AN EVALUATION OF LEAST SQUARE MODELS FOR TIDAL HARMONIC ANALYSIS AND PREDICTION IN APAPA DOCKYARD, LAGOS, NIGERIA

ABSTRACT

The harmonic method  of tidal  analysis  has  been  considered  as  more  accurate than  the  non-harmonic methods. This is because the precise knowledge of the astronomic tidal constituents of a river channel enhances a comprehensive analysis of the tidal behavior in relation to the astronomic constituents which are the predominant factors responsible for tidal patterns and variation. Over the years, the Ordinary Least Squares technique (OLS) has been used for tidal harmonic analysis. However, due to issues of performance speed, computational effectiveness and operational limitations base on confidence interval, the use of the Iteratively Re-weighted Least Squares (IRLS) is becoming increasingly prominent. This study presents a comparative analysis of both methods of tidal harmonic analysis using 12months sea level observational data taking at 10minutes interval at Dockyard Tide Gauge (Lagos state). A computational experiment based on some sort of hierarchical data input scenario has been conducted. The 12months data has been spited into four different data types being; (i) short term data without gaps (5months), (ii) short term data with gaps (4% missing data), (iii) multi-months data without gaps (12months) and (iv) multi-months data with gaps (5% missing data). The harmonic constants for each tidal constituent in all four observational scenarios were computed using both methods and the corresponding confidence interval and SNR determined for each method using the UT-tidal analysis hydrographic tool in MATLAB as designed by Codiga (2012). Tidal values at specific times based on the determined constituent values were thereafter predicted. It was discovered that there is no significant difference in tidal prediction between the results obtained from the use of both the OLS and the IRLS method for analyzing data not exceeding five months at 95% confidence interval with the OLS method performing better than the IRLS for short term data period. However, in the multi- months data period, the IRLS method performed better that the OLS method in both scenarios (complete data and data set with omissions). The study concludes that while long term data might be better for river tidal characteristics determination with values of 1.457m and 1.6495m for the Mean Low Water Springs (MLWS) and MHWS respectively, short term data is best for tidal prediction.

CHAPTER ONE

1.0         INTRODUCTION

1.1        Background to Study

Tides are the periodic movements of the sea waters i.e. periodic rises and falls in water level due to changes in the attractive forces of the Moon and Sun upon the rotating Earth (Badejo,

2017). The rise and fall of tide is accompanied by horizontal movement of the water called tidal current. It is necessary to distinguish clearly between tide and tidal current, although the relation between them is complex and variable. While tide is the vertical rise and fall of the water, tidal current is the horizontal flow. Tides are caused by the balancing effects of gravitational pull of the sun and moon on the rotation of the Earth. Generally, tides are classified according to the tidal pattern they exhibit. Locations in the ocean where there are two high then two low tides daily are classified as having semidiurnal tides while locations that have only one high tide followed by one low tide daily have diurnal tide (Reddy, 2001). Although few cases of mixed tidal pattern exist, most water bodies have semidiurnal tides. The times and amplitude of the tides at the coast are influenced by the alignment of the Sun and Moon, by the pattern of tides in the deep ocean and by the shape of the coastline and near-shore bathymetry (Mellor, 1996).

The principal tidal forces are generated by the Moon and Sun (Consoli, 2013). The Moon is the main tide-generating body. Due to its greater distance, the Sun‟s effect is only 46 percent of the Moon‟s. Conventionally, observed tides differ considerably from the tides predicted  by  equilibrium  theory  since  size,  depth,  and  configuration  of  the  basin  or waterway, friction, landmasses, inertia of water masses, Coriolis acceleration, and other factors are neglected in this theory. Nevertheless, equilibrium theory is sufficient to describe

the magnitude and distribution of the main tide-generating forces across the surface of the

Earth.

The gravitational effect of the Moon on the surface of the Earth is the same when it is directly overhead as when it is directly underfoot. Because the direction of the Earth’s rotation is in consonance with the Moon’s orbit around the Earth, the period of the Moon is slightly above day (about 24 hours and 50minutes). In the course of this daily movement, the Moon passed overhead and underfoot once, hence the period of strongest tidal forcing is

12hours and 25 minutes. It is important to note that the position of the Moon in relation to the Earth (overhead or underfoot) does not necessarily determine the high tides although, the period of the forcing determines the time between high tides.

Asides the moon, the gravitational attraction of the Sun on the Earth also allows for some secondary tidal effects. Whenever the Earth, Moon and Sun are aligned, the tidal effects of the Sun and Moon (loosely described as the M and S tidal constituents) reinforce themselves thereby causing higher high waters and lower low waters. This near-alignment occurs at the full moon and new moon; with the recurring extreme tides termed as spring tides while those with the smallest range are called the neap tides. The neap tides occur around the first and last quarter moons.

