MIKE 21. Tidal Analysis and Prediction Module. Scientific Documentation

Transcription

1 MIKE 1 Tidal Analysis and Predictin Mdule Scientific Dcumentatin MIKE 017

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3 PLEASE NOTE COPYRIGHT This dcument refers t prprietary cmputer sftware, which is prtected by cpyright. All rights are reserved. Cpying r ther reprductin f this manual r the related prgrammes is prhibited withut prir written cnsent f DHI. Fr details please refer t yur DHI Sftware Licence Agreement. LIMITED LIABILITY The liability f DHI is limited as specified in Sectin III f yur DHI Sftware Licence Agreement : IN NO EVENT SHALL DHI OR ITS REPRESENTATIVES (AGENTS AND SUPPLIERS) BE LIABLE FOR ANY DAMAGES WHATSOEVER INCLUDING, WITHOUT LIMITATION, SPECIAL, INDIRECT, INCIDENTAL OR CONSEQUENTIAL DAMAGES OR DAMAGES FOR LOSS OF BUSINESS PROFITS OR SAVINGS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION OR OTHER PECUNIARY LOSS ARISING OUT OF THE USE OF OR THE INABILITY TO USE THIS DHI SOFTWARE PRODUCT, EVEN IF DHI HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. THIS LIMITATION SHALL APPLY TO CLAIMS OF PERSONAL INJURY TO THE EXTENT PERMITTED BY LAW. SOME COUNTRIES OR STATES DO NOT ALLOW THE EXCLUSION OR LIMITATION OF LIABILITY FOR CONSEQUENTIAL, SPECIAL, INDIRECT, INCIDENTAL DAMAGES AND, ACCORDINGLY, SOME PORTIONS OF THESE LIMITATIONS MAY NOT APPLY TO YOU. BY YOUR OPENING OF THIS SEALED PACKAGE OR INSTALLING OR USING THE SOFTWARE, YOU HAVE ACCEPTED THAT THE ABOVE LIMITATIONS OR THE MAXIMUM LEGALLY APPLICABLE SUBSET OF THESE LIMITATIONS APPLY TO YOUR PURCHASE OF THIS SOFTWARE. MIKE 017

4 CONTENTS MIKE 1 Tidal Analysis and Predictin Mdule Scientific Dcumentatin 1 Intrductin... 1 Scientific Backgrund....1 General.... Tide Generating Frces and Tidal Ptential Ddsn's Develpment Astrnmical Argument and Greenwich Phase Lag Satellite Cnstituents and Ndal Mdulatin The Representatin f Tidal Currents Sme Criteria f Applicatin General Selectin f Cnstituents Inference Prgram Descriptin IOS Methd General intrductin Time series representatin Rayleigh criterin Step-by-step calculatin Analyses f Tidal Height Analysis f Tidal Height using the Admiralty Methd Analysis f Tidal Height using the IOS Methd Predictins f Tidal Height Predictin f Tidal Height using the Admiralty Methd Predictin f Tidal Height using the IOS Methd Acknwledgement References APPENDICES APPENDIX A Cnstituent Data Package APPENDIX B Glssary i

5 Intrductin 1 Intrductin The present manual is intended fr the user f the tidal analysis and predictin prgrams included in the MIKE 1 Tlbx. These prgrams cnstitute a necessary tl when dealing with castal engineering prblems, in particular regarding the definitin f bundary cnditins and the calibratin and validatin f hydrdynamic mdels, as well as the lng-term predictin f tidal levels and currents. They are als fundamental fr the interpretatin f large-scale circulatin prcesses, thrugh the calculatin f c-tidal and c-range lines and the analysis f the individual prpagatin f tidal cnstituents acrss the study znes. The prgrams are based n sme f the mst advanced wrks n tidal analysis (Ddsn, Gdin) and include facilities fr ndal mdulatin f the effects f satellites n the main cnstituents and fr inference f the main tidal cnstituents that cannt be included in the analysis due t the duratin f the recrd. The Rayleigh criterin is used fr the selectin f cnstituents frm a standard data package cmpsed by 69 cnstituents, 45 f which are f astrnmical rigin. Additinally, 77 shallw water cnstituents can be requested. The amplitudes and phases are calculated via a least squares methd, which enables the treatment f recrds with gaps. Fr the calculatin f frequencies, ndal factrs and astrnmical arguments, the prgram is based n Ddsn's tidal ptential develpment and uses the reference time rigin f January 1, 1976 fr the cmputatin f astrnmical variables. As an ptin t the abve-described general apprach, prgrams based n the Admiralty Methd have als been implemented. In this methd nly the fur main cnstituents M, S, 0 1 and K 1 are explicitly cnsidered, and crrective factrs are allwed t take int accunt the effects f a number f astrnmical and shallw water generated cnstituents. The necessary scientific backgrund fr the practising engineer is given in Chapter, which is supplemented by a glssary f terms n tidal thery, included at the end f the manual. Criteria f applicatin are discussed in Chapter 3, with particular emphasis n the chice f cnstituents and n the interpretatin f the inference f cnstituents. The descriptin f the prgrams is presented in Chapter 4. The Tidal Analysis and Predictin Mdule f MIKE 1 cmprises the fllwing fur cmplementary prgrams fr the analysis and predictin f tidal heights and currents: TIDHAC - Analysis f tidal heights TIDCAC - Analysis f tidal currents TIDHPC - Predictin f tidal heights TIDCPC - Predictin f tidal currents. In Chapter 5 examples f Analyses f Tidal Height using bth the Admiralty and the IOS Methd are given. Finally, Chapter 6 demnstrates examples f Predictins f Tidal Height using bth Admiralty and the IOS Methd. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 1

6 Scientific Backgrund Scientific Backgrund.1 General Understanding the principal prcesses invlved in the generatin and prpagatin f cean tides is fundamental fr the apprpriate applicatin f prgrams fr the analysis f tidal heights and currents, as well as fr the interpretatin and use f the calculated tidal cnstituents. The present chapter intends t give the basic scientific backgrund fr the user f the Tidal Analysis and Predictin Mdule f MIKE 1. Mre detailed and advanced accunts can be fund in the references listed at the end f this manual. Fr thse nt familiar with the specific terminlgy f tides cmmnly emplyed and used in this manual, a glssary f terms is als included. Oceanic tides are generated by the cmbined effects f the gravitatinal tractive frces due t the mn and the sun (accrding t Newtn's law f gravitatin), and f the centrifugal frces resulting frm the translatin f the earth arund the centres f gravity f the earth-mn and earth-sun systems. The resultants f these frces ver the whle earth balance each ther. Hwever, their distributin ver the earth's surface is nt unifrm, as shwn in Figure.1 (a), the resultant at each pint being the effective tide generating frce. The hrizntal cnstituent f these frces is the main agent in cnnectin with the generatin f tides, and is accrding t Ddsn and Warburg (1941) called the tractive frce, whse distributin is presented in Figure.1 (b). The tide generating frces are peridic, with perids determined by the celestial mvements f the earth-mn and earth-sun systems 1. The main species f tidal cnstituents can be summarised as fllws: Semi-diurnal cnstituents that are a cnsequence f the earth's rtatin, causing tw high waters and tw lw waters t ccur during ne cmplete revlutin at a certain place n the surface. The lunar principal cnstituent, M, represents the tide due t a fictitius mn circling the equatr at the mean lunar distance and with cnstant speed. In a similar way, we can define the slar principal cnstituent, S. The deviatins f the real mvements with respect t the regular rbits f the fictitius celestial bdies abve cnsidered (e.g. deviatins in distance, speed and eccentricity) are accunted fr thrugh the intrductin f additinal semi-diurnal cnstituents (e.g. N, K ). 1 Accrding t Laplace in Celestial Mechanics, the water surface mvements, under the influence f a peridic frce, are als peridic and have the same perid f the acting frce. DHI - MIKE 1 Tidal Analysis and Predictin Mdule

