**JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. C12, PAGES 28,785-28,804, DECEMBER 15, 2000 **

**Fourier and wavelet analyses of TOPEX/Poseidon-derived **

**sea **

**level anomaly **

**over the South China Sea: A contribution **

**to the South China Sea Monsoon Experiment **

**Cheinway **

**Hwang and Sung-An **

**Chen **

**Department **

**of Civil Engineering, **

**National **

**Chiao **

**Tung **

**University, **

**Hsinchu, **

**Taiwan **

**Abstract. **

**We processed **

**5.6 years **

**of TOPEX/Poseidon **

**altimeter **

**data **

**and **

**obtained **

**time **

**series **

**of **

**sea **

**level anomaly **

**(SLA) over **

**the South **

**China **

**Sea **

**(SCS). **

**Fourier **

**analysis **

**shows **

**that sea **

**level **

**variability **

**of the SCS contains **

**major **

**components **

**with periods **

**larger **

**than 180 days **

**and is **

**dominated **

**by the annual **

**and semiannual **

**components. **

**Tidal aliasing **

**creates **

**30-180 day **

**components **

**that can **

**be misinterpreted **

**as wind-induced **

**variabilities. **

**Continuous **

**and **

**multiresolution **

**wavelet **

**analyses **

**show **

**that the SLA of the SCS has **

**monthly **

**to interannual **

**components **

**of time-varying **

**amplitudes, **

**and **

**the **

**regional **

**slope **

**of SLA **

**is 8.9 **

**mm **

**yr **

**'1, **

**which **

**may **

**be caused **

**by the decadal **

**climate **

**change. **

**Coherences **

**of SLA with wind stress **

**anomalies **

**(WSA) **

**and sea **

**surface **

**temperature **

**anomalies **

**(STA) are significant **

**at the annual **

**and semi-annual **

**components. **

**At periods **

**of 2-5 years **

**the wavelet **

**coefficients **

**of SLA, WSA, and STA have **

**the **

**same **

**pattern, **

**but WSA leads **

**SLA, and STA follows **

**SLA. The zero **

**crossing **

**of SLA in spring **

**is **

**highly **

**correlated **

**with the onset **

**of the summer **

**monsoon. **

**The interannual **

**variability **

**of SLA is **

**correlated with E1Nifio-Southern Oscillation, and most important is that when the El Nifio-like**

**wavelet coefficients of SLA over the warm pool northeast of Australia or the SCS change**

**curvature **

**from negative **

**to positive, **

**an E1 Nifio is likely **

**to develop. **

**This is a contribution **

**to the **

**South China Sea Monsoon Experiment (SCSMEX). **

**1. Introduction **

**The South China Sea (SCS) is the largest marginal sea in the **

**western Pacific with a total area of 2,590,000 km 2. Figure 1 shows **

**the countries, waters, major islands and bottom features, and major **

**depth contours around the SCS. The South China Sea Monsoon **
**Experiment (SCSMEX) is an international project to study the **

**monsoons over the SCS. The participating countries include most **

**countries in east Asia and southeast Asia, Australia, and the United **

**States. Its purpose is to "better understand the key physical **
**processes in the onset, maintenance, and variability of the **

**the summer and winter monsoons over the SCS and to E1 **
**Nifio-Southem Oscillation (ENSO) will also be studied. **

**2. Sea Level Anomaly From TOPEX/Poseidon **

**T/P is a satellite altimeter mission specifically designed to**

**measure the height of the sea level with a repeat period of 9.9156 **

**days. We used the T/P Version **

**C Geophysical **

**Data Records **

**(GDRs) from Archiving, **

**Validation, **

**Interpretation **

**of Satellite **

**Oceanographic Data (AVISO) [1996] to generate corrected sea **
**surface heights (SSHs) from cycle 10 (December 26, 1992) to **