Tides or rather water level information are useful for several purposes. Tidal information is very useful for navigation, harbour and near shore engineering constructions, Long term climate studies, flood management, hydrographic survey, oil exploration and exploitation activities(Badejo, 2017).

Tidal analysis and prediction is very useful for planning, management and decision making in Ports and Harbour operations. Amongst others, tidal analysis and prediction is useful for:

i.         flood forecasting and monitoring

ii.        Storm surge warning

iii.       Study of Tsunami effects

iv.        Prediction of upwelling and fishing operations

v.         Study of horizontal and vertical crustal movements with the aid of GPS and Gravimeter for determining the secular sea level changes.

vi.        Study of Precise Inter-island drift.

vii.       Study of the effect of rise in sea level caused by global green House effect and other phenomenon.

viii.     The hourly tidal data are used on a global basis for applications such as altimeter calibration and assimilation of sea level data numerical models.

ix.        Local behaviour of sea (Chart datum, Mean Sea Level, Lowest low & highest High tide etc.) are required for major construction activities along the coast, e.g., Harbour development (Jian, 2003).

x.         Efficient utilization of energy generation potential of hydro-dams (Jian, 2003)

All over the world, studies of tidal variation, analysis and prediction is an important aspect of hydrographic surveys given its established importance to navigational security as well as national economy. In Nigeria, issues of safety on our territorial waters (inclusive of the inland water ways) are strictly the jurisdiction of the Nigerian Navy Force (NNF). While the NNF is charged with security issues, compliance of safety rules especially in dredging and water channelization is entrusted in the Nigerian Inland Waterways Agency (NIWA). The NNF achieves this aim through the Nigerian Navy Hydrographic Office (NNHO).

The Nigerian Navy Hydrographic Office (NNHO) is the coordinating center for all national hydrographic matters in Nigeria. Its primary responsibility is to produce charts, coordinate all the hydrographic surveys that are carried out on the Nigerian waters and prepare annual tables of tidal prediction across the Nigerian water ways. In the discharge of its duties, several organs both within and outside the NNF contribute either in whole or part. Some of these agencies include the National Emergence Management Agency (NEMA), Office of the Surveyor General of the Federation (OSGOF) etc.

Generally, two basic kinds of tidal analysis methods exist being the harmonic and nonharmonic methods. Of these two, the use of non-harmonic tidal prediction method is mostly utilized for two reasons. The first reason is to allow for optimal usage of the space in the Tide Tables. Daily predictions of the time and heights of all tides were included for all tide stations, then there would be too many pages. On the other hand, recording the differences in tidal time and height at a station takes only one line in the conventional tide table thus a total of about 60 stations per page. Therefore, by calculating tidal time and height  difference  between  stations  via  the  non-harmonic  comparison  method,  several stations could be accommodated for in the tide tables. Secondly, in thepast, most stations had only few data thereby making it nearly impossible to carry out a reliable harmonic analysis.

Nevertheless, the greatest prediction accuracy and also the greatest understanding of the tidal dynamics of any water body can only be obtained from harmonic analysis (Parker, 2007). It is against this backdrop that this study presents a harmonic analysis of part of the Lagos Lagoon using data obtained from an automatic tide guage at Dockyard Naval cantonment (near East mole), Apapa, Lagos.

1.2        Statement of Research Problem

Mathematically, the ordinary least squares (OLS) method exhibits a poor performance in the presence of outliers. A reliable alternative to the OLS is given by the robust regression technique by iteratively updating the weights. This yields the iteratively re-weighted least squares (IRLS). The IRLS has been extensively used in statistics, geodetic, geophysics and harmonic analysis literatures (Huber and Ronchetti, 2009; Codiga, 2011).

Codiga (2011) presented the use of iteratively reweighted least squares (IRLS) technique for analysis and prediction of tides. The development was motivated by the need to carry out tidal analysis on a long span sequence of irregularly spaced data. Since the datasets are irregularly spaced, determination of observational weights as in the case of the OLS (or weighted least squares (WLS) as the case may be) becomes a herculean task. He therefore proposed  that  an  iteratively  re-weighted  system  would  efficiently  continue  to  impose weights on each observation in the sequence until convergence is achieved. Although, the author claims this technique is very efficient for analysis of multi staged data covering more than one-year, there are only few documented research on the performance of the model for short time series tidal observations.

Furthermore, there are limited documented scientific investigation to validate the comparative strength of IRLS over the OLS in short term and long term tidal analysis. This paper therefore presents a comparative evaluation of the computational reliability of the IRLS and the OLS techniques of tidal analysis and prediction with a view to identifying circumstances that might warrant the need for IRLS in either short term or long span data using five months and twelve months data respectively.

