The fit is done in terms of sine and cosine components at the indicated tidal frequencies, with the amplitude and phase being calculated from the resultant coefficients on the sine and cosine terms.
tidem( t, x, constituents, infer = NULL, latitude = NULL, rc = 1, regress = lm, debug = getOption("oceDebug") )
t  A 

x  an optional numerical vector holding something that varies with
time. This is ignored if 
constituents  an optional character vector holding the names of tidal constituents to which the fit is done (see “Details” and “Constituent Naming Convention”.) 
infer  a list of constituents to be inferred from
fitted constituents according to the method outlined
in Section 2.3.4 of Foreman (1978).
If infer=list(name=c("P1","K2"), from=c("K1", "S2"), amp=c(0.33093, 0.27215), phase=c(7.07, 22.4) means that the amplitude of 
latitude  if provided, the latitude of the observations. If not
provided, 
rc  the value of the coefficient in the Rayleigh criterion. 
regress  function to be used for regression, by default

debug  an integer specifying whether debugging information is
to be printed during the processing. This is a general parameter that
is used by many 
An object of tidem, consisting of
constituent number, e.g. 1 for Z0
, 1 for SA
,
etc.
the regression model
a vector of constituent
names, in nonsubscript format, e.g. "M2
".
a vector of constituent frequencies, in inverse hours.
a vector of fitted constituent amplitudes, in metres.
a vector of fitted constituent phase. NOTE: The definition of phase is likely to change as this function evolves. For now, it is phase with respect to the first data sample.
a vector containing a sort of p value for each constituent. This is calculated as the average of the p values for the sine() and cosine() portions used in fitting; whether it makes any sense is an open question.
The tidal constituents to be used in the analysis are specified as follows; see “Constituent Naming Convention”.
If constituents
is not provided, then the constituent
list will be made up of the 69 constituents designated by Foreman as "standard".
These include astronomical frequencies and some shallowwater frequencies,
and are as follows: c("Z0", "SA", "SSA", "MSM", "MM", "MSF", "MF", "ALP1", "2Q1", "SIG1", "Q1", "RHO1", "O1", "TAU1", "BET1", "NO1", "CHI1", "PI1", "P1", "S1", "K1", "PSI1", "PHI1", "THE1", "J1", "SO1", "OO1", "UPS1", "OQ2", "EPS2", "2N2", "MU2", "N2", "NU2", "GAM2", "H1", "M2", "H2", "MKS2", "LDA2", "L2", "T2", "S2", "R2", "K2", "MSN2", "ETA2", "MO3", "M3", "SO3", "MK3", "SK3", "MN4", "M4", "SN4", "MS4", "MK4", "S4", "SK4", "2MK5", "2SK5", "2MN6", "M6", "2MS6", "2MK6", "2SM6", "MSK6", "3MK7", "M8")
.
If the first item in constituents
is the string
"standard"
, then a provisional list is set up as in Case 1, and then
the (optional) rest of the elements of constituents
are examined, in
order. Each of these constituents is based on the name of a tidal
constituent in the Foreman (1978) notation. (To get the list, execute
data(tidedata)
and then execute cat(tideData$name)
.) Each
named constituent is added to the existing list, if it is not already there.
But, if the constituent is preceded by a minus sign, then it is removed
from the list (if it is already there). Thus, for example,
constituents=c("standard", "M2", "ST32")
would remove the M2
constituent and add the ST32 constituent.
If the first item is not "standard"
, then the list of
constituents is processed as in Case 2, but without starting with the
standard list. As an example, constituents=c("K1", "M2")
would fit
for just the K1 and M2 components. (It would be strange to use a minus sign
to remove items from the list, but the function allows that.)
In each of the above cases, the list is reordered in frequency prior to the
analysis, so that the results of summary,tidemmethod()
will be in a
familiar form.
Once the constituent list is determined, tidem
prunes the elements of
the list by using the Rayleigh criterion, according to which two
constituents of frequencies \(f_1\) and \(f_2\) cannot be
resolved unless the time series spans a time interval of at least
\(rc/(f_1f_2)\).