7 Scientific Backgrund Figure.1 Ocean tides. a) Tide generating frces; b) Tractive frces; c) Influence f the declinatin f the mn/sun; d) Influence f the relative psitin f the mn and the sun e) lunar changes: fractin illuminated, distance and declinatin Diurnal cnstituents that are explained by the declinatin f the mn and the sun, and accunt fr the imprtant differences many times bserved between tw successive high r lw waters, Figure.1 (c). The perids and relative amplitudes f the main tide cnstituents that accunt fr 83% f the ttal tide generating frce (Ddsn, 1941) are presented in Table.1. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 3

8 Scientific Backgrund Table.1 Main tidal cnstituents Cnstituents Name Symbl Perid (hrs) Amplitude (M =100) Principal Lunar M Principal Slar S Larger Elliptical Lunar N Luni-Slar Declinatinal K Luni-Slar Declinatinal K Principal Lunar O Principal Slar P The relative psitin f the earth, the mn and the sun is respnsible fr the frtnight variatins bserved in the tidal ranges that reach a relative maximum (spring tides) when the three bdies are clsest t be in line (new mn and full mn), and a relative minimum (neap tides) when the lunar and slar frces are ut f phase and tend t cancel (first and last quarter), as it is shwn in Figure.1 (d). The spring-neap tidal cycles are nt prduced by a specific tide generating frce, but by the cmbined effects f the M and S cnstituents that are assciated with the apparent mvements f the mn and the sun. This can be easily verified by cnsidering the water level variatin due t the cnstituents M and S that can be expressed by h t cs t t M cs S (.1) cnsidering unit amplitudes and that there is n phase difference at the adpted time reference. Using a trignmetric transfrmatin, we get h t M S t at cs t t (.) where a and t t cs (.3) S M (.4) which shws that the superpsitin (withut interactin) f the tw semidiurnal cnstituents is still a semidiurnal wave with a frequency given by the average f the frequencies f the riginal waves and an amplitude that varies as a functin f Δω (it can be shwn that fr tw generic cnstituents f amplitudes a 1 and a, the amplitude f the mdulated wave varies between the sum - the cnstituents are in phase - and the DHI - MIKE 1 Tidal Analysis and Predictin Mdule 4

9 Scientific Backgrund difference - the cnstituents are in ppsitin - f the amplitudes f the riginal waves). The perid f the cmpund wave will then be given by T (.5) which cnsidering that ω M = E-4 rad/s and ω S = E-4 rad/s, is fund t be T = days, i.e. apprximately half f the syndic r lunar mnth, that is the time between tw successive new r full mns. In practice, the bserved spring and neap tides lag the maximum and minimum f the tidal frces, usually by ne r tw days. The tidal cycles are als dependent n the variatin f distance alng the elliptical rbits (the frces are higher at the apgee and aphelin than at the perigee and perihelin), and n the variatin f the declinatins f the mn and the sun (the diurnal tides increase with the declinatin, while the semidiurnal tides reach a maximum when the declinatin is zer). Fr example during a lunar syndic perid, the tw sets f spring tides usually present different amplitudes, Figure.1 (d). This difference is due t the varying lunar distance. One cmplete cycle frm perigee t perigee takes an anmalistic mnth f 7.6 days, the crrespnding tide generating frces varying by ± 15%. It is als interesting t refer that during the equinxes, March and September, due t the small declinatin f the sun, the slar semidiurnal frces are maximised, with the result that spring tidal ranges usually are significantly larger than average spring tidal ranges; thse tides are called equinctial spring tides. It is als imprtant t mentin that the plane f mtin f the mn is nt fixed, rtating slwly with respect t the ecliptic. The ascending nde at which the mn crsses the ecliptic frm suth t nrth mves backwards alng the ecliptic at a nearly unifrm rate f per mean slar day, cmpleting a revlutin in years. These mvements can be cmpletely mdelled thrugh the intrductin f additinal cnstituents with different amplitudes and angular speeds. Nevertheless fr their characterisatin the analysis f a lng perid f years wuld be necessary, which is fr mst cases impracticable. This leads t the use f crrectin terms that accunt fr the effect f thse cnstituents, which therefre riginally are called ndal crrectin factrs. Finally, the tw alternative reference systems currently used t define the astrnmical c-rdinates are briefly presented (Figure.). Fr a terrestrial bserver a natural reference system cnsists in expressing the psitin f a celestial bdy in terms f its angular distance alng the equatr frm the Vernal equinx, the right ascensin, tgether with its angular distance nrth and suth f the equatr measured alng the meridian, the declinatin. This system is called the equatrial system. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 5

10 Scientific Backgrund Figure. Astrnmical angular c-rdinates The plane f the earth's revlutin arund the sun can als be used as a reference. The celestial extensin f this plane, which is traced by the sun's annual apparent mvement, is called the ecliptic. The pint n this plane chsen as a zer reference is als the Vernal equinx. The psitin f a celestial bdy is defined in this system by its lngitude, which is the angular distance eastward alng the ecliptic, measured frm the Vernal equinx, and by its latitude, measured psitively t the nrth f the ecliptic alng a great circle cutting the ecliptic at right angles.. Tide Generating Frces and Tidal Ptential As described in the previus chapter, the tide generating frces are given by the resultant f the frces f attractin, due t the mn and the sun, and f the centrifugal frces due t the revlutin f the earth arund the centres f gravity f the earth-mn and earthsun systems. Cnsider the earth-mn system and a particle f mass m lcated at P 1 n the earth's surface, as shwn in Figure.3. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 6

11 Scientific Backgrund Figure.3 Diagram f the earth-mn system Accrding t Newtn's law f gravitatin, the frce f attractin at P 1 is given by F a m mm G (.6) R r m where G is the gravitatinal universal cnstant, ( Nm Kg - ). Because all pints f the earth travel arund the centre f mass f the earth-mn system in circles, which have the same radius, the distributin f the centrifugal frces is unifrm. Nting als that at the centre f gravity f the earth the centrifugal and attractin frces have t balance each ther, it is immediate t cnclude that the centrifugal frces are given by m m Fc G (.7) R m m The tide generating frce, F t, at P 1 will then be F t G m m r m Fa Fc 3 (.8) Rm which gives a net frce twards the mn. Similar cnsideratins shw that fr a particle at P the gravitatinal attractin is t weak t balance the centrifugal frce, which gives rise t a net frce away frm the mn with the same intensity as the frce at P 1. It is als easy t prve that the net frce at P 3 is directed twards the centre f the earth, with half the intensity f the tide generating frces acting at P 1 and P. This riginates an equilibrium shape (assuming static cnditins) which is slightly elngated alng the directin defined by the centres f the mn and f the earth. In the derivatin f this result the apprximatin 1/ 1 / R r / R 1; r / R 1/60 m m r m 1 r / R m has been used, since DHI - MIKE 1 Tidal Analysis and Predictin Mdule 7