**monsoon **

**over **

**southeast **

**Asia **

**and **

**southern **

**China" **

**[Lau, **

**1997, **

**p. cycle **

**219 (August **

**29, 1998). **

**The **

**first **

**nine **

**cycles **

**were **

**not **

**used **

**599]. **

**We have **

**joined **

**SCSMEX **

**to investigate **

**the **

**characteristics **

**of because **

**of a pointing **

**problem **

**[Fu et al., 1994]. **

**The **

**orbit **

**we used **

**sea level variability over the SCS derived from the in the T/P GDRs is based on the Joint Gravity Model 3 (JGM3) **

**TOPEX/Poseidon **

**(T/P) altimeter. **

**We will perform **

**Fourier **

**and gravity **

**field [Tapley **

**et al., 1996] **

**and **

**has **

**an accuracy **

**of about **

**4 **

**wavelet **

**analyses **

**of the T/P-derived **

**sea level time series. cm. The dry tropospheric **

**and **

**inverse **

**barometric **

**corrections **

**were **

**Compared **

**to Fourier **

**analysis, **

**wavelet **

**analysis **

**is a recently based on the European **

**Centre for Medium Range Weather **

**developed **

**tool that **

**is best **

**suited **

**for analyzing **

**phenomena **

**with Forecasts **

**(ECMWF)model. **

**The **

**wet **

**tropospheric **

**corrections **

**were **

**time-varying **

**frequencies **

**and amplitudes. **

**An extensive **

**body **

**of directly **

**taken **

**from **

**the TOPEX **

**Microwave **

**Radiometer **

**(TMR) **

**literature **

**associated **

**with **

**wavelet **

**analysis **

**has **

**been **

**developed **

**over measurements. **

**For the ionospheric **

**correction, **

**TOPEX uses **

**its **

**the past decade. **

**The lectures **

**by Daubechies **

**[1992] **

**provide dual **

**frequency **

**measurements **

**and **

**Poseidon **

**is based **

**on the **

**model **

**readers **

**with **

**both **

**an introductory **

**and **

**an in-depth **

**understanding **

**of of Doppler **

**orbitography **

**and radiopositioning **

**integrated **

**by **

**wavelets. **

**A practical **

**guide **

**to wavelet **

**analysis **

**is given **

**by in satellite **

**(DORIS). The CSR3.0 **

**ocean **

**tide model **

**[Eanes **

**and **

**Totfence **

**and Compo **

**[1998]. **

**A collection **

**of applications **

**of Bettadpur, **

**1995] **

**was used **

**to detide **

**the data. **

**In particular, **

**an **

**wavelets in geophysics and oceanography is given by in oscillator drift correction has been applied to the TOPEX range **
**Foufoula-Georgiou and Kumar [1994]. Moreover, we will measurements [AVISO, 1996]. To produce SLAs for the **

**compute **

**the frequency **

**response **

**functions **

**and **

**coherence **

**functions **

**subsequent **

**analyses, **

**we first formed **

**stacked **

**along-track **

**SSHs **

**by **

**[Bendat **

**and Piersol, **

**1993] between **

**sea level, wind, and sea averaging **

**the point SSHs from the 210 T/P cycles. **

**When **

**surface **

**temperature **

**over **

**the SCS **

**to see **

**the degrees **

**of interaction **

**averaging, **

**the gradient **

**of a geoid computed **

**from the Earth **

**among these signals. The relationships of sea level variability to Gravitational Model 1996 (EGM96) to harmonic degree 360 **
**[Lemoine et al., 1998] was used to reduce point SSHs to the **
**nominal ground tracks selected to correspond with the tracks of **

**Copyright **

**2000 **

**by **

**the **

**American **

**Geophysical **

**Union. **

**cycle **

**92 for **

**this **

**study. **

**A point **

**SLA **

**was **

**computed **

**as **

**Paper number 2000JC900109. **

**0148-0227/00/2000JC900109509.00 ** **zlh = h- h , ** **(1) **

**28,786 ** **HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA **
**105 ø **
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**Figure 1. The South China Sea and its surrounding countries and waters. Also given are selected depth contours **
**(solid lines), the ground tracks of TOPEX/Poseidon (dots), major bottom features, and islands. Squares indicate tide **
**gauge stations. **

**where h and h are point and averaged SSH, respectively. An **

**area-averaged SLA over a given area is the simple mean of all **

**point SLAs in the area, and the associated time is the central time **

**of the T/P cycle. In computing the simple mean we rejected outlier **

**SLAs using Pope's [ 1976] • - test. An outlier point SLA satisfies **

**= **

**. **

**(2) **

**where v i and • ** **are the residual of a SLA (observation minus **

**area mean) and the residual's a posterior standard error, **

**respectively, a is the confidence level, which is 95% in this **

**paper, **

**n is the number **

**of point **

**SLAs, **

**and *a;•,,-2 is the critical **

**, value with degrees of freedom of 1 and n -2. Outlier rejection **

**was performed iteratively until no outlier was found, and the final **

**area-averaged SLA was computed from the "clean" set of point **
**SLAs. On the basis of Figure 1, SCS depths range from less than **
**200 m over the continental shelf to 6000 m at the center of the **

