# tidem

##### Fit a Tidem (Tidal Model) to a Timeseries

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.

##### Usage

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

##### Arguments

- t
Either a

`sealevel`

object (e.g. produced by`read.sealevel`

or`as.sealevel`

) or a vector of times. In the former case, time is part of the object, so`t`

may not be given here. In the latter case,`tidem`

needs a way to determine time, so`t`

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

`t`

is a`sealevel-class`

object, in which case it is inferred as`t[["elevation"]]`

.- constituents
an optional vector of strings that name 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 (1977) [1]. 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 (1997) [1], 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 (1977) [1] discussion of a Fortran analysis code and also in Pawlowicz et al. (2002) [4] discussion of the T_TIDE Matlab code. Rounded to the 0.1mm resolution of values reported in [1] and [2], the`tidem`

results have root-mean-square amplitude difference to Foreman's 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`sl`

.- 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.

##### Details

The tidal constituents to be used in the analysis are specified as follows; see “Constituent Naming Convention”.

**Case 1**. 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")`

.**Case 2**. 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 (1977) 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.**Case 3**. 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 (1977) code [1], 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
constitutents. 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 [1], this timeseries is too short to resolve the
`SA`

constituent, so that `SA`

will not be in the resultant.
Similarly, Table 2 of [1] dictates the removal of
`PI1`

, `S1`

and `PSI1`

from the list. And, finally,
Table 3 of [1] dictates the removal of
`H1`

, `H2`

, `T2`

and `R2`

. Also, since Table 3
of [1] indicates that `GAM2`

gets subsumed into `H1`

,
and if `H1`

is already being deleted in this test case, then
`GAM2`

will also be deleted.

A list of constituent names is created by the following:

data(tidedata) print(tidedata$const$name)

**The text should include discussion of the (not yet performed) nodal
correction treatment.**

##### Value

An object of `tidem-class`

, consisting of

constituent number, e.g. 1 for `Z0`

, 1 for `SA`

,
etc.

the regression model

a vector of constituent
names, in non-subscript 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.

##### Bugs

This function is not fully developed yet, and both the form of the call and the results of the calculation may change.

Nodal correction is not done.

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.

##### Constituent Naming Convention

`tidem`

uses constituent names that follow the convention
set by Foreman (1977) [1]. This convention is slightly different
from that used in the T-TIDE package of Pawlowicz et al.
(2002) [4], with Foreman's `UPS1`

and `M8`

becoming
`UPSI`

and `MS`

in T-TIDE. As a convenience,
`tidem`

converts from these T-TIDE names to the
Foreman names, issuing warnings when doing so.

##### Agreement with `T_TIDE`

results

The `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.

##### References

1. Foreman, M. G. G., 1977. Manual for tidal heights analysis and prediction. Pacific Marine Science Report 77-10, Institute of Ocean Sciences, Patricia Bay, Sidney, BC, 58pp.

2. Foreman, M. G. G., Neufeld, E. T., 1991. Harmonic tidal analyses of long time series. International Hydrographic Review, 68 (1), 95-108.

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

4. 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.

##### See Also

Other things related to `tidem`

data: `[[,tidem-method`

,
`[[<-,tidem-method`

,
`plot,tidem-method`

,
`predict.tidem`

,
`summary,tidem-method`

,
`tidedata`

, `tidem-class`

,
`tidemAstron`

, `tidemVuf`

##### Examples

```
# NOT RUN {
library(oce)
# The demonstration time series from Foreman (1977),
# also used in T_TIDE (Pawlowicz, 2002).
data(sealevelTuktoyaktuk)
tide <- tidem(sealevelTuktoyaktuk)
summary(tide)
# AIC analysis
extractAIC(tide[["model"]])
# 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)
summary(m)
# }
```

*Documentation reproduced from package oce, version 0.9-23, License: GPL (>= 2)*