12 Scientific Backgrund Instead f wrking directly with the tide generating frces it is advantageus t use their gravitatinal ptential which, being a scalar prperty, allws simpler mathematical manipulatin. A ptential functin Ω is thus defined thrugh the relatinship F (.9) The gravitatinal ptential created at a distance R by a particle f mass M is then M G R (.10) which is in accrdance with Newtn's law f gravitatin, as it can easily be seen in M R F G i r (.11) Cnsidering the lcatin f the generic pint P n the earth's surface (Figure.4), the gravitatinal ptential can be written in terms f the lunar angle, the radius f the earth r and the distance R m between the earth and the mn, resulting in Gmm MP Gm R m m 1 r R m cs r R m ½ (.1) Figure.4 Lcatin f pint P n earth's surface which can be expanded as 3 Gm m r r r 1 L... 1 L L 3 3 Rm Rm Rm Rm (.13) where the terms L n are Legendre plynmials given by L 1 cs (.14) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 8

13 Scientific Backgrund L ½ 3cs 1 (.15) 3 L ½ 5cs 3cs (.16) 3 The first term in Equatin (.13) is cnstant and s has n assciated frce. The secnd term generates a unifrm frce parallel t OM, which is balanced by the centrifugal frce. This can be easily cnfirmed by differentiating with respect t r cs m G m r cs R m (.17) The third term is the mar tide generating term. The furth and higher terms may be neglected because r/r m 1/60. The effective tide generating ptential can therefre be written as r G m 3cs 1 m (.18) 3 R m It is cnvenient t express the lunar angle in suitable astrnmical variables as shwn in Figure.5, which are: the declinatin f the mn nrth f the equatr, d m the nrth latitude f P, p the hur angle f the mn C m, which is the difference in lngitude between the meridian f P and the meridian f the sub-lunar pint V. Figure.5 Lcatin f P using astrnmical variables These angles can be related thrugh the spherical frmula DHI - MIKE 1 Tidal Analysis and Predictin Mdule 9

14 Scientific Backgrund cs sin p sin dm cs p cs dm cs Cm (.19) Substituting this relatin in (.18) and cnsidering that the sea surface is nrmal t the resultant f the earth's gravity and f the tidal generating frces, Equatin (.18) becmes m r m m c C0 p 1 p cs 3 t sin ½ C t sin C t p (.0) where ξ is the elevatin f the free surface and the time dependent cefficients are given by 3 3 sin 0 t dm m r R C (.1) ½ C 3 t 3 sin dm cs Cm (.) 1 m 4 r R 3 r C t 3 cs dm cs C Rm 4 (.3) m This is the equatin f the Equilibrium Tide, which can be defined as the elevatin f the sea surface that wuld be in equilibrium with the tidal frces if the earth were cvered with water t such a depth that the respnse t the tidal generating frces wuld be instantaneus. The three cefficients characterise the three main species f tidal cnstituents: The lng-perid species, which cnstituents are generated by the mnthly variatins in lunar declinatin d m. It has a maximum amplitude at the ples and zer amplitude at latitudes 3516', nrth and suth f the equatr. The diurnal species, includes the cnstituents with frequencies clse t ne cycle per day (cs C m ), and is mdulated at twice the frequency f the lunar declinatin, with the maximum amplitude happening when the declinatin is a maximum. Spatially it has maximum amplitude at 45 latitude and zer amplitude at the equatr and the ples, the variatins nrth and suth f equatr being in ppsite phase. The semidiurnal species, includes the cnstituents with a frequency clse t tw cycles per day (cs C m ), and is als mdulated at twice the frequency f the lunar declinatin, but has a maximum amplitude when the declinatin is zer (equatr), and zer amplitude at the ples. The Equilibrium Tide due t the sun can be represented in an analgus frm t Equatin (.0), with m m, R m and d m replaced by m s, R s and d s. The resulting amplitudes are smaller than thse f the lunar tides by a factr f 0.46, but the essential interpretatins are the same. The Equilibrium Tide thery is a basis fr the definitin f the harmnic frequencies by which the energy f the bserved tides is distributed, and is als imprtant as a reference DHI - MIKE 1 Tidal Analysis and Predictin Mdule 10

15 Scientific Backgrund DHI - MIKE 1 Tidal Analysis and Predictin Mdule 11 fr the bserved phases and amplitudes f the tidal cnstituents. Its practical use is based n Ddsn's develpment and n the astrnmical variables described in the fllwing sectin..3 Ddsn's Develpment Ddsn (191) used the fllwing astrnmical variables in his develpment f the tidal ptential: s(t) the mean lngitude f the mn h(t) the mean lngitude f the sun p(t) the lngitude f the lunar perigee N'(t) the negative f the lngitude f the ascending nde N p'(t) the lngitude f the perihelin. These lngitudes are measured alng the ecliptic eastward frm the mean Vernal equinx. Ddsn rewrte the tidal ptential cnsidering the first fur plynmials in equatin (.13) and substituting fr the abve variables in the equivalent f equatin (.0), arriving t the fllwing general expressin ' ' sin ' ' cs 6 6, ' 6 6, 3 0 p n N m p l h k s i B G p n N m p l h k s i A G n m l k n m l k i i n m l k n m l k i i i s m T (.4) where τ is the number f mean lunar days frm the adpted time rigin, is the latitude f the bserver, G i and G' i are the gedetic cefficients f Ddsn, and the integers i,, k, l, m and n the s-called Ddsn numbers. Each term f the develpment (.4) is called a cnstituent. It has amplitude A r B and is characterised by the cmbinatin f integers i,, k, l, m and n, r equivalently by the frequency n m l k i f (.5) where ω 1, ω, ω 3, ω 4, ω 5 and ω 6 represent respectively the rates f change f τ, s, h, p, N' and p' per mean slar hur r per mean lunar day depending n the chice f the time variable. The perids and frequencies f these basic astrnmical mtins are presented in Table.. The astrnmical argument V (see Sectin.4) is defined by ' p' n m N p l h k s i V (.6)

16 Scientific Backgrund Table. Basic perids and frequencies f astrnmical mtins Perid Frequency f Angular Speed symbl in radiance rate f change f Mean Slar Day 1.00 mean slar days 1.00 cycles per mean slar day degrees per mean slar hur 0 c s Mean Lunar Day c m Sidereal Mnth s Trpical Year Julian Years h Mn's Perigee p Regressin f Mn's Ndes N' Perihelin p' Ddsn called a set f cnstituents with a cmmn i a species and a set f cnstituents within the same species with a cmmn a grup. By extensin a set f cnstituents with a cmmn i k is usually called a subgrup. It is f practical imprtance t mentin that subgrups can be separated by analysis f 1-year recrds. The values f the astrnmical variables in the cnstituent data package used in the prgram were calculated by Freman (1977) frm pwer series expansin frmulae presented in the Explanatry Supplement t the Astrnmical Ephemeris and the American Ephemeris and Nautical Almanac (1961), and in the Astrnmical Ephemeris (1974). The reference time rigin t r adpted is 000 ET 3 1 January In the prgram, the astrnmical variables at time t, which are used in the calculatin f the astrnmical argument V and f the mdel factrs u and f (see Sectin.5), are btained thrugh a first rder apprximatin; e.g. s(t) = s(t r ) + (t-t r ) ds(t r )/dt. Prvided they are cnsistent with ther time dependent calculatins, t and t r can be chsen arbitrarily. It is then pssible t write: 3 Ephemeris Time (ET), see Glssary DHI - MIKE 1 Tidal Analysis and Predictin Mdule 1

