Pruebas

5.3.1 Preparation of altimeter data

In this chapter, we are interested in determ ining the pure ocean tide, w hich is

related to the tide sensed by the altim eter, by:

4 = 4 + z , + z » 5.3

w here Zi is the ocean loading tide, corrected for using the CSR 3.0 tide m odel [Eanes,

1994], and Zb is the Earth body tide, corrected for using the C artw right-Tayler-Edden

tables [Cartwright and Tayler, 1971; Cartwright and Edden, 1973]. Za is also know n

as the altim etric or geocentric tide.

The altim etric sea surface height can be expressed as:

= Zrnss+

ZpEsi^)

V(t)+

w here z,„„ is the m ean sea surface height determ ined for each ground-track using the m ethod described in section 4.6, ri{t) is the tim e-varying sea surface height anom aly containing the ocean tide residual error, Ô£o is the residual orbit error after detrending

and Ô£ is the sum o f all additional m easurem ent and geophysical correction errors.

ZpEsit) is the a priori ocean tide m odel, FES95.2.1 [Le P rovost et a l , 1998], w hich is required before the residual orbit errors can be rem oved by the detrending process. This is then added back in to give the sea surface height anom aly, Az(t), to be used in this analysis (neglecting the additional error terms):

M 0 = ^/t£5 ( 0 + t7 ( 0 5.5

T he sea surface heig ht anom alies used in this w ork are taken from cro sso v er

locations, giving two m easurem ents o f Az(t) at each point. These m easurem ents are

then g ridded into 0 .5 °x 0 .5 ° cells ready for further analysis. The m ain reason for

usin g d a ta from cro sso v e r lo c atio n s is th a t th e ascen d in g and d esc e n d in g m easurem ent tim es for ERS-1 and 2 at crossover locations are separated by a tim e very close to (n 4- 1/2) days, w here n is an integer. T herefore, the sem idiurnal co n stitu en ts are sam pled alm ost in phase, w hereas the diu rnal co n stitu en ts are sam pled alm ost in antiphase. This additional source of inform ation is the m ain reason why the diurnal constituents, K, and ? i, can be separated from the annual signal.

despite the fact that they have an alias period o f alm ost exactly 1 y ear [A ndersen,

1994].

5.3.2 The response method

In general, tw o m ethods are available for the extraction o f tidal constituents from an altim etric sea surface height tim e series. The first m ethod is the classical harm onic

analysis, developed by L ord K elvin and Sir G eorge D arw in [Munk and C artw right,

1966]. It m akes use o f the fact that the tidal signal is a com plex harm onic, that is the sum o f m any sim ple harm onics each with their own am plitude, phase and period. E quation 5.6 shows the ocean tide, Ç, expressed at each location (0, X) by a com plex harm onic.

Ç {0 ,X ,t) = ^ [ H l ^ { e , X ) c o s c o j + H 2 „ (e ,X )s in COj] 5.6

where co„ are the frequencies o f the principal tidal constituents and t is the U niversal

Tim e. T he co nv en tion al am plitude, H„, and the G reenw ich phase lag, G„, are

obviously derivable from Hl„ and H2„. Several authors have used this approach to

determ ine the am plitudes and phases o f the m ain tidal constituents from altim etry, including W agner et a l [1994] and M a et a l [1994].

H ow ever, w hen this analysis is used, the R ayleigh criterion lim its the capability to extract inform ation at frequency separations o f less than 1/AL, w here AL is the length o f the data series [A ndersen, 1994]. From the alias periods given in table 5.2, we w ould therefore require a tim e series o f observations w ith a length greater than 8 years to separate the M2 and N2 constituents. An even longer tim e series w ould be

req u ired to separate K, and P i, and detectio n o f the S2 signal w o u ld rem ain

im possible. In order to use ERS-1 and 2 data for tidal analysis, a different approach is therefore required.

The m ethod used in this work is the response m ethod o f M unk a nd C artw right [1966],

w hich assum es a sm ooth adm ittance function across each tidal band. (The adm ittance function is the am plitude and phase m odifying function relating the recorded tide to the input potential). A n advantage o f this m ethod is that the adm ittance function is described by a sm all num ber o f param eters w ithin each tidal band. These param eters are in principle independent of the Rayleigh criterion, and therefore a solution for any co nstituen t can be inferred. For ERS-1 and 2, it is a far superior m eth od to the

harm onic analysis, as it allows the S2 constituent to be inferred. The m ethod has been

used to determ ine tidal constituents using data from G eosat [C artw right and Ray,

1990], T O PEX /PO SE ID O N [Ma et a l , 1994; D esai and Wahr, 1995; M atsum oto et

a l , 1995], and ERS-1 [Andersen, \99A\ Andersen, 1995].

In the response m ethod approach, equation 5.6 is replaced by:

i ; ( e ,x ,t) = ' ^ ' ^ [ u , { e , X ) d " ' \ t -2k ) + v , ( e , x W ’' \ t -2k)] 5 . 7

m = ] k = - K

w here m denotes the tidal species (1 or 2), and and 6^"'^ are the real and

im aginary term s in the tim e-dependent com ponent o f the tide-generating potential,

lagged by 2k days. It generally regarded sufficient to lim it ^ to 1.

