Admittance and Minor Constituent Inference

Admittance and Minor Constituent Inference#

Tidal atlases (FES2022, GOT5.6) typically provide harmonic constants for only a subset of significant constituents. Many minor constituents, while small individually, contribute meaningfully to the total tide. Rather than requiring every constituent to be explicitly mapped, PyFES infers the amplitudes and phases of minor constituents from those of nearby major constituents using the concept of admittance.

The Admittance Concept#

The ocean’s response to tidal forcing at a given location can be characterised by a complex transfer function \(Z(f)\), called the admittance, which relates the observed tidal amplitude and phase to the equilibrium (theoretical) forcing amplitude:

\[Z_k = \frac{H_k^{\text{obs}}}{H_k^{\text{eq}}} \, e^{i G_k}\]

where \(H_k^{\text{obs}}\) is the observed amplitude, \(H_k^{\text{eq}}\) is the equilibrium amplitude from the tidal potential development, and \(G_k\) is the phase lag.

The physical insight underlying inference is that the admittance varies smoothly with frequency within each tidal species. Constituents close in frequency share similar propagation and resonance characteristics, so their admittances can be interpolated from a few well-determined reference constituents.

Inference Modes#

PyFES offers four inference modes, configurable via with_inference_type(). These are generic and can be used with either prediction engine.

Zero Inference (ZERO)#

No inference is performed. Only constituents explicitly provided in the tidal atlas are used for prediction. Minor constituents not in the atlas contribute nothing to the predicted tide. This mode is useful for diagnostic purposes or when the atlas already contains a comprehensive set of constituents.

Linear Inference (LINEAR)#

The admittance is linearly interpolated between reference constituents within each species. For each species, three reference frequencies are used:

  • Diurnal: \(Q_1\), \(O_1\), \(K_1\)

  • Semidiurnal: \(N_2\), \(M_2\), \(S_2\)

For a minor constituent at frequency \(f\) between two reference frequencies \(f_a\) and \(f_b\):

\[Z(f) = Z(f_a) + \frac{f - f_a}{f_b - f_a} \left[Z(f_b) - Z(f_a)\right]\]

The inferred amplitude and phase are then:

\[H^{\text{inf}} = |Z(f)| \times H^{\text{eq}}, \qquad G^{\text{inf}} = \arg(Z(f))\]

This mode is recommended for GOT atlases (GOT4.10, GOT5.5, GOT5.6).

Spline Inference (SPLINE)#

The admittance is interpolated using a smooth spline through the reference constituents. The spline uses additional reference points compared to linear inference:

  • Diurnal: \(Q_1\), \(O_1\), \(K_1\)

  • Semidiurnal: \(2N_2\), \(N_2\), \(M_2\), \(K_2\)

The spline provides a smoother admittance curve, which typically yields more accurate inference for minor constituents far from the reference frequencies.

This mode is recommended for FES atlases (FES2014, FES2022).

Fourier Inference (FOURIER)#

The admittance is represented as a low-order Fourier series (Munk–Cartwright approach) fitted to the reference constituents:

\[Z(f) = \sum_{n=0}^{N} c_n \, e^{i n \alpha(f)}\]

where \(\alpha(f)\) is a normalised frequency coordinate and the coefficients \(c_n\) are determined by fitting to the reference admittances. This approach is particularly effective when the admittance has a systematic frequency dependence across the species.

Equilibrium Amplitudes and Love Numbers#

The inference calculation requires the equilibrium (theoretical) amplitude \(H_k^{\text{eq}}\) for each constituent. These values derive from the tidal potential development (see The Tide-Generating Potential) and are stored as constants in PyFES.

For diurnal constituents, the equilibrium amplitudes are corrected by the frequency-dependent Love number \((1 + k_l - h_l)\), where \(k_l\) and \(h_l\) are the load Love numbers at frequency \(l\). This correction accounts for the elastic deformation of the solid Earth in response to the tidal load.

Selecting an Inference Mode#

The choice of inference mode should match the tidal atlas:

# Recommended for FES atlases
settings = pyfes.FESSettings().with_inference_type(pyfes.SPLINE)

# Recommended for GOT atlases
settings = pyfes.PerthSettings().with_inference_type(pyfes.LINEAR)

# No inference (diagnostic)
settings = pyfes.FESSettings().with_inference_type(pyfes.ZERO)

# Fourier inference
settings = pyfes.FESSettings().with_inference_type(pyfes.FOURIER)

References#

  • Ray, R. D. (1999). A Global Ocean Tide Model From TOPEX/POSEIDON Altimetry: GOT99.2. NASA Technical Memorandum 209478.

  • Munk, W. H. & Cartwright, D. E. (1966). Tidal Spectroscopy and Prediction. Philosophical Transactions of the Royal Society of London A, 259(1105), 533–581.

  • Simon, B. (2013). Marées Océaniques et Côtières (943-MOC), Ch. VI.

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