1.3        Aim and Objectives

The aim of this study is to perform harmonic tidal analysis of the tides at Apapa Dockyard Tide guage station (TGS), Lagos, Nigeria using OLS and IRLS models with a view to identify the preferred model for tidal prediction given different observational scenarios.

The objectives of the study are to:

i.         Determine the harmonic constituents for Apapa Dockyard TGS using OLS and IRLS techniques.

ii. Examine the performance of the OLS and IRLS techniques under various data scenarios.

iii.      Examine the confidence interval and error estimates of the determined constituents.

iv.       Predict tides at specified times given for the different data scenarios based on the determined harmonic constituents.

1.4        Research Questions

1.   How can the OLS and IRLS models be implemented for determination of tidal harmonic constituents?

2.   Under what data scenario can the OLS and IRLS techniques perform to optimum capacity for constituent determination?

3.   What  relationship  exists  between  the  computed  constituents’  confidence  interval  for harmonic constituents and the Signal to Noise ratio (SNR)?

4.   What kind of data scenario is best for tidal prediction?

1.5        Justification of the Study

Tidal prediction and analysis of water bodies is an essential task for safe navigation and flood hazard mitigation. Despite numerous studies across the world, local efforts on tidal prediction in Nigeria is limited to the national tide tables prepared by the Nigerian Navy Hydrographic Office (NNHO). These national tide tables because they were generated by non-harmonic tidal  analysis do not provide the amplitude, phase and  phase lag of the astronomic tidal constituents that make up the analyzed tides. Although the tables have served well for navigational purposes, they are grossly deficient for advanced scientific applications in hydrography such as

(i) flood forecasting in the maritime domain.

(ii) atmospheric studies.

(iii) oceanic tidal loading determination in Nigeria‟s maritime domain.

Another drawback to non-harmonic tidal analysis is that, it ignores the modulation of the perihelion which is effectively constant over historical time of about 18.6 year time series (Pawlowicz, 2002). This modulation value is required to resolve all the listed frequencies (because there is a minimum of one distinct wavelength for each constituent which differs from other constituents).In order to handle this, it is assumed that the phase/amplitudes of response  sinusoids  (having  similar  frequencies)  are  equal  to  those  of  the  equilibrium response (Kowalick and Luick, 2019).

Furthermore, the non-harmonic analysis does not allow for an easy determination of the characteristics of the resulting phase/amplitude in a deterministic way. Generally, the fit would include elements of the confidence interval for the deterministic part (Pawlowicz, 2002).

For these reasons, the harmonic analysis and prediction approach is a preferred method of tidal analysis and prediction. Nevertheless, several harmonic analysis techniques have continued to evolve in response to growing computational demands for tidal analysis and prediction. Two major harmonic analysis techniques are the Ordinary Least Squares (OLS) and  the  Iteratively  Reweighted  Least  Squares  (IRLS).  This  study  therefore  presents harmonic analysis of the tides at an Automatic Tide Guage station located at Dockyard, Apapa using both the ordinary least squares (OLS) and the Iteratively Reweighted Least Squares (IRLS) technique with a view to evaluating the computational effectiveness, performance speed and operational limitations of both models based on the confidence interval generated from each method.

Apapa Dockyard station was chosen for this study because of the availability of data. Efforts to access data from other tide stations proved abortive as such, the study utilized only the Apapa station data which was available to the researcher at the time of the study. The outcomes of this study would be very relevant and important to the Nigerian Navy, The inland waterways authority and hydrographers generally.

1.6         Study Area

The Lagos lagoon has a length of more than 50 km and width of between 3 to 13 km wide. It is separated from the Atlantic Ocean by a long sand spit of about 5 km wide with swampy margins on the lagoon side. Lagos state has a surface area of approximately 6,354.7sq km. With the exception of the Commodore channel, the lagoon is fairly shallow and is usually plied by ferries, boats, barges and daughter-vessels. The Lagos Lagoon averages 2-4 m deep, but is 10 m deep in the entrance at the Commodore channel (Okusipe, 2008). Lagos Lagoon discharges its water content into the Atlantic Ocean through the Lagos harbour. The Lagos harbour or Commodore Channel is 0.5 km to 1 km wide and 10 km long. The Lagos port is located at Apapa in a broad western branch off the main channel of the harbour. The Lagos Lagoon is tidal, water from the Atlantic Ocean moves into the lagoon during high tides and recedes during low tides. The Lagos lagoon is affected by a powerful long shoredrift. It is fed by several rivers, the most important of which are the Ogun, Ona/Ibu, Oshun, Shasha and Oni.

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