5. Harmonic Analysis

5. Harmonic Analysis
5.1 Laplace principle
We found in our previous reflections that the tides are influenced by the regional basin geometry
and hence develop their local characteristic. Although the frequencies of the oscillations are
defined by astronomic periods and hence remain the same all over the earth, tidal sea level
changes are modulated by local phase shifts and amplitude modifications. Due to the constant and
known angular frequencies tidal elevations are in principle be predictable, if the local
characteristics such as amplitudes, phase shifts and relevant number of tidal constituents are
known. These can be estimated from long observation time series (tide gauge). We saw in some
examples that long periodic modulation of the tidal potential by a single frequency can be
represented as the superposition of harmonic functions with similar frequencies each with
constant amplitude:
(132)
Laplace principle (18 th Century): Tidal elevations can be expressed by superposition of harmonic
functions oscillating with the same frequencies as identified for the developed tidal potential.
(133)
2 local constants are used in the upper expression: Hi and δi. These constants are named harmonic
constants and they are estimated by harmonic analysis specifically for a respective location and
tidal record. The prerequisite for estimation of the harmonic constants is the existence of a long
period surface elevation record F(t). Such time series have traditionally been recorded
continuous, typically in an interval of one hour.
We now search for an approximation H(t) of the form:
(134)
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with defined constants Hi and δi such as H(t) best fits the observed elevation time series F(t). The
number of harmonic contributions considered (n+1) specifies the degree of approximation with
the upper limit being the number of harmonic contributions currently estimated (396 after
Doodsen, 1921). Furthermore, the practical issue of a finite length of the observational time scale
defines a limitation for the number of components to consider (and consequently for the
accuracy). This is especially the case for the long periodic components in the spectra. The best fit
of Hi and δi can be estimated by the least square fit method. The difference between fit and
observed time series is estimated for each time step:
(135)
and we search for the values Hi and δi resulting in minimum values for the sum of squared
differences
(136)
For practical purposes the time series H(t) is presented in the form:
(137)
We than can interpret the quadratic sum as a function of coefficients Ai and Bi and search for the
minimum of G(A0,Ai,Bi ) by finding the zero of the derivatives of G with:
(138)
These conditions give 2n+1 equations for 2n+1 unknowns. The respective equations have the
form:
(139)
since
(140)
follows
(141)
or in detail
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m
m
∑ F(t)cosσ i
t = ∑[A 0
+ ∑ A i
cos(σ i
t) + ∑ B i
sinσ i
t]cosσ i
t
t=1
t=1 t=1
i=1
m
m
n
n
= ∑ A 0
cosσ i
t + ∑cosσ i
t∑ A i
cosσ i
t + ∑cosσ i
t∑B i
sinσ i
t
i=1
i=1
i=1
i=1
i=1
n
m
n
(142)
€
Corresponding sets of equations are received by using the last equation of (138). m is in general a
quite large number (with hourly time step already 8760 for every year), hence solving the upper
equations is a huge task and a fast solver with high performance are needed. For an almost
complete estimation, about 390 harmonic components are needed. To reach a high degree of
accuracy about 115 semi-diurnal, 160 diurnal and 100 long periodic components are needed to be
considered. Less accurate with a still sufficient degree of accuracy can be achieved using a much
lower number of harmonic components.
5.2 Tidal prediction with 7 components
From a tidal table from a nearby observation point, we can find out the frequencies and the
harmonics constants for the 7 most prominent spectral lines. If we neglect the long-periodic
contributions, especially the half-year and yearly contributions, the 7 most important
contributions are K2, M2, N2 and S2 (semi-diurnal) and K1, O1 and P1 (diurnal).
Tidal prediction will be performed based on the same principles as used in the previous section,
i.e. we apply a formulae of the type:
(155)
Time is thereby counting from beginning of every year, phase δ i is therefore related to a
calculation of time which begins at New Year every year. The general phase δ i will be separated
into two contributions, (V0+u)i and χi. (V0+u)i is the equilibrium phase of the tidal component
(in the tidal potential) to the 0-meridian at New Year and χi is the epoch of the tide, the own
anomaly of the tidal component from the equilibrium tidal phase. It is common praxis to further
split up the epoch of a tide into a local contribution and a contribution depending on geographic
length.