**basin **

**and **

**at the Manila **

**Trench. **

**From **

**numerical **

**models **

**[Shaw **

**and **

**Chao, 1994] and dritter data [Hu, 1998] the SCS has two distinct **

**HWANG **

**AND CHEN: **

**FOURIER **

**AND WAVELET **

**ANALYSES **

**OF SOUTH **

**CHINA **

**SEA **

**3OO**

**250**

**-250**

**-300**

**1993**

**SLA1 (Northern SCS)**

**-- SLA2 (Southern SCS)**

**---- SLA3 (all SCS)**

**... SLA4 (continental shelf of SCS) **

**I i i ** **I ** **J i i , i i i i i i , , I **

**1994 **

**1995 **

**1996 **

**1997 **

**1998 **

**1999 **

**Year **

**Figure **

**2. Time **

**series **

**of SLA **

**in four **

**areas **

**of the **

**SCS. **

**28,787 **

**altimeter-observed **

**SSHs **

**over **

**shallow **

**water **

**will **

**have **

**larger **

**errors **

**Table **

**1 shows **

**the **

**statistics **

**in generating **

**the **

**four **

**SLA **

**time **

**series. **

**than **

**those **

**over **

**the **

**deep **

**ocean. **

**To **

**see **

**the **

**spatial **

**characteristics **

**of The **

**large **

**percentage **

**of rejected **

**SLAs **

**in SLA4 **

**is due **

**to the **

**large **

**the **

**SCS **

**sea **

**level, **

**we **

**computed **

**SLA **

**time **

**series **

**in four **

**subareas **

**ocean **

**tide **

**model **

**error, **

**distortions **

**of altimeter **

**footprints, **

**and **

**of the SCS: **

**The first one **

**(SLA1) **

**is over **

**the deep **

**ocean **

**(depths **

**> **

**200 m) between **

**14.5 **

**ø and 22øN, **

**the **

**second **

**(SLA2) **

**is over **

**the **

**deep **

**ocean **

**between **

**5 ø and 14.5øN, **

**the **

**third **

**(SLA3) **

**is over **

**the deep **

**ocean **

**between **

**5 ø and 22øN **

**(this **

**area **

**combines **

**the **

**first and the second areas), and the last (SLA4) is over the**

**continental **

**shelf **

**(defined **

**as the area **

**with depths **

**<200 **

**m) and **

**between **

**14.5 **

**ø and 22øN. Figure **

**2 shows **

**the four SLA time **

**series. **

**Note **

**that **

**the time **

**series **

**in Figure **

**2 and **

**other **

**figures **

**in this **

**paper **

**have **

**the beginning **

**of each **

**year **

**(January **

**1) labeled. **

**In **

**general, **

**SLA1, **

**SLA2, **

**and **

**SLA3, **

**which **

**are **

**over **

**the **

**deep **

**ocean, **

**agree **

**very **

**well **

**with **

**each **

**other **

**but **

**are **

**significantly **

**different **

**from **

**SLA4. **

**SLA3 **

**behaves **

**basically **

**as the average **

**of SLA1 **

**and **

**SLA2. **

**wind-driven events. Figure 3 shows the standard errors of the area **

**means in the four SLA time series. The standard error of an area **

**mean, tyro, is computed by **

**O'm **

**= **

**I( **

**• V, **

**t=l**

**2 **

**)/n(n **

**-- **

**1) **

**, **

**(3) **

**where v t and n are defined **

**in (2). By this definition **

**a SLA **

**standard error will be affected by the variability of the sea surface **
**under study. It appears that the SLA4 error is not random and has **