Finally, tidem
looks in the remaining constituent list to check
that the application of the Rayleigh criterion has not removed any of the
constituents specified directly in the constituents
argument. If
any are found to have been removed, then they are added back. This last
step was added on 20171227, to make tidem
behave the same
way as the Foreman (1978) code, as illustrated in his
Appendices 7.2 and 7.3. (As an aside, his Appendix 7.3 has some errors,
e.g. the frequency for the 2SK5 constituent is listed there (p58) as
0.20844743, but it is listed as 0.2084474129 in his Appendix 7.1 (p41).
For this reason, the frequency comparison is relaxed to a tol
value of 1e7
in a portion of the oce test suite
(see tests/testthat/test_tidem.R
in the source).
A specific example may be of help in understanding the removal of unresolvable
constituents. For example, the data(sealevel)
dataset is of length
6718 hours, and this is too short to resolve the full list of constituents,
with the conventional (and, really, necessary) limit of rc=1
.
From Table 1 of Foreman (1978), this timeseries is too short to resolve the
SA
constituent, so that SA
will not be in the resultant.
Similarly, Table 2 of Foreman (1978) dictates the removal of
PI1
, S1
and PSI1
from the list. And, finally,
Table 3 of Foreman (1978) dictates the removal of
H1
, H2
, T2
and R2
, and since that document
suggests that GAM2
be subsumed into H1
,
then if H1
is already being deleted, then
GAM2
will also be deleted.
A summary of constituents may be found with:
data(tidedata) print(tidedata$const)
This function is not fully developed yet, and both the form of the call and the results of the calculation may change.
The reported p
value may make no sense at all, and it might be
removed in a future version of this function. Perhaps a significance level
should be presented, as in the software developed by both Foreman and
Pawlowicz.
tidem
uses constituent names that follow the convention
set by Foreman (1978). This convention is slightly different
from that used in the TTIDE package of Pawlowicz et al.
(2002), with Foreman's UPS1
and M8
becoming
UPSI
and MS
in TTIDE. To permit the use of either notation,
tidem()
uses tidemConstituentNameFix()
to
convert from TTIDE names to the
Foreman names, issuing warnings when doing so.
T_TIDE
resultsThe tidem
amplitude and phase results, obtained with
tidem(sealevelTuktoyaktuk, constituents=c("standard", "M10"), infer=list(name=c("P1", "K2"), from=c("K1", "S2"), amp=c(0.33093, 0.27215), phase=c(7.07, 22.40))),
are identical the T_TIDE
values listed in
Table 1 of Pawlowicz et al. (2002),
after rounding amplitude and phase to 4 and 2 digits past
the decimal place, to match the format of the table.
Foreman, M. G. G., 1978. Manual for Tidal Currents Analysis and Prediction. Pacific Marine Science Report. British Columbia, Canada: Institute of Ocean Sciences, Patricia Bay.
Foreman, M. G. G., Neufeld, E. T., 1991. Harmonic tidal analyses of long time series. International Hydrographic Review, 68 (1), 95108.
Leffler, K. E. and D. A. Jay, 2009. Enhancing tidal harmonic analysis: Robust (hybrid) solutions. Continental Shelf Research, 29(1):7888.
Pawlowicz, Rich, Bob Beardsley, and Steve Lentz, 2002. Classical tidal
harmonic analysis including error estimates in MATLAB using T_TIDE
.
Computers and Geosciences, 28, 929937.