17 Scientific Backgrund DHI - MIKE 1 Tidal Analysis and Predictin Mdule 13 f t t t V t s t i t t t t p n t N m t p l t h k t s t i t p n t N m t p l t h k t s t i t V r t t r r r r r r r r... ' ' ' ' (.7) fr every frequency..4 Astrnmical Argument and Greenwich Phase Lag On the basis f Ddsn's harmnic develpment, tidal levels can then be represented by a finite number f harmnic terms f the frm N g t V a cs 0 (.8) where fr the generic cnstituent, a is the amplitude, V the astrnmical argument and g the phase lag n the Equilibrium Tide. If, fr practical purpses, a lcal time rigin t is cnsidered, we can write using the previus develpment, see equatin (.7): n g V t a cs 0 (.9) where r r t t t V t V V (.30) In rder t increase the efficiency f the cmputatins, t is chsen t be the central hur f the recrd t be analysed. The astrnmical argument V(L,t) f a tidal cnstituent can be viewed as the angular psitin f a fictitius star relative t lngitude L at time t. This fictitius star is the causal agent f the cnstituent, and is assumed t travel arund the equatr with an angular speed equal t that f its crrespnding cnstituent. Althugh the lngitudinal dependence can be easily calculated, fr histrical reasns L is generally assumed t be the Greenwich meridian, and V is reduced t a functin f ne variable.

18 Scientific Backgrund The Greenwich phase lag is then given by the sum f the astrnmical argument fr Greenwich and the phase φ f the bserved cnstituent signal, as btained frm a sinusidal regressin analysis, such that N 0 a cs t (.31) φ being a functin f the time rigin adpted. Cmparing equatins (.9) and (.31), it turns ut that g V (.3) In this analysis it is implicit that all bservatins have been recrded r cnverted t the same time reference. T avid pssible misinterpretatin f phases due t the use f cnversins between different time znes, it is a recmmended practice t cnvert all bservatins t GMT. The practical imprtance f the Greenwich phase lag is that it is a cnstant, thus independent f the time rigin. Physically it can be interpreted as the angular difference between the Equilibrium Tide and the bserved tide as schematically shwn in Figure.6. Figure.6 Relatin between the Equilibrium tide and the Observed tide If the tidal predictin is t be referred t a lcal meridian L, and given the fact that the astrnmical argument V is calculated in relatin t Greenwich, the fllwing crrectin is necessary: g L L g (.33) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 14

19 Scientific Backgrund This can be easily verified nting that the psitin f the bserved tide in relatin t L is L given by V g, V being the astrnmical argument referred t L. It then fllws that: V L L g V L g V L g (.34).5 Satellite Cnstituents and Ndal Mdulatin Ddsn's (191) develpment f the tidal ptential cntains a very large number f cnstituents. The large recrd length necessary fr their separatin makes it practically impssible t analyse all the cnstituents simultaneusly. In general terms, fr each f the main cnstituents included in a tidal analysis nly an apparent amplitude and phase are btained due t the effects f the nn-analysable cnstituents, which depend n the duratin f the perid cnsidered. Adustments are then necessary s that the true amplitudes and phases f the specified cnstituents are btained. The standard apprach t this prblem is t frm clusters cnsisting f all cnstituents with the same first three Ddsn numbers. The mar cnstituent (in terms f tidal ptential amplitude) lends its name t the cluster and the lesser cnstituents are called satellites. The adustments calculated in these cnditins are exact fr 1-year analysis, and are currently called ndal mdulatin. Hwever, it is wrth mentining that this designatin is actually nt crrect. It has been used befre the advent f mdern cmputers t designate crrectins fr the mn's ndal prgressin that were nt incrprated int the calculatins f the astrnmical argument fr the main cnstituent. Thus, althugh the term ndal mdulatin will be retained thrugh this wrk because f its generalised use, the term 'satellite mdulatin' is mre apprpriate because the adustments abve discussed are due t the presence f satellite cnstituents differing nt nly in the cntributin f the lunar nde t their astrnmical argument, but als in the lunar and slar perigee. In rder t make the "ndal mdulatin" crrectin t the amplitude and phase f a main cnstituent, it is necessary t knw the relative amplitudes and phases f the satellites. It is cmmnly assumed that the relatinships fund fr the Equilibrium Tide (tidal ptential) als hld fr the actual tide cnstituents that are clse in frequency. That is, the tidal ptential amplitude rati f a satellite t its main cnstituent is assumed t be equal t the crrespnding tidal heights' amplitude rati, and the difference in tidal ptential phase equals the difference in the bserved tidal phase. Due t the presence f satellites in a given cluster r subgrup, the signal at the frequency ω is the result f a sin A V g Aka k sinv k g k k a csv g (.35) fr the diurnal and terdiurnal cnstituents, and DHI - MIKE 1 Tidal Analysis and Predictin Mdule 15

20 Scientific Backgrund a cs A V g Aka k csv k g k k a sinv g (.36) fr the slw and semidiurnal cnstituents. Single subscript refers t the mar cntributr while k and subscripts refer t satellites riginating frm tidal ptential terms f the secnd and third rder, respectively. A is the element f the interactin matrix, which represents the interactin between a main cnstituent and its satellites. The abve expressins can be written in a simplified frm as f t a V t u t cs g (.37) where the ndal crrectin factrs fr amplitude and phase f and u can be evaluated thrugh the frmulae (Freman, 1977) f t 1 k A k k A r k k r k sin cs k k k k ½ (.38) u t k k A k k sin cs k k k k k k arctan (.39) A r r where Δ k = V k -V and α k is 0 if a k and a have the same sign r ½ therwise. Fr the analysis f N + 1 cnsecutive bservatins, Δt time units apart, the terms f the interactin matrix A k are given by A k N 1t k / N 1sin t / sin (.40) k where ω and ω k are the frequencies f the main cntributr and f its satellites. It can be shwn that A k 1. Fr practical purpses it is assumed in the present prgrams that A k = 1. In the develpment f equatin (.37) is als assumed that g = g k = g l and that r k = a k /a, and r l = a l /a are given by the rati f the tidal equilibrium amplitudes f the satellites t the amplitude f the mar cntributr. Using expressins (.38) and (.39), it is then pssible t calculate f and u fr any cnstituent at any time. The lw frequency cnstituents have nt been subect t ndal mdulatin due t the fact that lw frequency nise can be as much as ne rder f magnitude greater than the satellite cntributins fr the analysed signal. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 16