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Equilibrium phase can be found in standard tables, normally published for many years in
advance. In the following table, you find harmonic constants for Bergen together with the
equilibrium phase and epoch for 1.1.1990.
Component Frequency Amplitude Epoch Vo+u
K1 15,04107 3,2 cm 170 16,7
O1 13,94304 3,0 cm 17 240,1
P1 14,98593 1,1 cm 152 349,6
K2 30,08214 4,6 cm 337 213,9
S2 30 15,8 cm 335 0
M2 28,98410 43,9 cm 298 259,4
N2 28,43973 8,4 cm 270 324,3
Using the upper table as a starting point, we are able to perform a tidal prediction for a full year.
For a more precise estimation we’ll perform the analysis for a month and start every month under
consideration of a new equilibrium tidal phase at the beginning of the respective month (found in
additional tables).
6 Appendix
6.1 Harmonic analysis of short time series
In case we would like to find a mathematical representation of tidal contributions from a much
shorter time series, e.g. for observations of tidal currents, which in contrast to the surface
elevation, are typically observed only during a short period of instruments deployment, the
previous type of analysis can not be applied. Here we need another approach to receive a
mathematic representation for the periodic contributions. Although surface elevation records are
typically available for long time periods, limitations using the previous analysis method could
arise as well in case of analysing daily variations of surface elevation. If the temporal variability
of observations is related to tidal forcing, it is near by hand to introduce moon hours (to simplify)
as time variable, which we’ll use in the following. We furthermore have to consider, that in
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addition to the tidal forcing from the tidal potential, real surface elevations are partly caused by
resonance, which can occur on shorter time scales than 12h. Hence, we’ll find periodic
oscillations with periods shorter than 12 hours. This might create a significant contribution of
short periodic disturbances, which we have to account for to enable realistic prediction. We
therefore need to include representation of higher harmonic components, which cannot be found
from celestial periods.
We assume that the observed height X(t) can be represented by the following series
approximation:
(143)
where σ is the basic diurnal frequency with 2π/24 rad/h and t is measured in moon hours. Hence
we consider diurnal and semidiurnal tidal variations and high frequency disturbances (resonance
and unresolved long periodic variability). We multiply the equation with cos (nσt) and integrate
over one (moon) day. As generally known, we’ll have only a contribution from the first term of
the series (A n –coefficient), namely:
(144)
with introduction of ω=nσt (144) reduce to
(145)
The amplitudes A n can therefore be estimated by the following equation:
(146)
Accordingly we’ll find that
In practice, the variable will be reported as an hour value (or discrete). Hence, we need to
discretize the above form. Below we’ll find the expression for the daily basic diurnal frequency
(n=1):
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(147)
(148)
Similarly we get the coefficients for the semi-diurnal components (n=2):
(149)
(150)
Components of higher harmonic frequencies can be estimated in the same way.
Removing of trends
For the upper analysis we assumed that the observed disturbance behave periodic. In case this is
not true, we’ll fail in calculating the amplitudes using the upper method. We can avoid this by
correcting the observation material from irregularities. Particularly, trends have to be removed
from the observations, i.e. A0 is not a constant, but varies with observation period A0= A0(t). We
compensate for such trends by reconstructing a data set without such a trend but conserving the
main periodic features of the time series. For the daily oscillation we’ll achieve this by:
replace X0 by (X0 +X23)/2 and
replace X1 by (X1+X22)/2 etc.
in the cosine summation, but use the half-difference for the sine summation.
Filtering
In oceanography we use a series method to remove high frequency disturbances. In case of long
time series, such disturbances are normally no problem and it is not necessary to apply particular
methods to remove these before performing the upper analysis. However, for shorter time series,
these disturbances might cause failure of the upper analysis and we need to remove them before
we perform a harmonic analysis. Here we won’t discuss the more advanced methods for filtering,
but introduce a simple filter, i.e. the moving averaged. This method is widely applied in
oceanography and particularly used for analyzing tidal records. In dependence of the
observational material which is typically available in hourly resolution, appropriate filtering of
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high frequency disturbances can be achieved by creation of an average for every hour in the
form:
(151)
The interval used for averaging depends on the frequencies of the disturbances, but normally 4
hours are sufficient. While performing filtering, we need to account for that we’ll flatten as well
the tidal signal while averaging, and hence we have to compensate for this effect. The flattening
of signals will be stronger for higher frequencies, i.e. compensation factors for semi-diurnal
periods will be stronger than for diurnal periods.