**periodic **

**components **

**arising **

**from ocean **

**tide model **

**errors. **

**A **

**Table 1. Statistics of Area-Averaged SLA over the SCS **

**Number of selected points **

**Percentage of rejected points, % **
**Mean standard error, mm **

**North **
**727 **
**0.76 **
**3.49 **
**South **
**997 **
**1.66 **
**2.75 **

**All ** **Continental Shelf **

**1726 ** **132 **

**1.15 ** **3.73 **

**28,788 ** **HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA **
**2O **

**" **

**• ,, **

**SLA1 **

**I I, **

**i **

**, I **

**', ', ... **

**SLA2 1 **

**',**

**"**

**',**

**''**

**',**

**... SLA3**

**',, " **

**" t **

**, t t **

**', ,, -- SLA4 ,,1 **

**11**

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**I I**

**I I i •**

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**II I**

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**15 **

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**',, ,I,,," ,l',,' **

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**I t' , **

**:,,I, / **

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_{I•1• ,, II }', I**II**

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_{1 }**•**

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_{II II }**..I**

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**"' **

**', **

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**',¾"::' '"'"' **

**10 **

**30 **

**50 **

**70 **

**90 **

**110 130 150 170 190 210 **

**TOPEX/POSEIDON **

**cycle **

**Figure 3. Estimated errors (1 standard deviation) of the area-averaged SLA in four areas of the SCS. **

**spectral **

**analysis **

**shows **

**that **

**the spectrum **

**of the SLA4 **

**error **

**is SLA1 **

**is much **

**larger **

**than **

**that **

**of SLA2 **

**because **

**of the higher **

**almost **

**"flat," **

**except **

**at two distinct, **

**large **

**components: **

**the largest latitude **

**of the northern **

**SCS, which is more sensitive **

**to the **

**component **

**has **

**a period **

**of 59 days **

**and **

**an amplitude **

**of 1.2 mm, seasonal **

**variation **

**in solar **

**insulation **

**than **

**the **

**southern **

**SCS. **

**Figure **

**and **

**the **

**next **

**one **

**has **

**a period **

**of 27 days **

**and **

**an amplitude **

**of 0.8 5 shows **

**the filtered **

**SLA **

**time **

**series **

**that **

**were **

**computed **

**using **

**a **

**mm. **

**These **

**two components **

**have **

**to do with **

**the T/P tidal **

**aliasing **

**Gaussian **

**filter **

**with **

**a wavelength **

**of 1 year; **

**see **

**Wessel **

**and **

**Smith **

**discussed **

**below. **

**In addition, **

**the SLA2 **

**error **

**is smaller **

**than **

**the [1995] **

**for the **

**definition **

**of the **

**Gaussian **

**filter **

**and **

**its **

**wavelength. **

**SLA1 **

**error **

**because **

**SLA2 **

**uses **

**more **

**data **

**points **

**than **

**SLA1, **

**and Clearly, **

**the annual **

**components **

**over **

**the northern, **

**southern, **

**and **

**the **

**northern **

**SCS **

**has **

**a much **

**larger **

**sea **

**surface **

**variability **

**than **

**the continental **

**shelf **

**parts **

**of the **

**SCS **

**have **

**summer **

**peaks **

**at different **

**southern **

**SCS **

**[Hwang **

**and **

**Chen, **

**2000]. **

**times **

**of the **

**year. **

**Comparing **

**the **

**phases **

**of the **

**annual **

**components, **

**3. Fourier Spectra **

**of Sea Level Anomaly **

**3.1. Periodograms **

**To see the overall characteristics of the sea level variability **
**over the SCS in the frequency domain, we performed Fourier **
**transforms to obtain periodograms of SLA1, SLA2, SLA3, and **
**SLA4 (Figure 4). Table 2 shows the amplitudes and periods of the **

**first 10 largest components of the four SLA series. All four SLA **

**we find that summer peaks for SLA1 lead those of SLA2 by 21 **

**days and SLA4 by 76 days. In the four SLA time series, there are **

**two strong interannual components with periods of 1036 and 2072 **

**days, which will be investigated in connection to ENSO. Again, in **

**all aspects, SLA3 behaves essentially as the average of SLA1 and **

**SLA2. **

**3,2. ** _{Tidal Aliasing in TOPEX/Poseidon }

**SLA4, being over the continental shelf, has a distinct Fourier **

**series **

**have **

**strong **

**components **

**with **

**a period **

**of 345 **

**days, **

**which spectrum **

**compared **

**to those **

**of SLA1-SLA3. **

**SLA4 **

**has **

**strong **

**HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA ** **28,789 **
**50 **
**40 **
**.-. 30 **
**E **
**E **
**E **

**<t: 20 **

**SLA1**

**SLA2**

**SLA3**

**... SLA4**

**10 ':i **

**. **

**?., **

**:: **

**i !iii!i .;',.,..**

**• !**

**0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 **

**Frequency **

**(cycle/year) **

**Figure **

**4. Periodograms **

**of SLA **

**time **

**series **

**in four **

**areas **

**of the SCS. **

**Table 2. Periods and Amplitudes of the 10 Leading Components of the Four SLA Time Series Over the SCS **

**North ** **South ** **All ** **Continental shelf **

**(Period, days Amplitude, mm) (Period, days Amplitude, mm) (Period, days Amplitude, mm) (Period, days Amplitude, mm) **

**345 ** **39 ** **345 ** **18 ** **345 ** **26 ** **63 ** **45 **
**296 ** **15 ** **2072 ** **16 ** **2072 ** **13 ** **345 ** **44 **
**230 ** **10 ** **1036 ** **8 ** **296 ** **10 ** **61 ** **28 **
**173 ** **10 ** **259 ** **7 ** **414 ** **8 ** **1036 ** **19 **
**414 ** **10 ** **414 ** **7 ** **159 ** **7 ** **414 ** **19 **
**2072 ** **9 ** **159 ** **6 ** **259 ** **7 ** **2072 ** **18 **
**188 ** **8 ** **296 ** **6 ** **173 ** **6 ** **188 ** **18 **
**159 ** **8 ** **49 ** **5 ** **1036 ** **6 ** **173 ** **13 **
**130 ** **8 ** **35 ** **5 ** **130 ** **5 ** **65 ** **12 **
**oo ** **• ** **1 1 **
**138 ** **7 ** **•" ** **5 ** **Ioo ** **5 **

**28,790 ** **HWANG AND CHEN' FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA **

**1993 ** **1994 ** **1995 ** **1996 ** **1997 ** **1998 **

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**13_ 6 ... **

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**..• ** **- ß . - - - . . : : : : - : - I ** **: : . : : - : - ß ß : I ** **I : o ** **ß . : : - : - . : : t : : ß . : t : : ' L I ** **I **

**1993 ** **1994 ** **1995 ** **1996 ** **1997 ** **1998 **

**Time (year) **

**I **

**0 ** **5 ** **10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115120125130 **

**Plate 1. Wavelet coefficients of (a) SLA, (b) wind stress anomaly, and (c) sea surface temperature anomaly over the **
**$CS. The period of a wavelet coefficient is in the sense of Fourier analysis. The unit of coefficients is arbitrary. A **
**coefficient of 130 means that the corresponding wavelet has the maximum similarity with the signal,f(t); a coefficient **
**of 0 means that the wavelet has the maximum similarity with -fit); a coefficient of 65 implies the least similarity **

**HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA ** **28,791 **
**1980 ** **1982 ** **1984 ** **1986 ** **1988 ** **1990 ** **1992 **
**80 ' ** **' ** **' ** **[ ** **' **
**: **