21 Scientific Backgrund.6 The Representatin f Tidal Currents The representatin f tidal currents is traditinally dne using rectangular c-rdinates and cmplex variables. In the present case the east/west and nrth/suth directins are adpted as the x and y directins. Assuming that bth cnstituents f the current are made f a peridic cnstituent and f a set f tidal cnstituents with frequencies ω, the verall signal can be represented by a cmplex variable Z(t) which can be expressed as Z M t X t X cs t i Y M t Y cs t 1 1 (.41) Freman (1978) shws that the cntributin f a generic cnstituent t Z(t), that we here call Z (t), is given by Z where t Z t Z t a i i t a exp i i t exp (.4) CX SY CY SX ½ a (.43) CX SY CY SX ½ a (.44) CY SX arctan CX SY (.45) CY SX arctan CX SY (.46) with CX SX CY X cs (.47) X sin (.48) Y cs (.49) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 17

22 Scientific Backgrund SY Y sin (.50) Examinatin f this expressin reveals that tw vectrs are assciated t each cnstituent, Z t Z has length t and Z t, rtating with an angular speed f ω radians per hur. a, rtates cunter-clckwise, and is at radians cunter-clckwise frm the psitive x axis at t=0 (Figure.7). Z t has length psitive x-axis at time t = 0. a, rtates clckwise and is at radians cunter-clckwise frm the The net rtatinal effect is that the cmpsite vectr Z (t) rtates cunter-clckwise if > a, clckwise if a < a, and mves linearly if a = a. a Figure.7 Current Ellipse Ntatin. a) Dimensins f a cnstituent ellipse; b) Cnfiguratin at t=0 Expressin (.4) can be further transfrmed int Z t exp i a a i a a sin cs t t (.51) shwing that ver a time perid f π/ω hurs, the path f the cmpsite vectr traces ut an ellipse (which degenerates in a line segment when a = a ), whse respective mar and minr semi axis lengths are a + a and inclinatin frm the psitive x-axis is / a - radians. a, and whse angle f As an aid t understand the develpment and meaning f Greenwich phases fr tidal currents, it is cnvenient t extend the cncept f fictitius stars previusly used in tidal DHI - MIKE 1 Tidal Analysis and Predictin Mdule 18

23 Scientific Backgrund elevatin thery. Similarly, it is assumed that each pair f rtating vectrs Z + and Z -, is attributable t tw fictitius stars which mve cunter-clckwise and clckwise respectively, at the same speed as the assciated cnstituent, arund the periphery f a 'celestial disk' tangential t the earth at the measurement site. It is als cnsidered that at time t, the angular psitin f the cunter-clckwise rtating star, S +, the star respnsible fr Z +, is V(t ) radians cunter-clckwise frm the psitive x axis, where V(t ) is the same astrnmical argument, relative t Greenwich, as the ne f the cnstituent in the tidal ptential. Simultaneusly, the angular psitin f the clckwise rtating star, S -, is assumed t be at V(t ) radians clckwise frm the psitive x-axis. As a cnsequence, the cnstant phase angles g + and g - by which S + and S - lead (r lag) the respective vectrs Z + and Z -, Figure.8( a), can be termed Greenwich phases and are defined by g V t (.5) g V t (.53) Figure.8 Definitin f Greenwich Phase Lags a) Fictitius stars related t E-W axis b) Fictitius stars related t mar semi-axis Prvided that the cmplex variable cnditin is satisfied- (the psitive imaginary axis has t be 90 cunter-clckwise frm the psitive real x-axis), the chice f a rectangular crdinate system fr the riginal current measurements is arbitrary. Since all angles s far have been specified with respect t the east/west axis, there is a crrespnding arbitrariness in the phases f Z and Z. Hwever, it is pssible t btain invariant phases fr these vectrs by referring angular measurements t a mar semi-axis f the cnstituent ellipse, see Figure.8 (b). Tw different fictitius stars, Z + and Z -, similar t S + and S -, can nw be visualised, their angular psitins at t=0 being nw V(t ) radians frm OA in the apprpriate directins. This apprach has the advantage that the phase f bth rtating vectrs relative t their respective stars, can nw be expressed as a single Greenwich phase angle g, given by g V t (.54) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 19

24 Scientific Backgrund r g g g (.55) In a similar way t the interpretatin previusly given fr the Greenwich phase lag, g can nw be viewed as the interval by which the instant f maximum current (when Z and Z cincide alng OA) lags the simultaneus transit f the fictitius stars, at OA. and It can als be prved that the maximum current ccurs at times g V t t n, n..., 1, 0, 1,... (.56) The ambiguity due t the divisin by in expressin (.54) is avided in the prgram by impsing the cnditin that the nrthern mar semi-axis f the cnstituent always be used as the reference axis (Freman, 1977). DHI - MIKE 1 Tidal Analysis and Predictin Mdule 0

25 Sme Criteria f Applicatin 3 Sme Criteria f Applicatin 3.1 General The analysis f a time series is based n the fact that any cntinuus and peridic functin, such as t xt T 0, 1,,... x (3.1) where T is the perid, can be represented by a trignmetric Furier series, given by x X 0 t X cs t Y sin t 1 which can be written in a mre cnvenient way as x t A A t where 1 0 cs A / is the average f the time series 0 X 0 (3.) (3.3) A X Y is the amplitude f harmnic arctan Y / X is the phase f harmnic and is the angular frequency f harmnic. T Hwever, as explained in Sectin.4, in tidal analysis it is cnvenient t refer the phase f each cnstituent t the psitin f a fictitius star travelling arund the equatr with the angular speed f the crrespnding cnstituent, the time series being then represented by the frmula 0 t a a t V g 1 x 0 cs (3.4) 0 where a, V and g are the amplitude, the astrnmical argument at the adpted time rigin t=0 and the Greenwich phase lag f the cnstituent with frequency ω, respectively. Frm the develpment f Ddsn's tidal ptential it is knwn that the tidal cnstituents tend t gather in grups with very clse frequencies, usually with a dminant cmpnent that cntains mst f the energy f the grup. Due t the great number f cnstituents in DHI - MIKE 1 Tidal Analysis and Predictin Mdule 1

26 Sme Criteria f Applicatin each grup it is nt practical fr the mdulatin f a tidal signal t include explicitly all the cnstituents with very small amplitudes. Therefre a main cnstituent is selected, and the effects f its satellites are incrprated as crrectins t the amplitude and phase f the main cnstituents. The mdulatin f a tidal signal can then be represented by 0 t a f t a t V u t 1 x 0 cs g (3.5) where the crrectin factrs f (t) and u (t) represent the effects f the satellites n the amplitude and phase f each main cnstituent. In ther wrds each grup f cnstituents is mdelled by a sinusidal functin with a time varying amplitude and phase, given by t t a A f (3.6) t t 0 g V u (3.7) 0 The values f V, f (t) and u (t) are universal fr any tidal cnstituent, and are therefre independent f the time series t be analysed. Taking these cnsideratins int accunt, the general methdlgy used fr the analysis f tidal time series can be summarised as fllws: a. Selectin f the main cnstituents t be included in the analysis. The Rayleigh criterin is usually used and will be presented in detail in next Sectin. b. Calculatin f the fictitius amplitudes and phases A and φ, thrugh the applicatin f sinusidal regressin techniques fr the slutin f the system f equatins (3.3). c. Inference f main grup cnstituents which are nt analysable ver the duratin f the available time series. The basis fr inference will be presented in Sectin d. Calculatin f the astrnmical argument f each cmpnent V at the time rigin adpted, as well as f the crrectin factrs f and u. The universal frmulae used have been presented in Sectin.5. e. Calculatin f the amplitudes and Greenwich phase lags f the main cnstituents analysed r inferred, thrugh the applicatin f relatins (3.6) and (3.7), is A a (3.8) f t c where t c is taken as the central hur f the ttal perid analysed. DHI - MIKE 1 Tidal Analysis and Predictin Mdule