Other things related to tides:
[[,tidemmethod
,
[[<,tidemmethod
,
as.tidem()
,
plot,tidemmethod
,
predict.tidem()
,
summary,tidemmethod
,
tidedata
,
tidemclass
,
tidemAstron()
,
tidemVuf()
,
webtide()
library(oce) # The demonstration time series from Foreman (1978), # also used in T_TIDE (Pawlowicz, 2002). data(sealevelTuktoyaktuk) tide < tidem(sealevelTuktoyaktuk)#> Note: the tidal record is too short to fit for constituents: SA SSA MSM MF SIG1 RHO1 TAU1 BET1 CHI1 PI1 P1 S1 PSI1 PHI1 THE1 SO1 OQ2 2N2 NU2 GAM2 H1 H2 MKS2 LDA2 T2 R2 K2 MSN2 SO3 MK4 SK4 2MK6 MSK6summary(tide)#> tidem summary #>  #> #> Call: #> tidem(t = sealevelTuktoyaktuk) #> RMS misfit to data: 0.7808454 #> #> Fitted Model: #> Freq Amplitude Phase p #> Z0 0.00000 1.98062 0.00 < 2e16 *** #> MM 0.00151 0.21213 263.34 0.0051 ** #> MSF 0.00282 0.15606 133.80 0.0062 ** #> ALP1 0.03440 0.01523 334.96 0.7368 #> 2Q1 0.03571 0.02458 82.69 0.6516 #> Q1 0.03722 0.01579 65.74 0.7541 #> O1 0.03873 0.07641 74.23 0.1262 #> NO1 0.04027 0.02903 238.14 0.3716 #> K1 0.04178 0.13474 81.09 0.0262 * #> J1 0.04329 0.02530 7.32 0.5977 #> OO1 0.04483 0.05310 235.75 0.2729 #> UPS1 0.04634 0.02980 91.73 0.6272 #> EPS2 0.07618 0.02115 184.60 0.6769 #> MU2 0.07769 0.04189 83.23 0.3727 #> N2 0.07900 0.08377 44.52 0.0723 . #> M2 0.08051 0.49041 77.70 0.3465 #> L2 0.08202 0.02132 35.21 0.7301 #> S2 0.08333 0.22024 137.48 3.1e07 *** #> ETA2 0.08507 0.00713 246.04 0.8902 #> MO3 0.11924 0.01484 234.97 0.7426 #> M3 0.12077 0.01226 261.57 0.8020 #> MK3 0.12229 0.00492 331.60 0.9172 #> SK3 0.12511 0.00234 237.67 0.9680 #> MN4 0.15951 0.00917 256.47 0.8475 #> M4 0.16102 0.01257 291.79 0.7544 #> SN4 0.16233 0.00830 270.86 0.8659 #> MS4 0.16384 0.00103 339.36 0.9842 #> S4 0.16667 0.00468 299.56 0.9135 #> 2MK5 0.20280 0.00127 310.16 0.9793 #> 2SK5 0.20845 0.00455 104.00 0.9172 #> 2MN6 0.24002 0.00353 271.22 0.9371 #> M6 0.24153 0.00173 158.87 0.9681 #> 2MS6 0.24436 0.00564 306.12 0.8938 #> 2SM6 0.24718 0.00227 298.91 0.9555 #> 3MK7 0.28331 0.00857 212.25 0.8508 #> M8 0.32205 0.00304 42.38 0.9497 #>  #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> * Processing Log #> #>  20200721 16:53:09 UTC: `create 'tidem' object` #>  20200721 16:53:09 UTC: `tidem(t = sealevelTuktoyaktuk)`#> [1] 71.0000 606.0823# Fake data at M2 t < seq(0, 10*86400, 3600) eta < sin(0.080511401 * t * 2 * pi / 3600) sl < as.sealevel(eta) m < tidem(sl)#> Note: the tidal record is too short to fit for constituents: SA SSA MSM MM MSF MF ALP1 2Q1 SIG1 Q1 RHO1 O1 TAU1 BET1 NO1 CHI1 PI1 P1 S1 PSI1 PHI1 THE1 J1 SO1 OO1 UPS1 OQ2 EPS2 2N2 MU2 N2 NU2 GAM2 H1 H2 MKS2 LDA2 L2 T2 S2 R2 K2 MSN2 ETA2 MO3 SO3 MK3 SK3 MN4 SN4 MS4 MK4 S4 SK4 2MN6 2MS6 2MK6 2SM6 MSK6summary(m)#> tidem summary #>  #> #> Call: #> tidem(t = sl) #> RMS misfit to data: 9.24624e08 #> #> Fitted Model: #> Freq Amplitude Phase p #> Z0 0.00e+00 9.69e10 0.00 0.88 #> K1 4.18e02 3.85e09 42.50 0.77 #> M2 8.05e02 1.00e+00 266.40 <2e16 *** #> M3 1.21e01 4.55e09 113.49 0.72 #> M4 1.61e01 1.88e09 179.90 0.92 #> 2MK5 2.03e01 1.44e09 175.25 0.93 #> 2SK5 2.08e01 2.65e09 197.77 0.85 #> M6 2.42e01 1.83e09 200.43 0.89 #> 3MK7 2.83e01 1.74e09 195.70 0.90 #> M8 3.22e01 1.67e09 0.53 0.92 #>  #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> * Processing Log #> #>  20200721 16:53:09 UTC: `create 'tidem' object` #>  20200721 16:53:09 UTC: `tidem(t = sl)`