**(a) , i **

**i **

**'. **

**i :, **

**:**

**'**

**'**

**•**

**'" j ... :i ... j ... -_!**

**L ...**

**70 4 ... 1**

**i ... !**

**! ... i**

**:**

**i ... t**

**!**

**• ... !**

**i .... i**

**t**

**•**

**:**

**:**

**t **

**• I ! I i ' **

**i ß ' " **

**! **

**ß **

**60 ... **

**l ... **

**i ... , ... **

**7 **

**... **

**.T- **

**... **

**r **

**... **

**T' **

**t ... **

**t**

**•**

**!**

**_c:**

**1**

**i**

**,**

**I**

**!**

**i**

**o 50**

**... •.-x- œ ... .:.- ... ***

**E**

**.t ... -X,'- ... • ...**

**1994**

**1996**

**I**

**i**

**2O **

**I **

**o 50**

**E**

**.o 40**

**i**

**1998**

**'**

**80**

**I**

**I**

**.... 40**

**E**

**- 30**

**,i ! **

**,.. **

**, **

**• **

**• **

**,: x i **

**ix i **

**,' **

**.x { **

**30 ... **

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**i**

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**' **

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**ß**

**•'**

**•**

**..-" l**

**"**

**r**

**!**

**...**

**: ....**

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**.,.--, ...**

**1980**

**1982**

**1984**

**1986**

**1988**

**1990**

**1992**

**1994**

**1996**

**1998**

**Time (year) **

**lO**

**8o**

**- 70**

**60**

**- 5o**

**- 4o**

**- 30**

**20**

**10**

**0**

**5**

**10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115120125130**

**Plate 2. Wavelet coefficients of (a) NINO3 SST and (b) extended SLA over the SCS. Period and interpretation of **
**wavelet coefficients are the same as those given in Plate 1. The centers of "E" and "L" mark the locations of the peaks **
**of E1Nifio and La Nina, and the center of "X" marks the location where the E1 Nifio-like wavelet coefficients of SLA **

**28,792 ** **HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA **
**100 ** **... ** **, ... ** **, ... ** **, ... ** **, ... ** **, ... **
**5O **
**-5O **
**Northern SCS **
**Southern SCS **
**Continental shelf of SCS **
**-100 ** **... ** **' ... ** **' ... ** **' ... ** **' ... ** **' ... **
**1993 ** **1994 ** **1995 ** **1996 ** **1997 ** **1998 ** **1999 **
**Year **

**Figure 5. Time series of smoothed SLA showing the annual variation of sea level in the northern, southern, and **
**continental shelf of SCS. **

**relatively weak in the other SLA series. The amplitudes of the length we used is 1 year. Then the phases and the amplitudes of the **
**components with periods of 173 and 188 days in SLA4 reach 1-2 errors at given tidal frequencies were computed by a least squares **

**cm and are considered large compared to the same components in harmonic analysis. In general, CSR3.0 has the least error at **

**the others. These components may well be aliased tidal signals Dongsha, which is surrounded by deep waters. At Haikuo the **

**arising **

**from the error of the CSR3.0 **

**tide model **

**we used **

**in largest **

**error **

**occurs **

**at K•, while **

**at Taiping **

**and **

**Dongsha **

**the largest **

**generating SLAs. Table 3 lists the tidal aliasing periods of 11 errors occur at M 2. Although Haikuo is located on the continental**

**major ocean tides for T/P, which were computed using [Parke et shelf and has the largest error at K•, the strongest components in **
**al., 1987; Schlax and Chelton, 1994] ** **SLA4 are those with periods of 60-62 days, which we believe are **

**T(f) = **

**P **

**fp- [fp **

**+ 0.5] **

**' **

**(4) **

**where f is the tidal frequency, p = 9.9156 days is the T/P repeat **

**period, and [fp + 0.5] is the integer part of OCp +0.5). For T/P the **

**due to aliased M 2 and S2. Moreover, SLA1 has a relatively strong **

**173 day component, which becomes smaller in SLA3. SLA3, **

**being the average of SLA1 and SLA2, contains less tidal aliasing **
**effect because it has no significant components with periods close **
**to the tidal aliasing periods. In practice, if the phase of an aliased **
**tide is spatially smooth, then the tidal aliasing effect may be **
**reduced by averaging SLAs over an area larger than the **
**aliasing periods of the major ocean tides range from 28 to 183 days **

**and **

**can **

**be **

**mistakenly **

**associated **

**with **

**monthly, **

**intraseasonal, **

**and wavelength **

**of this **

**aliased **

**tide. **

**Formulae **

**for computing **

**the **

**semiannual **

**sea **

**level **

**variability **

**generated **

**by wind **

**and **

**other **

**wavelengths **

**of aliased **

**tides **

**are **

**given **

**by Schlax **

**and **

**Chelton **

**forcings. **

**Table **

**4 shows **

**the **

**amplitudes **

**and **

**phases **

**of the **

**CSR3.0 **

**[1994]. **

**For **

**example, **

**the **

**wavelength **

**of the **

**aliased **

**M2 **

**for **

**T/P **

**is **

**9 ø The widths of the northern SCS and the southern SCS are just **
**tide model errors at Haikuo, Dongsha, and Taiping tide gauge ** **ß **

**stations, **

**whose **

**locations **

**are given **

**in Figure **

**1. To determine **

**the about **

**the wavelength **

**of the aliased **

**M2, and hence **

**SLA1 **

**and **

**CSR3.0 **

**model **

**error **

**at a tide **

**gauge **

**station, **

**we first **

**subtracted **

**the SLA2 **

**may **

**marginally **

**escape **

**the contamination **

**of the aliased **

**M 2 **

**CSR3.0 modeled values from the measured values to get the errors. tide. Because the whole SCS extends more than 9 ø , SLA3 is less **

**HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA ** **28,793 **

**Table 3. Tidal Aliasing Periods of TOPEX/Poseidon for **

**11 Major Ocean Tides **

**Tide ** **Aliasing Period, days **

**M2 ** **62 **
**S2 ** **59 **
**N2 ** **50 **
**K2 ** **87 **
**O2 ** **46 **
**K• ** **173 **
**Q• ** **69 **
**Mm ** **28 **
**Mf ** **36 **
**Ssa ** **183 **

**the continental shelf of the SCS is < 9 ø , hence, along with the fact **
**that the M2 error and its phase variation are large, SLA4 contains a **

**strong component with period equal to the aliasing period of M2. **

**For comparison, in Figure 6 we show the periodograms of SLA4, **

**as well as SLA series over the Taiwan Strait, the East China Sea, **

**and the Yellow Sea, which are all located on the continental shelf **

**of east Asia. All the SLA series in Figure 6 contain strong **

**components with periods of about 60 days, which are due to the **

**aliased M2 and S2 tides. Comparing the periods of the dominant **

**cmnponents in Figure 6 and the aliasing periods in Table 3, we **

**believe that the T/P-derived SLAs over the Taiwan Strait and the **
**East China Sea are affected by almost all aliased major ocean tides. **

**As an example, we find that the Rms difference between CSR3.0 **

**modeled and observed tidal values over 1993-1994 at the Taichung **

**tide gauge station is 50 cm. The Taichung station is located on the **
**central west coast of Taiwan, and there the tidal amplitude is 3 m. **

**This explains why SLAs over the Taiwan Strait are so seriously **

**affected by the aliased tides. **

**Wind has components with periods of 30-60 days [Philander, **
**1990], and in particular, the wind over the SCS has a component **

**lOO **

**II **

**- Continental **

**shelf **

**of SCS **

**II ** **- ** **Taiwan Strait **

**III **

**,. East **

**China **

**Sea **

**. ] **

**I**

**.**

**Yellow **

**Sea **

**so 'i **

**,- **

**0 ** **2 ** **4 ** **6 ** **8 ** **10 ** **12 ** **14 ** **16 ** **18 ** **20 **

**Frequency (cycle/year) **

**Figure 6. Comparison of periodograms of TOPEX/Poseidon-derived SLA in four areas of the continental shelf of **

**28,794 ** **HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA **

**with **

**a period **

**of 180 **

**days **

**due **

**to the **

**summer **

**and **

**winter **

**monsoons **

**be easily **

**used **

**to identify **

**the **

**scale **

**of the **

**signal **

**and **

**its location **

**on **

**(here **

**we consider **

**only **

**the **

**magnitude **

**of wind; **

**see **

**below). **

**All the **

**t axis. **

**The **

**constant **

**1/•a in (5) is to ensure **

**that **

**the **

**norm **

**of **

**these wind components may give rise to sea level variability of the **

**same **

**frequencies. **

**Because **

**of **

**the **

**closeness **

**between **

**the **

**periods **

**of •((t- b)/a) is equal **

**to the norm **

**of •(t), i.e., **

**the **

**wind **

**components **

**and **

**the **

**aliased **

**periods **

**of tides, **

**there **

**is **

**The **

**wavelet **

**function **

**•(t) must **

**have **

**a **

**considerable **

**uncertainty **

**as to whether **

**certain **

**components **

**in the compact **

**support **

**and satisfy **

**the admissibility **

**condition, **

**which **

**four SLA time series **

**are due **

**to aliased **

**tides **

**or are due **

**to wind. demands **

**that the mean **

**of •(t) up to a certain **

**order **

**of moment **

**However, **

**if a component **

**is due **

**to an aliased **

**tide, **

**then **

**this be zero. **

**Furthermore, **

**small **

**scale **

**corresponds **

**to high **

**frequency **

**component **

**should **

**show **

**consistent **

**amplitude **

**throughout **

**the **

**entire **

**and large **

**scale **

**corresponds **

**to low frequency. **

**Thus **

**C(a,b) **

**time **

**span **

**of a time **

**series. **

**To detect **

**the **

**time-variation **

**of the represents **

**the **

**time-scale **

**structure **

**of a signal. **

**amplitude of a component will rely on wavelet analysis discussed ** _{Since data are always }_{given in a discrete }_{form, the continuous }

**below. Moreover, to have less tidal aliasing effect and to have a **

**wavelet transform is approximated as **

**time series that represents the overall behavior of the SCS sea level, **

**only **

**SLA3 **

**will **

**be **

**used **

**in **

**the **

**following **

**analysis. **

**1 2v-• **

**nat- **

**b•, **

**= • ** **5• f(nzlt)•( ** **)zlt , ** **(6) **

**Cj'k **

**•j .=0 **

**aj **

**4. Wavelet Analysis of Sea Level Anomaly **

**In the Fourier **

**analysis **

**performed **

**in Section **

**3.1, a time **

**series where **

**N is the number **

**of records **

**and zlt is the sampling **

**interval. **

**is regarded as a stationary signal, so that in theory the spectral Equation (6) can be evaluated by algorithms such as fast Fourier **
**content of a segment of the series should be equivalent to that of transform [Totfence and Compo, 1998]. The discrete scales and **
**any other segment. However, many signals, including SLA, are translations are selected as **

