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. The scheme was devised for hourly data; for other sampling schemes, please see “Application to non-hourly data”.

```
tidem(
t,
x,
constituents,
infer = NULL,
latitude = NULL,
rc = 1,
regress = lm,
debug = getOption("oceDebug")
)
```

- t
either a vector of times or a sealevel object (as created with

`read.sealevel()`

or`as.sealevel()`

). In the first case,`x`

must be provided. In the second case, though, any`x`

that is provided will be ignored, because sealevel objects contain both`time`

and water`elevation`

, and the latter is used for`x`

.- x
an optional numerical vector holding something that varies with time. This is ignored if

`t`

is a sealevel object, because it is inferred automatically, using`t[["elevation"]]`

.- 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`

is`NULL`

, the default, then no such inferences are made. Otherwise, some constituents are computed based on other constituents, instead of being determined by regression at the proper frequency. If provided,`infer`

must be a list containing four elements:`name`

, a vector of strings naming the constituents to be inferred;`from`

, a vector of strings naming the fitted constituents used as the sources for those inferences (these source constituents are added to the regression list, if they are not already there);`amp`

, a numerical vector of factors to be applied to the source amplitudes; and`phase`

, a numerical vector of angles, in degrees, to be subtracted from the source phases. For example, Following Foreman (1998), if any of the`name`

items have already been computed, then the suggested inference is ignored, and the already-computed values are used.`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

`P1`

will be set as 0.33093 times the calculated amplitude of`K1`

, and that the`P1`

phase will be set to the`K1`

phase, minus an offset of`-7.07`

degrees. (This example is used in the Foreman (1978) discussion of a Fortran analysis code and also in Pawlowicz et al. (2002) discussion of the T_TIDE Matlab code. Rounded to the 0.1mm resolution of values reported in Foreman (1978) and Pawlowicz et al. (2002), the`tidem`

results have root-mean-square amplitude difference to Foreman's (1978) Appendix 7.3 of 0.06mm; by comparison, the results in Table 1 of Pawlowicz et al. (2002) agree with Foreman's results to RMS difference 0.04mm.)- latitude
if provided, the latitude of the observations. If not provided,

`tidem`

will try to infer this from the first argument, if it is a sealevel object.- rc
the value of the coefficient in the Rayleigh criterion.

- regress
function to be used for regression, by default

`lm()`

, but could be for example`rlm`

from the`MASS`

package.- debug
an integer specifying whether debugging information is to be printed during the processing. This is a general parameter that is used by many

`oce`

functions. Generally, setting`debug=0`

turns off the printing, while higher values suggest that more information be printed. If one function calls another, it usually reduces the value of`debug`

first, so that a user can often obtain deeper debugging by specifying higher`debug`

values.

An object of tidem, consisting of

- const
constituent number, e.g. 1 for

`Z0`

, 1 for`SA`

, etc.- model
the regression model

- name
a vector of constituent names, in non-subscript format, e.g. "

`M2`

".- frequency
a vector of constituent frequencies, in inverse hours.

- amplitude
a vector of fitted constituent amplitudes, in metres.

- phase
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.

- p
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 shallow-water 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,tidem-method()`

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
\(rc/(f_1-f_2)\).

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 2017-12-27, 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 `1e-7`

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:

The framework on which `tidem()`

rests on the assumption of data
that have been sampled on a 1-hour interval (see e.g. Foreman, 1977).
Since regression (as opposed to spectral analysis) is used to infer
the amplitude and phase of tidal constituents, data gaps do not pose
a serious problem. Sampling intervals under an hour are also not a
problem. However, trying to use `tidem()`

on time series that are
sampled at uniform intervals that exceed 1 hour can lead to results
that are difficult to interpret. For example, some drifter data are
sampled at a 6-hour interval. This makes it impossible to fit for the
S4 component (which has exactly 6 cycles per day), because the method
works by constructing sine and cosine series at tidal frequencies and
using these as the basis for regression. Each of these series will have
a constant value through the constructed time, and regression cannot handle
that (in addition to a constant-value constructed series that is used to fit
for the Z0 constituent). `tidem()`