27 Sme Criteria f Applicatin The accuracy btained in the mdulatin f tidal time series using the abve methdlgy is bviusly dependent n the length f the time series available. The fllwing aspects shuld be taken int cnsideratin: a. The thery f ndal r satellite mdulatin is based, in the present prgram, n the cnstituents that have the same first three Ddsn numbers. The main cnstituents f each grup are nly analysable fr time series with at least the duratin f 1 year. Fr time series with shrter duratins, nt all the main cnstituents can be analysed, which means that the calculated values A and φ btained by sinusidal regressin fr the specified main cnstituents, include nt nly the effect f the satellites in each grup but als the effects f the nn-analysed main cnstituents. Therefre additinal crrectin factrs wuld be needed t btain the crrect mdulatin f the analysed main cnstituents. Hwever, in practice it is nt pssible t calculate these additinal factrs, due t the fllwing reasns: - The tidal ptential ratis fund fr the equilibrium tide cannt be assumed t be representative f the amplitude ratis f cnstituents that belng t different grups. The assumptin is that lcal effects are nly equivalent fr cnstituents with very clse frequencies. - The spreading f energy f a nn-analysed cnstituent by the neighburing analysed main cnstituents depends n the set f cnstituents selected fr analysis. The errrs intrduced by using ndal mdulatin factrs fr the analysis f perids ther than ne year bviusly depend n the duratin f the time series. In particular, they increase with the decrease f the duratin f the time series. Depending n the distributin f the energy by frequencies, sme cmpnents will be mre accurate than thers, and it is nt pssible t establish general rules fr the evaluatin f the errrs. It shuld be kept in mind that, with the assumptins f the present tidal analysis prgrams, it is nly pssible t make accurate frecasts n the basis f the analysis f time series cvering perids f 1 year r lnger. b. The speed f prpagatin f a prgressive lng wave is apprximately prprtinal t the square rt f the depth f water in which it is travelling. Thus, shallw water has the effect f retarding the trugh f a wave mre than the crest, which distrts the riginal sinusidal wave shape and intrduces harmnic signals that are nt predicted in the tidal ptential develpment. The frequencies f these harmnics can be fund by calculating the effect f nn-linear terms in the hydrdynamic mmentum equatins (Gdin, 197). When shallw water effects are imprtant and the frequencies f main and nnlinear rigin cnstituents are such that they cannt be separated ver the available time series, the analysed values f the amplitude and phase f the main cnstituent will be affected by the presence f the shallw water cnstituent. Therefre, the calculated ndal mdulatin f the main cnstituent will nt be representative f the glbal effects f the nn-reslved cnstituents (shallw water cnstituents and satellites). A series f successive analysis can be dne t check the effectiveness f the ndal crrectins and the presence f shallw water cnstituents. c. The frmulatin f the tidal analysis prgrams des nt accunt fr nn-tidal generated cnstituents, as fr example seiches and meterlgical waves. It is cnvenient that the time series t be analysed by the present prgrams have been previusly filtered frm nn-tidal effects, when they are expected t strngly interact with the main cnstituents t be analysed. If necessary, the analysis f nn-tidal effects can be btained thrugh a Furier analysis f the crrespnding filtered time DHI - MIKE 1 Tidal Analysis and Predictin Mdule 3

28 Sme Criteria f Applicatin series, r thrugh the direct specificatin f specific nn-tidal frequencies t be used tgether with the tidal cnstituents in a sinusidal regressin analysis. This prcedure is especially interesting fr the treatment f relatively shrt perids (e.g. 15 days) that, frm an engineering pint f view, are representative f the main phenmena f interest, having als the practical advantage f prviding simultaneusly with the calculatin f the tidal cnstituents, the separatin f tidal and nn-tidal effects. One f the practical interests f the present prgrams fr engineering applicatins is the shrt term frecast fr statins lcated in a certain study zne, in rder t prvide a cmpatible set f data fr the calibratin and verificatin f mathematical mdels. In fact, due t limitatins f time and resurces, it is usually nt pssible t deply simultaneusly the necessary instruments fr measuring time series f levels and currents. Even fr thse instruments perating simultaneusly, imprtant gaps may ccur, which can als be filled thrugh the applicatin f the tidal analysis prgrams. In these applicatins the previus remarks cncerning the accuracy f the calculated amplitudes and phases shuld be kept in mind. In the ideal situatin f having simultaneus time series f the same duratin fr all the pints selected fr a study, n frecast is necessary, and the fictitius values btained directly frm the sinusidal regressin analysis are enugh fr mst f the needs f castal engineering applicatins. In particular they represent crrectly the verall effects f the interactin between all the main and satellite cnstituents fr the perid analysed, which means that this perid can be simulated with very gd accuracy. 3. Selectin f Cnstituents Given tw sinusidal waves f frequencies ω i and ω, their interactin riginates a cmpsed wave with a frequency given by the average f the frequencies f the parent sinusidal waves and with a variable amplitude. As shwn in Sectin.1 the amplitude variatin has a frequency equal t the difference between the frequencies f the riginal waves. As a cnsequence the resulting phenmenn has a perid f repetitin, the syndic perid, given by T s i (3.9) The syndic perid T s f tw sinusidal waves represents then the minimum duratin necessary fr their separatin, that is, given a time series f duratin T r, the fllwing cnditin must be fulfilled Tr T s (3.10) Taken a tidal cnstituent as the cmparisn cnstituent, Rayleigh used the abve cnditins t decide if whether r nt a specific cnstituent shuld be included in the analysis f a certain time series. If ω 0 is the frequency f the cmparisn cnstituent, a cnstituent f frequency ω 1 will be included in the analysis if 0 1 T r RAY (3.11) where RAY is user defined, and cmmnly knwn as the Rayleigh number. RAY = 1 crrespnds f using the minimum perid necessary fr the separatin f the tw cnstituents, i.e. the syndic perid. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 4

29 Sme Criteria f Applicatin In rder t determine the Rayleigh cmparisn pairs, it is imprtant t cnsider, within a given cnstituent grup (e.g. diurnal r semi-diurnal), the rder f cnstituent inclusin in the analysis as T r (the duratin f the recrd t be analysed) increases. In this cntext the fllwing criteria fr the selectin f cnstituents fr a tidal analysis have been used in the present prgram. d. within each cnstituent grup, when pssible, the cnstituent selectin is made accrding t the rder f the decreasing magnitude f the tidal ptential amplitude (as calculated by Cartwright, 1973); e. when pssible, the candidate cnstituent is cmpared with whichever f the neighburing, already selected cnstituents, that are nearest in frequency; f. when there are tw neighburing cnstituents with cmparable tidal ptential amplitudes, rather than waiting until the recrd length is sufficient t permit the selectin f bth at the same time (i.e. by cmparing them t each ther), a representative f the pair t which crrespnds the earliest inclusin, is adpted. This will give infrmatin sner abut that frequency range, and via inference, will still enable sme infrmatin t be btained n bth cnstituents. Taking int accunt the abve criteria, the Rayleigh cmparisn pairs chsen fr the lw frequency, diurnal, semi-diurnal and terdiurnal cnstituent grups are given in Table 3.1 t Table 3.4, respectively. Figures given fr the length f recrd required fr cnstituent inclusin assume a Rayleigh number 1.0. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 5