F or altim etry work, the expansion o f the tides is generally carried out in term s o f the orthotide functions, P ^ ^ \t) and Q ^p \t) o f G roves and R eynolds [1975], w hich are sim ply linear com binations o f the functions a ^ " ' \ t -2k) and b ^ " ' \ t -2k ) appearing in equation 5.7. The transform ations betw een these two sets o f function are described in detail in A ppendix A o f C artw right and Ray [1990], and are such that the orthotide functions are nearly orthogonal in tim e. This has the im plication that the results o b tained by this m etho d are ind ependent o f the nu m ber o f term s used in the expansion.

The expansion o f the ocean tides in equation 5.7 can now be expressed as:

2 2K

m = \ j = 0

w here and are the coefficients to be determ ined from the altim eter data,

representing a total o f 12 coefficients, w ith 6 in each o f the diurnal and sem idiurnal bands.

One further m odification is m ade to the standard response m ethod here, w hich is to

solve sim u ltan eo u sly for the annual signal. T his appro ach w as d ev elo p ed by

A n d ersen [1994] to account for the fact that the K, and P, constituents have an alias period o f exactly 1 year for the ERS missions. The final expansion o f the sea surface height anom aly, Az, is therefore:

Az(e,A,O = i;X[c/J”'(6i,A)P“ (O + V<”>(0,A)ôf>w]

m = lk=0 5 . 9

+flj {d, X) cos( i//^f ) + « 2 ( 0, A ) sin( y/t)

w here a^ÇO, X) and <2 2(0, A) are the am plitudes o f the annual signal w ith frequency y/.

W ith these tw o additional coefficients, we are therefore solving for a total o f 14 param eters, a procedure requiring the inversion o f 1 4 x 14 norm al m atrices. Equation 5.9 w as solved for each 0 .5 °xO .5° bin containing the sea surface h eight anom alies from the crossover locations, assum ing the tidal coefficients to be constant over these sm all areas. F or a unique solution, at least 14 observations are required in each grid cell. H ow ever, for the purposes of this work, the inversion w as only carried out if m ore than 30 observations w ere available, resulting in a total o f 9860 usable grid cells.

The tide potentials required to define the orthotide functions at tim e t w ere com puted by a harm onic expansion o f the 30 largest spectral lines w ithin both the diurnal and

sem idiurnal bands, taken from the tables o f C artw right a n d T ayler [1971] and

C artw right and Edden [1973]. This ensures an adequate definition o f the m ajor tidal constituents and their nodal modulations.

A ndersen [1994] investigated the use of the response m ethod to generate a tide m odel

from ERS-1 data, and found that the S2 constituent is resolved rem arkably well, even

though it is inferred from other sem idiurnal constituents. H e also fo und that by solving for the annual signal sim ultaneously, very good estim ates o f the diurnal constituents could be made, even though Ki and P, both have an alias period o f alm ost exactly one year.

5.3.3 Deriving amplitudes and phases of tidal constituents

For m ost practical applications o f a tide m odel, we require the am plitudes and phases o f the in divid ual tidal constituents. In this section, we show how these can be

com puted at each grid location from the coefficients and The resulting

am plitudes and phases w ill later be used in com parisons o f the ERS m odel w ith existing tide m odels and tide gauge data.

The coefficients, H \ and 7/2, o f the conventional representation o f a sim ple harm onic

given in equation 5.6 are derivable from the real and im aginary parts o f the com plex adm ittance function, Z { œ) = X{cû) + iY(cû), for a given species o f frequency co:

m = ( - ï ) ” H X ((ü)

H 2 = { - lT * 'H Y { m )

The adm ittance function corresponding to equation 5.7 is:

K

Z{co) = X(co) + iY(co) = + iv,^)Qxp(-2icok)

5.10

5.11

5.12

k = - K

A fter orthogonalisation o f the potential functions as described above, Z{co) can be

com puted from the orthotide coefficients given in table 5.3, and the coefficients U( m )

and V(ni).

X{cû) — PqqUq+[p^Q Pi + [ / ? 2 0 P2 1^ ( ^ ) 5.13

^ ( ^ ) = Poo^o +[Pio - P i A c o W i + [P20 - P2lC{^) + 5.14

w here c{co) = 2coscoAt, and s{cû) = 2sinu)A r. (The superscript m has been dropped

for clarity). Equations 5.13 and 5.14, and the values in table 5.3 are taken from

C artw right and Ray [1990] w ith the corrections described by M atsum oto et al. [1995] applied.

Coefficient Diurnal (m = 1) Sem idiurnal (m = 2)

Poo 0.0298 0.0200 PlO 0.1408 0.0905 Pll 0.0805 0.0638 P2 0 0.6002 0.3476 P21 0.3025 0.1645 ^21 0.1517 0.0923

Table 5.3 O rthotide coefficients

The am plitudes and phases o f the m ajor sem idiurnal and diurnal constituents (M? and K] respectively) are derived from the altim eter data using this m ethod, and the results are shown in figure 5.1.

0 10 20 30 40 50 60 70 80 90 100 cm 0 36 72 108 144 180 216 252 288 324 360 deg 0 5 10 15 20 25 30 35 40 45 50 cm 0 36 72 108144 180 216 252 288 324 360 deg

Figure 5.1 Amplitude (a) and phase (b) of the M j constituent derived from altimeter data. Similarly for the K, constituent, (c) and (d).