tracks (27 ascending and 27 descending ground3. Acquisition of harmonics on T/P tracksCummins et al. (2001) separated the barotropic and internal tide by using an eighth-order polynomial to fit608J OURNAL O F A TM OSPHE

30、RIC AND O CEA NI C T EC HNOLOGYVOLUME 28TABLE 2. Amplitude difference and vector difference between the T/P solutions and the data of tidal stations for the M2, S2, N2, and K2constituents (cm).StationT/PM2S2N2K2Distance (km)8N/8E8N/8EDHDDHDDHDDHDS1 S2 S3 S4 S5 S6 S728.22/182.6316.75/190.4823.87/193.

31、7121.96/200.6421.43/202.2120.90/203.5319.73/204.9328.42/182.1816.50/191.0823.72/194.1321.76/200.7721.93/203.5321.16/203.8718.46/205.00RMS49.8369.4843.4525.29147.4244.86142.153.342.600.230.290.871.490.051.733.562.624.970.292.551.562.792.951.470.962.731.580.190.591.131.451.531.023.291.740.910.722.221.

32、830.270.760.370.020.620.570.180.470.570.761.112.650.660.591.721.360.271.821.750.080.460.530.721.030.272.112.060.920.470.561.361.31the 7 yr of T/P altimeter data in the North Pacific. Dushaw (2002) used a 300-km running mean of the along-track T/P altimeter data to filter near Hawaii, showing that th

33、e result of filtering H cosG and H sinG, where H is am- plitude and G isphaselag, isbetterthanthatoffiltering H and G.Based on these previous studies, we use suitable-order polynomials to fit the values of H cosG and H sinG along every T/P track in this section, respectively. First, transform amplit

34、ude H and phase lag G to the form H cosG and H sinG. Then, use the nth(n 5 1, 2, 3, . .)- order polynomial to fit the values of H cosG and H sinG along every track, respectively. Finally, the values of amplitude H and phaselag G atanypointon T/Ptracks can be acquired by coordinate transformation. He

35、re, we fit the values of HcosG for the M2 tide along a T/P track as an example.The polynomial is defined asf(y) 5 a 1 a y 1 a y2 1 1 a yn,where yk (k 5 1, 2, 3, . , N) is the latitude of obser- vationpointontheT/Ptrack. Letthepartial differential with respect to the parameters a0, a1, a2, . . . , an

36、 of the cost function equal to zero, and we obtainN(a 1 a y 1 a y 211 a yn ) H cosG 50kk01 k2nkkk51 Ny (a 1 a y 1 a y211 a yn ) HcosG 50k01 k2knkkkk01 k2knkkk k51N2n:yk(a 01 a 1yk 1 a2yk1 1 a y )n Hk cosGk 5 0k.nk51(4)(2)Let us introduce the following notations:012nwhere yisthelatitude ofanypointont

37、heT/P track. The cost function isNNNyi1 j 2 ,b 5yi 1 H cosG ,M 5kki, jikkk51k51(5)i 5 1, 2, . . . , n 1 1, j 5 1, 2, . . . , n 1 1,1 a y2 11 a yn ) HcosG 2,J(a ) 5 (a 1 a yn01 k2knkkkk51(3)Thus, Eq. (4) is rewritten asTABLE 3. As in Table 2, but for K1, O1, P1, and Q1 constituents.StationT/PK1O1P1Q1

38、8N/8E8N/8EDistance (km)DHDHDHDHDDDDS128.22/182.63S216.75/190.48S323.87/193.71S421.96/200.64S521.43/202.21S620.90/203.53S719.73/204.9328.42/182.1816.50/191.0823.72/194.1321.76/200.7721.93/203.5321.16/203.8718.46/205.00RMS49.8369.4843.4525.29147.4244.86142.150.891.700.300.490.510.441.050.200.691.270.0

39、10.720.600.631.310.310.990.350.4062.090.580.851.942.730.951.251.930.830.830.220.741.381.491.672.911.061.081.491.201.550.180.440.710.840.361.610.930.930.421.890.681.22609APRIL 2011F A N E T A L .TABLE 4. Comparisons ofthemean value of theamplitudediffere

40、nce andphase-lag difference ofthe M2 tidebetween T/Psolutions and harmonics derived by polynomial fitting for all the tracks.Order3456789101112131415H (cm)G (8)0.723.670.613.410.553.160.703.050.522.940.512.910.502.780.492.720.492.620.492.660.644.830.492.890.562.81232MMMM3 23tracks to produce an empi

41、rical cotidal chart, is developed. We select any point p in the study area. Based on the ratio of the distances between this point and its two nearest ascending (descending) tracks, a series of points on the descending (ascending) tracks that have the same ratio with this point between the two ascen

42、ding (descending) tracks are obtained. By fitting the harmonics of these points with a suitable-order polynomial, the values of amplitude and phase lag of this point p are acquired. Here, we will describe thisfitting method in detail.a. Identify the four nearest tracks to point pA series of quadrang