tries to handle such problems by examining
the range of the constructed sine and cosine time-series, omitting any
constituents that yield near-constant values in either of these. Messages are
issued if this problem is encountered. This prevents failure of the regression,
and the predictions of the regression seem to represent the data reasonably well,
but the inferred constituent amplitudes are not physically reasonable. Cautious
use of `tidem()`

to infer individual constituents might be warranted, but
users must be aware that the results will be difficult to interpret. The tool
is simply not designed for this use.

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 T-TIDE package of Pawlowicz et al.
(2002), with Foreman's `UPS1`

and `M8`

becoming
`UPSI`

and `MS`

in T-TIDE. To permit the use of either notation,
`tidem()`

uses `tidemConstituentNameFix()`

to
convert from T-TIDE 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)))
```

match 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, respectively, to match the format of
that table.

Foreman, M G., 1977 (revised 1996). Manual for Tidal Heights Analysis and Prediction. Pacific Marine Science Report 77-10. British Columbia, Canada: Institute of Ocean Sciences, Patricia Bay.

Foreman, M. G. G., 1978. Manual for Tidal Currents Analysis and Prediction. Pacific Marine Science Report 78-6. 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), 95-108.

Leffler, K. E. and D. A. Jay, 2009. Enhancing tidal harmonic analysis: Robust (hybrid) solutions. Continental Shelf Research, 29(1):78-88.

Pawlowicz, Rich, Bob Beardsley, and Steve Lentz, 2002. Classical tidal
harmonic analysis including error estimates in MATLAB using `T_TIDE`

.
Computers and Geosciences, 28, 929-937.

Other things related to tides:
`[[,tidem-method`

,
`[[<-,tidem-method`

,
`as.tidem()`

,
`plot,tidem-method`

,
`predict.tidem()`

,
`summary,tidem-method`

,
`tidalCurrent`

,
`tidedata`

,
`tidem-class`

,
`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, MSK6
summary(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 < 2e-16 ***
#> 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 3.1e-07 ***
#> 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
#>
#> - 2023-09-21 15:07:16 UTC: `create 'tidem' object`
#> - 2023-09-21 15:07:16 UTC: `tidem(t = sealevelTuktoyaktuk)`
# AIC analysis
extractAIC(tide[["model"]])
#> [1] 71.0000 -606.0823
# Fake data at M2
library(oce)
data("tidedata")
M2 <- with(tidedata$const, freq[name=="M2"])
t <- seq(0, 10*86400, 3600)
eta <- sin(M2 * 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, MSK6
summary(m)
#> tidem summary
#> -------------
#>
#> Call:
#> tidem(t = sl)
#> RMS misfit to data: 4.082553e-15
#>
#> Fitted Model:
#> Freq Amplitude Phase p
#> Z0 0.00e+00 1.69e-16 0.0 0.54
#> K1 4.18e-02 5.35e-16 326.4 0.33
#> M2 8.05e-02 1.00e+00 266.4 <2e-16 ***
#> M3 1.21e-01 1.45e-16 65.8 0.82
#> M4 1.61e-01 5.37e-16 310.7 0.33
#> 2MK5 2.03e-01 1.79e-16 205.9 0.76
#> 2SK5 2.08e-01 4.61e-16 74.8 0.51
#> M6 2.42e-01 5.44e-16 189.9 0.49
#> 3MK7 2.83e-01 2.13e-16 236.6 0.70
#> M8 3.22e-01 5.99e-16 139.2 0.28
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> * Processing Log
#>
#> - 2023-09-21 15:07:16 UTC: `create 'tidem' object`
#> - 2023-09-21 15:07:16 UTC: `tidem(t = sl)`
```