**nonstationary and have time-varying frequencies and amplitudes. **

**For **

**example, **

**sea **

**level **

**variabilities **

**can **

**be due **

**to wind, **

**sea **

**surface **

**a• = jAt, j = 2,---,d **

**max **

**temperature **

**(SST), **

**ENSO, **

**and **

**other **

**forcings **

**that **

**have **

**different bk **

**= kAt, **

**k = 0,---,N-1 , **

**(7) **

**magnitudes and frequencies at different times. To see the **

**time-varying **

**components **

**of a signal, **

**a better **

**tool **

**than **

**Fourier **

**where **

**dmax **

**is a number **

**<N. **

**The **

**indexj **

**starts **

**from **

**2 because **

**2zlt **

**analysis **

**is wavelet **

**analysis. **

**In this **

**study **

**we employ **

**both **

**the is the **

**smallest **

**resolvable **

**scale. **

**The **

**choice **

**of dmax **

**depends **

**on **

**the **

**continuous **

**wavelet **

**transform **

**and **

**the **

**wavelet **

**multiresolution **

**spectral **

**content **

**of the **

**analyzed **

**signal, **

**[see **

**also **

**Torrenee **

**and **

**transform **

**to analyze **

**SLA and **

**other **

**data. **

**The continuous **

**Compo, **

**1998; **

**Kumar **

**and **

**Foufoula-Georgiou, **

**1994]. **

**Note **

**that **

**one-dimensional **

**wavelet **

**transform **

**of a signal, **

**f(t), is defined **

**as Torrenee **

**and **

**Compo **

**[1998] **

**and **

**Kumar **

**and **

**Foufoula-Georgiou **

**[Daubeehies, **

**1992] **

**[1994] **

**present **

**two **

**different **

**criteria **

**for **

**choosing **

**dmax. **

**1 ** **Plate l a shows the wavelet coefficients of SLA3 computed **

**C(a,b)=•aalf(t)•(t-b)dt **

_{a }

_{a }

**, **

**(5) **

**with **

_{1992] }

_{1992] }

**the **

**real **

**part **

**of **

**the **

**Morlet **

**wavelet, **

**defined **

**as **

**[Daubechies, **

**where **

**a and **

**b are **

**scale **

**and **

**translation, **

**respectively, **

**•(t) is the **

**•(t) ----II•-l/4e-t2/2 **

**COS(5t) **

**, **

**(8) **

**wavelet function, and C(a,b) is the wavelet coefficient. If • is a **

**comr•lex **

**function, **

**then **

**we must **

**use **

**the conjugate **

**of • in (5). The **

**Morlet **

**wavelet **

**is widely **

**used **

**in geophysical **

**research **

**such **

**as **

**Mathematically, **

**C(a,b) **

**is **

**the **

**projection **

**off(t) **

**on **

**•((t - b) **

**/ a) or seismic **

_{and frequency domains are given by }**data **

**analysis. **

**Properties **

**of **

**the **

**Morlet **

**wavelet **

_{Kumar }**in **

**the **

**time **

_{and }**the inner product of fit) and •((t-b)/a).**

**The greater the**

_{Foufoula-Georgiou [1994]. It is easy to show that the Morlet }**similarity **

**between **

**f(O and •((t - b) / a), the larger **

**the coefficient **

**wavelet **

**in (8) has **

**a compact **

**support **

**and **

**satisfies **

**the **

**admissibility **

**C(a,b). **

**Thus **

**large **

**absolute **

**values **

**of the wavelet **

**coefficients **

**can condition **

**up to any order **

**of moment, **

**see, for example, **

**the **

**Table 4. Amplitudes and Phases of CSR3.0 Tide Model Errors :•t Three Tide Gauge Stations of the SCS **

**Tide ** **Haikuo ** **Taiping ** **Dongsha **

**Amplitude, cm Phase, deg ** **Amplitude, crn Phase, deg ** **Amplitude, cm Phase, deg **

**M2 ** **9.5 ** **94.7 ** **17.1 ** **159.3 ** **8.7 ** **25.3 **
**S2 ** **4.4 ** **333.7 ** **6.5 ** **340.6 ** **2.6 ** **200.0 **
**N2 ** **3.0 ** **307.9 ** **3.4 ** **268.0 ** **1.1 ** **111.4 **
**K2 ** **3.0 ** **140.7 ** **1.2 ** **175.9 ** **0.5 ** **43.4 **
**O• ** **10.3 ** **285.9 ** **9.4 ** **47.8 ** **4.9 ** **179.9 **
**P• ** **4.1 ** **118.3 ** **6.5 ** **256.2 ** **1.7 ** **75.1 **
**K• ** **20.0 ** **84.4 ** **15.6 ** **267.9 ** **5.4 ** **32.7 **
**Q• ** **1.5 ** **29.5 ** **1.8 ** **152.2 ** **0.3 ** **281.8 **
**Mm ** **1.7 ** **237.4 ** **0.3 ** **6.2 ** **3.9 ** **256.4 **
**Mf ** **1.2 ** **314.3 ** **0.9 ** **321.9 ** **0.6 ** **317.7 **
**Ssa ** **3.3 ** **39.9 ** **1.5 ** **207.2 ** **4.4 ** **100.2 **

