The fit is done in terms of sine and cosine components at the indicated tidal frequencies (after possibly pruning to satisfy the Rayleigh criterion, as explained in phase 2 of the procedure outlined in “Details”), 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, see “Application to non-hourly data”.
tidem(
t,
x,
constituents,
infer = NULL,
latitude = NULL,
rc = 1,
regress = lm,
debug = getOption("oceDebug")
)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 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.
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.
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"]].
an optional character vector holding the names of tidal constituents to which the fit is done; see “Details” and “Constituent Naming Convention”.
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.)
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.
the value of the coefficient in the Rayleigh criterion.
function to be used for regression, by default
lm(), but could be for example rlm from the
MASS package.
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.
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.
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)))
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.
Dan Kelley
A summary of constituents used by tidem() may be found with:
data(tidedata)
print(tidedata$const)
A multi-stage procedure is used to establish the list of tidal constituents to be used in the fit.
Phase 1: initial selection.
The initial list tidal constituents (prior to the pruning phase described below) to be used in the analysis are specified as follows; see “Constituent Naming Convention”.
If constituents is not provided, then the constituent
list will be made up of the 69 constituents designated by Foreman as "standard".
These include astronomical frequencies and some shallow-water frequencies,
and are as follows: c("Z0", "SA", "SSA", "MSM", "MM", "MSF", "MF", "ALP1", "2Q1", "SIG1", "Q1", "RHO1", "O1", "TAU1", "BET1", "NO1", "CHI1", "PI1", "P1", "S1", "K1", "PSI1", "PHI1", "THE1", "J1", "SO1", "OO1", "UPS1", "OQ2", "EPS2", "2N2", "MU2", "N2", "NU2", "GAM2", "H1", "M2", "H2", "MKS2", "LDA2", "L2", "T2", "S2", "R2", "K2", "MSN2", "ETA2", "MO3", "M3", "SO3", "MK3", "SK3", "MN4", "M4", "SN4", "MS4", "MK4", "S4", "SK4", "2MK5", "2SK5", "2MN6", "M6", "2MS6", "2MK6", "2SM6", "MSK6", "3MK7", "M8").
If the first item in constituents is the string
"standard", then a provisional list is set up as in Case 1, and then
the (optional) rest of the elements of constituents are examined, in
order. Each of these constituents is based on the name of a tidal
constituent in the Foreman (1978) notation. (To get the list, execute
data(tidedata) and then execute cat(tideData$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, a user
might specify constituents=c("standard", "-M2", "ST32") to 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.
Phase 2: pruning based on Rayleigh's criterion.
Once the initial constituent list is determined during Phase 1,
tidem applies the Rayleigh criterion, which holds that
constituents of frequencies \(f_1\) and \(f_2\) cannot be
resolved unless the time series spans a time interval of at least
\(rc/(f_1-f_2)\). The details of the decision of which
constituent to prune is fairly complicated, involving decisions
based on a comparison table. The details of this process are provided
by Foreman (1978).
Phase 3: optionally overruling Rayleigh's criterion.
The pruning list from phase 2 can be overruled by the user. This
is done by retaining any constituents that the user has named
in the constituents parameter. This action
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).
Comments on phases 2 and 3
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.
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)
summary(tide)
# AIC analysis
extractAIC(tide[["model"]])
# 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)
summary(m)
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