As Haidong Pan notes, there are many tidal constituents in addition to (the usually dominant) M2 that could be examined for seasonal variability.  For lunar tides like O1 and N2, any analysis of seasonality should begin by checking for the presence of nearby spectral lines, in the manner laid out in my Note for M2.  In each case there are small astronomical constituents within the relevant tidal group, as well as compound tides and climate-driven lines at frequencies 1 or 2 cpy away from the central constituent. One simply needs to take care not to overlook an important contributor, such as a compound tide (of which there are many possible).

The solar tides, as Haidong Pan points out, are more problematic. For S2 there is semiannual modulation from K2, but also annual modulation from T2 and R2. Harmonic analysis is probably not recommended in this case, but some rough results can be obtained by a response analysis (e.g., Cartwright, 1968). With a response analysis, the gravitational parts of K2, T2, and R2 can be approximately determined from estimates of the mean S2 tide. Any residual modulations seen in S2 can then presumably be attributed to seasonal climate variability. But this is only a "rough" approach, because it overlooks the radiational forcing of S2, caused by loading by the S2 atmospheric tide, which itself has significant seasonal variability. The radiational forcing of the S2 ocean tide has been studied by Arbic (2005) and others. In the end, even with a response analysis, we may isolate a seasonal S2 signal, but it may not be clear whether it is originating in the ocean (e.g., from seasonal stratification) or in the atmosphere (from air tides).

The diurnal K1 is just as difficult, if not more so. Again a response analysis may be used to determine the gravitational parts of the neighboring constituents P1 and psi1. But the S1 constituent, 1 cpy from K1, is almost wholly radiational, with significant temporal variability. In monthly estimates of K1, modulations from S1 would be difficult to untangle from other seasonal changes.

For these reasons, analysis of lunar tides is more straightforward. Understanding their seasonality can still be difficult and even the spectral approach requires care. For example, O1 seasonality was briefly examined at station Lusi (China) in the course of a review of coastal tides (Ray et al., 2011). Small semiannual variations in O1 (of order 1%) will be induced by the linear constituent tau1, 2 cpy away, but monthly estimates of O1 at Lusi revealed significantly larger variations than that. Unfortunately, in our 2011 discussion, we overlooked the possible presence of the compound tide MP1 (which coincides with tau1). Because both M2 and P1 are large along the China coastline, the compound MP1 is almost certainly responsible for most of the semiannual oscillation seen in monthly estimates of O1 at Lusi. Nonetheless, there is still a significant annual variation in O1, which is probably induced by climatic changes in ocean stratification, in the same manner as seen for M2 in that region (Kang et al., 2002).

Finally, seasonality of the compound KO2 could indeed mimic seasonality in M2.  Presumably, in general, the effect is very small, but it is potentially at work in shallow-water regions with large diurnal tides and small M2.

References:

Arbic, B. K. (2005), Geophys. Res. Lett., 32, doi: 10.1029/2004GL021668.

Cartwright, D. E. (1968), Phil. Trans. R. Soc., 263, 1-55.

Kang, S. K. et al. (2002), J. Geophys. Res., 107, doi: 10.1029/2001JC000838.

Ray, R. D., G. D. Egbert, S. Y. Erofeeva (2011), in "Coastal Altimetry" (Eds: S. Vignudelli et al.), Springer, pp. 191-216.

I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.

One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of high-quality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars.  But there is no hard rule for the minimum amount of data needed before one can proceed.

Of course, it is implausible to study seasonal variability without data spanning most of a full year.  Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2.  One's interpretation of tidal variability would be sharply different in these two cases.

A simple Rayleigh criterion for separating constituents would also call for at least one year of data.  Yet that criterion is only a rough rule-of-thumb and depends on noise levels.  Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noise-free observations, but with real-world noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.

A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution.  The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps.  Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent. 

One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present.  Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted. 

The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.

I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.

One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of high-quality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars.  But there is no hard rule for the minimum amount of data needed before one can proceed.

Of course, it is implausible to study seasonal variability without data spanning most of a full year.  Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2.  One's interpretation of tidal variability would be sharply different in these two cases.

A simple Rayleigh criterion for separating constituents would also call for at least one year of data.  Yet that criterion is only a rough rule-of-thumb and depends on noise levels.  Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noise-free observations, but with real-world noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.

A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution.  The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps.  Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent. 

One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present.  Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted. 

The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.