**HWANG AND CHEN: FOURIER AND WAVELET ANALYSES OF SOUTH CHINA SEA ** **28,795 **

**Signal **

**al **

**a2 **

**a3 **

**a4**

**a5**

**a6 **

**a7**

**dl **

**d2 **

**d3 **

**d4 **

**d5 **

**d6 **

**d7 **

**Figure 7. Wavelet decomposition tree of a seven level multi- **

**resolution wavelet transform. **

**integration result by Gradshteyn and Ryzhik [1994, p. 531]. **

**According to Torrerice and Compo [ 1998, Table 1 ] the relationship **

**between the period p (or wavelength) of a Fourier component and **

**the scale a of the Morlet wavelet given in (8) is **

**4/ra **

**p = ** **= 1.232a , ** **(9) **

**5+42+52 **

**which **

**helps **

**to interpret **

**the wavelet **

**coefficients **

**in the Fourier **

**sense. Furthermore, to reduce the edge effect in the wavelet**

**a zone called "cone of influence (COI)" [Torrence and Compo, **
**1998] where the coefficients are less reliable than those in other **
**parts of the plot. However, according to Totfence and Compo **
**[1998, p. 67], if the time series is cyclic, there will be no COI. **
**Since the SLA time series (also the time series in Plates lb and lc) **
**is dominated by the annual cycle (see Figure 5) and is almost **
**cyclic, the COI of the wavelet coefficients in Plate 1 should be **
**negligible. Below is a summary of the SLA components identified **
**in Plate 1 a. **

**1. The 30-120 day components are strong from January 1993 to **
**January 1995 and weak after January 1995. The fact that these **
**components have different amplitudes at different times suggests **
**that SLA3 is not affected by the aliased M2 and S2 tides. **

**2. The semiannual component is identified as a local high in **
**spring and a local high in winter in each year at period of 6 months. **
**This component is due to the summer and winter monsoons of the **
**SCS. The peaks always occur in May to June (for the spring high) **
**and in November to December (for the winter high). In the springs **
**of 1995 and in the winter of 1997-1998, such semiannual highs **

**almost disappear. The time-varying amplitude of this component **

**shows no contamination of aliased K• and Ssa tides in SLA3. **

**3. The annual component creates the almost periodic wavelet **

**coefficients that can be easily identified in Plate 1 a. It has a peak in **
**summer, and the time span between two consecutive summer **

**peaks is about 12 months. The amplitude varies from one year to **

**another with the largest in 1996 and the smallest in 1994. **

**4. The wavelet coefficients corresponding to the interannual **

**components are rather smooth. At periods of 2-3 years, two bands **

**of highs occur between late 1993 and the summer of 1994 and **

**between the summer of 1996 and the spring of 1997. Notice that **

**the two bands of highs occur about 1 year before both the **

**1994-1995 and the 1997-1998 E1 Nifios. At periods > 3 years, a **

**band of highs begins in late 1995 and ends in the spring of 1997. In **
**fact, the spring of 1997 is the time when the 1997-1998 E1 Nifio **

**starts to develop. **

**Next we perform wavelet multiresolution transform on SLA3. **

**The multiresolution transform applies a wavelet matrix, consisting **

**of scaling coefficients, to a data vector hierarchically to obtain a **
**sequence of vectors containing approximations and details at **

**different levels; see, for example, Press et al. [1993, p. 594] and **
**Chui [1992, p. 20] for the computational algorithm. Thus, with the **

**wavelet multiresolution transform, a signal is decomposed into **

**components of different resolutions. For example, Figure 7 shows **

**the wavelet decomposition tree of a seven-level decomposition of a **
**signal. In Figure 7 we have the relationships a, = a,+•+d,+•, and **

**signal = a7+d7+do+ds+dg+ds+d2+d•. The higher the degree of **

**approximation, the lower the resolution. For example, a2 is coarser **
**than a•. This is also true of the details. In this study we used the **

**scaling coefficients of the Daubechies number 3 (db3) wavelet **

**[Daubechies, 1992] to form the wavelet matrix for the **

**multiresolution transform. The db3 wavelet has a compact support **

**length of five, and up to the third order derivative of its Fourier **

**transform at the origin (frequency = zero) is zero. Figure 8 shows a **

**seven-level decomposition of SLA3, which is summarized as **

**follows. **

**1. Detail d• detects five anomalously negative SLA values **
**between January 1993 and January 1995. The responsible T/P **
**cycles are 31, 41, 55, 65 and 79, when the Poseidon altimeter was **

**on. It also shows that between January 1993 and January 1995, the **

**monthly variability is much stronger than at any other time. **
**2. Approximations a2 and a3 show the seasonal variations of **
**SLA. Detail d3 shows the almost semiannual variation of sea level **

**related to the summer and winter monsoons. **