A useful 36-hours analysis method compensated for trends
In case of short time series, observed only over a couple of days, we cannot hope to extract many
daily or half daily or longer periodic components. Nevertheless, such a time series can be useful
to study the semi-diurnal and diurnal tidal characteristics and possible longer periodic
contributions, e.g. if an experiment is performed during spring and neap tide, it is possible to
represent the tidal phenomena as a composite of 2 respective components. This can be achieved
by using the following useful method, which uses series with at least 37 hourly values. The
calculation will be repeated every 12th or 24th hours in dependence of the length of the time
series. The method was developed for calculation by hand, but can be easily programmed on a
PC.
Starting point of calculation is the time series, i.e. the first 37 hour steps of the observed surface
elevation as a 6x7 matrix Mij:
X1 X7 X13 X19 X25 X31
X2 X8 X14 X20 X26 X32
X3 X9 X15 X21 X27 X33
Mij = X4 X10 X16 X22 X28 X34
X5 X11 X17 X23 X29 X35
X6 X12 X18 X24 X30 X36
X7 X13 X19 X25 X31 X37
From this matrix we can now calculate a vector S i by multiplication
Si=MijFj
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whereby Fj is a vector with the following 6 elements:
Fj =(‐1, 3, ‐4, 4, ‐3, 1)
Multiplication with Fj represents a way of averaging of the assumed harmonic characteristic
of the variable. The resulting vector S i is a consist of weighted sums, i.e. S1 is a weighted
sum with contribution of term 1, 7, 13, 19, 25 and 31. This particular weighting gives the
most weight to contributions which are in the centre of the time series. S i has 7 elements,
which cover a half period centred around the centre value of the time series X19.
Further smoothing can be reached by pair‐wise subtraction and addition of values which
are found in equidistance from centre value S4, hence:
M1=S1‐S7
N1=S1+S7
M2=S2‐S6
N2=S2+S6
M3=S3‐S5
N3=S3+S5
M4=S4‐S4
N4=2S4
M‐time series will now contain the respective sinus contribution, distributed by hour time
steps 3,2, 1 and 0 hours from central values X19. The respective N‐time series contains the
cosine contribution.
The actual harmonic analysis will be performed by defining values Y and Z by integration of
the reduced Si, to be multiplicated by cos(nσt) and sin (nσt), respectively:
(152)
(153)
The resulting heights Y and Z are both relative measures of contributions of Acos σt and Bsinσt.
The respective amplitudes A and B are received by reducing:
and (154)
This is without a compensation for high frequency noise. It is easy to understand that the
averaging from above as well removes trends from the observational material.
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6.2 Analysis of rotating tidal currents
Observing the currents in a fixed location in the sea, we’ll find tidal currents with overlaid
residual currents. By removing the residual current, we’ll realize that the tidal currents form a
periodic rotation of current vectors together with periodic variations in the current speed, both
usually not in phase. We separate such a time series by decomposition into 2 main current
directions, i.e. into an eastward component u(t) and a northward component v(t).
Assuming a tidal disturbance varying with the main frequency σ, which is typically semi-diurnal,
we can than estimate the periodic variation in the 2 (horizontal) current components by:
(156)
Or using an equivalent description:
(157)
Instead of the 4 coefficients M,N, P and Q we use 2 amplitude coefficients and 2 phase angles α
and β, with
(158)
We than can find that
(159)
and
(160)
Let us now discuss the combined tidal phenomena. First, we concentrate on the absolute current
speed V, with
(161)
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This equation tells us, that the current speed varies harmonically between 2 extremes. Maximum
values will occur when
(162)
Hence in case of:
(163)
or
(164)
Maximum value of V is A+B.
Analogous we can find the minimum values appears at
(165)
and hence
(166)
The minimum value is V=A-B.
The upper equations predict also the time when the components of tidal currents are at
maximum, i.e. along the large half axis. Currents are at minimum along the small half axis.
Estimation of position of the half axis will be found by in the following. Direction of the current
vector V(u(t),v(t)) varies in t with u(t) and v(t). Assuming the angle between the current vector
and the x-axis counting positively in anticlockwise direction, we’ll find the direction of the large
half axis by
(167)
This can be reduced to
(168)
The large half axis has therefore an orientation of:
(169)
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