30 Sme Criteria f Applicatin Table 3.1 Order f selectin f lng-perid cnstituents in accrdance with the Rayleigh criterin DHI - MIKE 1 Tidal Analysis and Predictin Mdule 6

31 Sme Criteria f Applicatin Table 3. Order f selectin f diurnal cnstituents in accrdance with the Rayleigh criterin DHI - MIKE 1 Tidal Analysis and Predictin Mdule 7

32 Sme Criteria f Applicatin Table 3.3 Order f selectin f semi diurnal cnstituents in accrdance with the Rayleigh criterin DHI - MIKE 1 Tidal Analysis and Predictin Mdule 8

33 Sme Criteria f Applicatin Table 3.4 Order f selectin f terdiurnal cnstituents in accrdance with the Rayleigh criterin As previusly mentined, shallw water cnstituents d nt have a tidal ptential amplitude and cnsequently, bective i) des nt apply t them. Hwever, a hierarchy f DHI - MIKE 1 Tidal Analysis and Predictin Mdule 9

34 Sme Criteria f Applicatin their relative imprtance has been used fr their selectin accrding t Gdin. A further criteria used by Freman is that n shallw water cnstituent shuld appear in an analysis befre all the main cnstituents frm which it is derived, have als been selected. Table 3.5 shws that this has been upheld fr all the shallw cnstituents included in the standard 69 cnstituent data package used in the present prgrams. The criteria utlined here shuld be fllwed when chsing the Rayleigh cmparisn cnstituent fr any f the additinal cnstituents included in the cnstituent data package. Table 3.5 Shallw water cnstituents included in the standard data package DHI - MIKE 1 Tidal Analysis and Predictin Mdule 30

35 Sme Criteria f Applicatin 3.3 Inference If the length f a tidal recrd is such that certain imprtant cnstituents cannt be included directly in the analysis, apprximate relatinships can be used fr the inference f their amplitudes and phases. Hwever, these frmulae are an 'a psteriri' crrectin t the values btained by sinusidal regressin and nly affect the cmpnent frm which the inference is made. Nting that the nn-inclusin f an imprtant cnstituent affects all the surrunding cnstituents, the accuracy f an inferred cnstituent will be greatly dependent n the relative magnitude and frequencies f the neighburing cnstituents. The errrs assciated with the spreading f energy f the missing cnstituents by the neighburing cnstituents due t the sinusidal regressin analysis cannt be cmpensated fr. Given a cnstituent with frequency ω t be inferred frm a cnstituent with frequency ω 1, and denting the fictitius amplitude and phase calculated fr the latter by A 1 ' and φ 1 ', the crrected and inferred values fr bth cnstituents can be fund thrugh the frmulae: A1 / S ' A1 C (3.1) arctan S / / ' 1 1 C (3.13) 1 r1 A1 f / f1 a (3.14) u V V 1 1 u 1 (3.15) where r a A / f / A f 1 1 / 1 (3.16) a1 g1 g (3.17) S r sin 1 f f r V u V u / 1 Tr sin T 1 (3.18) C 1 r cs r V u V u / 1 f f 1 Tr sin T 1 (3.19) In the prgram the amplitude rati r 1 and the Greenwich phase lag differences are user specified. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 31

36 Prgram Descriptin 4 Prgram Descriptin 4.1 IOS Methd General intrductin In the traditinal harmnic methd fr the descriptin f tide, such as the ne implemented in the IOS methd (by G Freman), the tidal variatin is described by harmnic cnstituents, except fr the nineteen yearly variatin f the tide caused by peridic changes in the lunar rbital tilt. Variatins caused by the nineteen yearly variatin in the lunar rbital tilt is described by amplitude and phase crrectins t tidal cnstituents (dented ndal crrectins). Ddsn tidal cnstituents describe sub- and super-harmnics as well as seasnal variatins. This methd is ptentially the mst detailed descriptin f the tide at a specific lcatin, and is therefre typically used fr lcatins where the tide is mnitred cntinuusly thrugh several years. The Rayleigh criterin is used fr the selectin f cnstituents frm a standard data package cmpsed by 69 cnstituents. The standard cnstituents include 45 astrnmical main cnstituents and 4 shallw water cnstituents. They are derived nly frm the largest main cnstituents, M, S, N, K, K 1 and O 1, using the lwest types f pssible interactin. Additinally, 77 shallw water cnstituents can be included if explicitly requested by the user. The shallw water cnstituents that can be included in the mdulatin f tidal time series, are derived frm the remainder main cnstituents cnsidering higher types f interactin. The amplitudes and phases are calculated via a least squares methd, which enables the treatment f recrds with gaps. Fr the calculatin f frequencies, ndal factrs and astrnmical arguments, the prgram is based n Ddsn's tidal ptential develpment and uses the reference time rigin f January 1, 1976 fr the cmputatin f astrnmical variables Time series representatin The general representatin f a tidal time series is made accrding t the harmnic develpment x N t f t a V t u t where 1 cs g (4.1) a, g are the amplitude and Greenwich phase lag, f (t), u (t) are the ndal mdulatin amplitude and phase crrectin factrs, and V (t) is the astrnmical argument, fr cnstituent. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 3

37 Prgram Descriptin The astrnmical argument V (t) is calculated as V t V t 0 t t 0 (4.) where t 0 is the reference time rigin. The first stage in the analysis f tidal recrds is dne via a least squares fit methd fr the cmputatin f the amplitudes A and phases f representative f the cmbined effects f the main cnstituents and respective satellites, accrding t x N t A cs t 1 (4.3) The time series t be analysed have t be recrded with a 1-hur interval, and gaps can be handled autmatically by the prgram. In rder t reduce the cmputatinal time, the time rigin is taken as the central hur f the recrd. Fr the purpse f frecasting, the values f the amplitudes and Greenwich phase lags f the main cnstituents, as well as the crrespnding time dependent crrectin factrs that accunt fr satellite interactin, are calculated via ndal mdulatin. The fllwing relatinships are used: a A / f t 0 (4.4) g t u t V 0 0 (4.5) where t 0 is the central hur f the tidal recrd. The prgram calculates the crrectin factrs fr all satellites that have the same first three Ddsn numbers, which implies that the mdulatin is nly fully effective fr recrds with ne year f duratin Rayleigh criterin The Rayleigh criterin is applied fr the chice f main cnstituents, the syndic perid being cnsidered as default (Rayleigh number equal t 1.0). Fr recrds shrter than 1 year, sme main cnstituents cannt be analysed, which implies the falsificatin f the amplitudes and phases f the neighburing cnstituents. It shuld be kept in mind that the prgrams cannt cmpensate fr the assciated errrs, because the ndal mdulatin suppses that all the main cnstituents have been included in the analysis. Nevertheless a crrectin can be made fr the nearest cmpnent in frequency, with simultaneus inference f the amplitude and phase f the nn-analysed cnstituent, when requested by the user. When applying equatin (4.1) fr predictin, the ndal crrectin factrs f(t) and u(t) are apprximated by a cnstant value thrughut the perid f a mnth, calculated at 00 hr f the 16 th day f the mnth. This apprximatin is currently used due t the small variatins experienced by thse factrs ver the perid f a mnth. It shuld be kept in mind that differences between analysed and predicted recrds may arise due t the fact that fr analysis the ndal crrectin factr are calculated nly fr the central hur f the whle DHI - MIKE 1 Tidal Analysis and Predictin Mdule 33