43、les in the study area are sur- rounded by ascending and descending tracks as shown in Fig. 1. If we want to acquire the harmonic constants of point p in the study area, we should first know the quadrangle in which this point is located. Meanwhile, thefour nearest tracks tothis pointare identified.As

44、 shown in Fig. 2, the necessary and sufficient conditions for point p(x, y) located in the quadrangle p1p2 3p4p arepoint p is below straight line p1p4 5 y , (y4 2 y1)/ x4 2 x1(x 2 x1) 1 y1, point p is above straight line p2 p3 5 y . (y3 2 y2)/(x3 2 x2)(x 2 x3) 1 y3,point p is above straight line p1p

45、2 5 y . (y2 2 y1)/ (x2 2 x1)(x 2 x2) 1 y2, andpoint p is belowstraight line p3p4 5 y , (y4 2y3)/ (x4 2 x3)(x 2 x4) 1 y4.If the above four conditions are satisfied, point p isb11,11,21,31,n11a07M2,1M2,2M2,3M2,n116ab66647 6 1 7 62 7MMMM53,13,23,33,n117 6 ab7.52 776636.7 45 45.baMn11,1Mn11,2Mn11,3Mn11,

46、n11n11n(6)Thevaluesofa , a , a ,. . . ,a areobtainedbysolvingthe0 1 2nabove linear Eq. (6), and substituting these parameters into Eq. (2) yields the values of the barotropic tide, that is,f(y) 5 a 1 a y 1 a2y 1 1 a yn.(7)012nSimilarly, wecan also getng(y) 5 a 1 a y 1 a y2 1(8)1 a ny ,012by progress

47、ing the values of H sin G for the M2 tidewith the similar method.According to the method above, we use 3rd15th-order polynomials tofitthe values of H cosG and H sinG for the M2 tide along every track, respectively. The mean values of the amplitude difference and phase-lag differ- ence of the M2 tide

48、 between the T/P solutions and har- monics derived by polynomial fitting for all the tracks are given in Table4.From Table 4, we can see that using a higher order in polynomial fitting does notlead toa smaller difference. The mean value of the amplitude difference for all of the tracks by 10th-, 11t

49、h-, 12th-, and 14th-order poly- nomial fitting are all equal to 0.49 cm, which are much betterthantheotherresults. Wecan obviously seethat the mean value of the phase-lag difference for all of the tracks by the 11th polynomial fitting is the smallest. Thus, we choose the 11th-order polynomial to fit

50、 thelocated in the quadrangle p p p p . The four nearest1 2 3 4trackstopoint p, the ascending (i, i 1 1) and descending ( j, j 1 1) tracks, are identified.TABLE 5. Amplitude difference and vector difference between harmonics derived from tidal stations and those obtained by this fitting method for M

51、2 tide (cm). values of H cosG and H sinG for the M tide along2every track, respectively. Then, the values of H cosGStation8N/8EDHDand H sinG for the M2 tide at any point on the tracks are acquired.28.22/182.6316.75/190.4823.87/193.7121.96/200.6421.43/202.2120.90/203.5319.73/204.932.022.540.091.511.3

52、02.033.092.012.622.541.624.183.965.663.103.60S1 S2 S3 S4 S5 S6 S7 RMS4. Development of the fitting methodIn this section, a new fitting method, which is char- acterized by the application the harmonics from T/P610J OURNAL O F A TM OSPHERIC AND O CEA NI C T EC HNOLOGYVOLUME 28FIG. 3. Point p islocate

53、d between twoadjacent ascending tracks (i, i 1)1, and n 2 m 1 1 descending tracks (m, m 1 1, . . . , j, j 1 1, . .,n) pass through the two ascending tracks. A series of points pm, pm11, . . . , pj, pj11, . . . , pn that have the same ratio with point p between thetwo adjacent ascending tracks lie on

54、 the n 1 (m 2 1) descending tracks, respectively.qFIG. 2. Point p is located in the quadrangle p1p2p3p4 that is sur- rounded by two adjacent ascending tracks (i, i 1 1) and two adja- cent descending tracks ( j, j 1 1). Four vertices of the quadrangle are p1(x1, y1), p2(x2, y2), p3(x3, y3), and p4(x4

55、, y4), respectively; d1, d2,d3,d4 arethedistancesbetween thepointpandfoursides(p1p4, p2p3, p1p2, p3p4), respectively.d 5 2 l(lpp )(lpp )(lp p )/p p ,1141 41 4where l 5 (pp1 1 pp4 1 p1p4)/2 is reckoned.Similarly, we can calculate the distances (d2, d3, d4) between point p and theother threesides( p2 p3, p1 p2, p3p4). Then, the distances (d1, d2) between point p and its two nearest ascending tracks (i, i 1 1) and the distances (d3, d4) between point p and its two nearest descending tracks ( j, j 1 1) are k