The permanent tide and the International Height Reference Frame IHRF

The International Height Reference System (IHRS) adopted by the IAG in 2015 contains two novelties. First, the mean-tide system is adopted for handling the permanent tide. Second, the reference level of height system is defined by the equipotential surface where the geopotential has a conventional value W0=62636853.4 m2s-2. This value was first determined empirically to provide a good approximation to the global mean sea level and then adopted as a reference value by convention. I analyse the tidal aspects of the reference level based on W0. The W0 is by definition independent of the tidal system adopted for the equipotential surface, but for different tidal systems, different functions are involved in the W of the equation W=W0. I find that the empirical determination of the adopted estimate W0 is inconsistent from the viewpoint of tidal systems. However, the adopted estimate and the consistent estimate from the same data round off to the same value. I propose a simplified definition of IHRF geopotential numbers that would make it possible to transform between IHRF and zero-tide geopotential numbers using a simple datum-difference surface. IHRF should adopt a conventional (best) estimate of the permanent tidal potential such as is contained in the IERS Conventions and use it as a basis for other conventional formulas. The tide-free coordinates of the ITRF, and tide-free Global Geopotential Models are central in the modelling of geopotential for the purposes of the IHRF. I present a set of correction formulas that can be used to move to the zero-tide model before, in the middle, or after the processing, and finally to the mean-tide IHRF. To reduce the confusion around the multitude of tidal systems I propose that modelling should primarily done in the zero-tide system, with the mean-tide potential as an add-on.

long-term average under tidal forcing. The potential field includes the potential of this "average Earth", and the time-average of the tidegenerating potential, although the latter is not generated by the Earth's masses.
3. The zero-tide concept is a "middle alternative", for the potential field quantities. The potential field is that of the "average Earth". The gravity field is generated only by the masses of the Earth (plus the centrifugal force). For the geometric shape of the Earth, the zero-tide concept is identical with the mean-tide concept. Ekman (1989) introduced the systematic thinking about the permanent tide, terming the three different cases as "concepts" as in the above, as did . Later Ekman (1996) used solely "cases". However, starting in the 1990s, there was a gradual shift in the terminology to "systems of permanent tide", within which the present author has also participated ). In retrospect, I think that this shift was unfortunate: The word "system" brings associations to geodetic systems like "coordinate reference systems", where formal transformations between systems are valid without consideration of the physical background of the operations. But the "systems of permanent tide" are not that kind of system. I will discuss the subject further in sections 3 and 6.
For the rest of the paper until section 6, "concept" is used.
There is a lacuna in the conventional 3-point taxonomy presented above: It gives the impression that it is only the "crust" or "topography" of the Earth which could be presented either at the tide-free or at the mean-tide (=zero-tide) position. For instance, the 3-D geometric shape of the Earth is normally represented by the tidefree International Terrestrial Reference Frame (ITRF) coordinates of the topography. But, what about the coordinate representation of intangible surfaces of the potential field such as the geoid, or geoid models? A moment's reflection shows that for the potential field quantities there are two tidal concepts present: the tidal concept of the potential, and the tidal concept of the coordinate representation. The two are logically independent of each other.
There is some danger of confusion: it may be tempting to think (not in very precise terms) of the mean-tide and the tide-free coordinates as two different coordinate systems. The misleading indication "Global Geopotential Models (GGMs) are given in ITRF coordinates" may channel the users' thoughts in this direction. But, there is only one coordinate system, the system that is also used everywhere in free space, and in which GGMs are given. The instantaneous position of reference points and other objects varies periodically because of the tides. In the mean-tide concept the coordinates are given at the time-averaged position. In the (conventional) tide-free concept, the coordinates are given at a conventional off-mean position within their tidal range. However, the coordinate system is the same in both cases. This line of thought is as valid for intangible surfaces as it is for concrete objects.
Obviously, by comparing tide-free positions and mean-tide positions one can write a nonlinear coordinate transformation simulating, to some extent, their relation. But the representation of space in the "tide-free coordinate system" would bring unsurmountable problems and normal physics would fail.
In the conventional 3-point taxonomy, it was tacitly assumed that the tidally different geoids would always be represented at their mean-tide positions; see for instance Fig. 1 in Mäkinen and Ihde (2009). The question is further discussed in the example at the end of section 5.