38 Prgram Descriptin recrd. In any case the calculatin f f, u and V is based n the values f the astrnmical variables s, h, p, N' and p' at the reference time January 1, 1976, and n the first term in their Taylr expansin, e.g. s t st t t t s t r r r (4.6) Step-by-step calculatin The steps perfrmed by the analysis prgrams are as fllws: 1. Data input cmprising: - cntrl and decisin parameters - time series (levels r nrth/suth and east/west current cmpnents). Calculatin f the middle hur t c f the analysis perid. 3. Calculatin at t c, f the ndal mdulatin crrectin factrs f and u and f the astrnmical arguments V, fr all the cnstituents in the cnstituent data package. 4. Determinatin via the Rayleigh criterin, f the cnstituents t be used in the least squares fits. 5. Cnstructin and slutin relative t time t c, f the least squares matrices (fr levels r fr each f the x (east/west) and y (nrth/suth) current cmpnents). Details f the fit, such as errr estimates, average, standard deviatin, matrix cnditin number and rt mean square residual errr values are given, tgether with the parameters C and S frm which the amplitudes and phases f the cnstituents can be cmputed, accrding t the relatinships given in equatins (4.7) and (4.8). 6. Inference fr the requested cnstituents nt included in the least squares fit, and adustment f the cnstituents used fr inference. 7. Ndal mdulatins f the analysed and inferred cnstituents, using the previusly cmputed ndal crrectin factrs f and u. Greenwich phase lags and cnstituent amplitudes are then btained (fr currents the amplitudes f the mar and minr semi-axis lengths). 8. Data utput. A C S (4.7) arctan S / C (4.8) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 34

39 Analyses f Tidal Height 5 Analyses f Tidal Height 5.1 Analysis f Tidal Height using the Admiralty Methd Figure 5.1 Specify a Name in the Tidal Analysis f Height frm the Tidal Tls in MIKE 1 Tlbx Figure 5. Specify Data Frmat f Time series t be analysed DHI - MIKE 1 Tidal Analysis and Predictin Mdule 35

40 Analyses f Tidal Height Figure 5.3 Select the time series t be analysed Figure 5.4 Specify the perid t be analysed DHI - MIKE 1 Tidal Analysis and Predictin Mdule 36

41 Analyses f Tidal Height Figure 5.5 Select the Analysis Methd. Fr shrt-range time series ptinal specify the Phase difference and Amplitude factr between M-S and K1-O1 Figure 5.6 If enabled, specify the name f Cnstituent utput file and Residual utput file (see utput examples) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 37

42 Analyses f Tidal Height Figure 5.7 Specify lcatin f lg file and execute the prgram Figure 5.8 Example f Residual utput file. The file includes the measured values, the calculated tide and the residual (the difference between the calculated and the measured tide) DHI - MIKE 1 Tidal Analysis and Predictin Mdule 38

43 Analyses f Tidal Height Figure 5.9 As an ptinal utput, yu can get the actual calculated cnstituent - an example is given abve 5. Analysis f Tidal Height using the IOS Methd Figure 5.10 Specify a Name in the Tidal Analysis f Height frm the Tidal Tls in MIKE 1 Tlbx DHI - MIKE 1 Tidal Analysis and Predictin Mdule 39

44 Analyses f Tidal Height Figure 5.11 Specify Data Frmat f Time series t be analysed Figure 5.1 Select the time series t be analysed DHI - MIKE 1 Tidal Analysis and Predictin Mdule 40

45 Analyses f Tidal Height Figure 5.13 Specify the perid t be analysed Figure 5.14 Select the Analysis Methd. Fr the IOS methd specify the Latitude and Rayleigh Criteria. DHI - MIKE 1 Tidal Analysis and Predictin Mdule 41

46 Analyses f Tidal Height Figure 5.15 Specify any interfered cnstituents Figure 5.16 Specify any additinal Cnstituents, which d nt have a specified Rayleigh cmparisn cnstituent DHI - MIKE 1 Tidal Analysis and Predictin Mdule 4

47 Analyses f Tidal Height Figure 5.17 Specify lcatin f lg file and execute the prgram. Figure 5.18 Example f Residual utput file. The file include the measured values the calculated tide and the residual, the difference between the calculated and the measured tide DHI - MIKE 1 Tidal Analysis and Predictin Mdule 43

48 Analyses f Tidal Height Figure 5.19 As an ptinal utput, yu can get the actual calculated cnstituent, see the example abve DHI - MIKE 1 Tidal Analysis and Predictin Mdule 44

49 Predictins f Tidal Height 6 Predictins f Tidal Height 6.1 Predictin f Tidal Height using the Admiralty Methd Figure 6.1 Specify a Name in the Tidal Predictin f Height frm the Tidal Tls in MIKE 1 Tlbx Figure 6. Specify the type f predictin t be using user defined cnstituents DHI - MIKE 1 Tidal Analysis and Predictin Mdule 45

50 Predictins f Tidal Height Figure 6.3 Specify the Statin Name and Lngitude and Latitude f the statin and the wanted predictin perid Figure 6.4 Specify the Amplitude and Phase fr each individual Cnstituent r select a premade cnstituents file DHI - MIKE 1 Tidal Analysis and Predictin Mdule 46

51 Predictins f Tidal Height Figure 6.5 Specify the utput file name Figure 6.6 Specify lcatin f lg file and execute the prgram DHI - MIKE 1 Tidal Analysis and Predictin Mdule 47

52 Predictins f Tidal Height Figure 6.7 A graphical file shwing the predicted tide will signify the terminatin f the prgram Figure 6.8 Example f generated tidal data DHI - MIKE 1 Tidal Analysis and Predictin Mdule 48

53 Predictins f Tidal Height 6. Predictin f Tidal Height using the IOS Methd Figure 6.9 Specify a Name in the Tidal Predictin f Height frm the Tidal Tls in MIKE 1 Tlbx Figure 6.10 Specify the type f predictin t be using user defined cnstituents DHI - MIKE 1 Tidal Analysis and Predictin Mdule 49

54 Predictins f Tidal Height Figure 6.11 Specify the Statin Name and Lngitude and Latitude f the statin and the wanted predictin perid Figure 6.1 Select the number f cnstituents and specify the Amplitude and Phase fr each individual Cnstituent, r select a pre-made cnstituents file DHI - MIKE 1 Tidal Analysis and Predictin Mdule 50

55 Predictins f Tidal Height Figure 6.13 Specify the utput file name Figure 6.14 Specify lcatin f lg file and execute the prgram DHI - MIKE 1 Tidal Analysis and Predictin Mdule 51

56 Predictins f Tidal Height Figure 6.15 A graphical file shwing the predicted tide will signify the terminatin f the prgram. Please nte, the seasnal variatin, which is due t the fact that mre than ne year's data has been analysed Figure 6.16 Example f generated tide data DHI - MIKE 1 Tidal Analysis and Predictin Mdule 5