Historical background, current tasks
The first time that the International Association of Geodesy (IAG) took a position on the permanent tide was at the XVII General Assembly of the IUGG (International Union of Geodesy and Geophysics) in Canberra in 1979. The tidefree concept was adopted in the IAG Resolution No. 15. This was a rapid response after Heikkinen (1979) had warned about the problems in the application of Stokes' formula that the use of mean-tide gravity, as implied by the Honkasalo (1964) correction, would cause. After this, several authors (e.g., Ekman 1979Ekman , 1981Groten 1980Groten , 1981 pointed out how the tide-free Earth is a problematic model for the actual Earth. At the XVIII General Assembly of the IUGG in Hamburg in 1983 the IAG then reversed its position: in its Resolution No. 9 the IAG recommended the zero-tide concept for the potential field quantities, and in its Resolution No. 16 the IAG recommended the mean-tide concept for the shape of the Earth. The tide-free quantities that are currently in use were not a response to the IAG Resolution No. 15 of 1979, in that they were mostly born unintentionally, rather than by weighing alternatives between different tidal concepts. With respect to the early tidal corrections to gravity and levelling ("luni-solar corrections"), the new quantities were almost inevitable. The correction was made using the total tidegenerating potential from (often simplified) ephemerides. It would have required a special effort to contemplate the permanent component and to care about it, never mind restore it. Later, when corrections to geodetic quantities were made using tidal spectroscopy, the method for many quantities usually was (and still is) to make, at the first step, a correction using the total tide-generating potential and then refine it for the most important waves. This is the method that has been applied in the International Earth Rotation and Reference Systems Service (IERS) Conventions, both for the geopotential and for the station positions, starting with McCarthy (1992). It was then very easy for the code-writers to overlook the fact that at the first step they also removed part of the Earth's presumed response to the permanent tide-generating potential. When Poutanen et al. (1996) pointed out that the ITRF coordinates are tide-free, the IERS Standards (McCarthy 1992) were still unambiguously prescribing mean-tide (=zero-tide) coordinates.
All three tidal concepts are currently used for referencing geodetic quantities.
ITRF coordinates are tide-free . Regional and national 3-D reference frames, such as the ETRFxx (realisations of the ETRS89) derive from the ITRF and are tide-free. Their great practical importance implies that tide-free 3-D coordinates will stay with us for a long time. GGMs are provided either tidefree or zero-tide or in both versions. Legacy national height systems from levelling are either tide-free (i.e., tide-free crust over tide-free geoid) or mean-tide (mean-tide crust over mean-tide geoid). National height systems that have been adopted since 2005 are zero-tide (mean-tide crust over zero-tide geoid), as is the regional height reference frame EVRF2007 (Sacher et al. 2009). The adoption of the mean-tide concept for IHRS is now encouraging others to follow suit: the EVRF2019 update of EVRF2007 is provided in both zero-tide and mean-tide versions (Sacher and Liebsch 2020). The International Gravity Standardization Net 1971 (IGSN71) is mean-tide (Morelli et al. 1974) but all modern gravity reference values since the 1980s are provided in zero-tide . For more history and detail, see e.g., Ekman (1989Ekman ( , 1996 and, especially for height systems, Mäkinen and Ihde (2009 'Note 1' in the resolution above refers to Ihde et al. (2015), now available also in Ihde et al. (2017), and 'note 2' above refers to Sánchez et al. (2015), expanded to a detailed paper by Sánchez et al. (2016). Observe that item 2 means that the potential W should be interpreted as the sum of the Newtonian potential of the masses of the Earth (including the potential of the permanent tidal deformation), the centrifugal potential of the Earth's rotation, and the time-average of the tidegenerating potential, although the last mentioned is not always considered part of the gravity field of the Earth as it is not generated by the masses of the Earth or by its rotation. In what follows, I discuss various aspects of the permanent tide in the International Height Reference Frame (IHRF), the realisation of the IHRS. As a height system does not exist in isolation from other geodetic quantities, the exposition will necessarily cover permanent-tide concepts in general.
Notation and units. I use the indices MT, ZT, and NT to indicate mean-tide, zero-tide, and tide-free (=non-tidal) quantities, respectively. For a quick assessment of the size of potential quantities from the perspective of say, the management of levelling networks, I occasionally use the "geopotential unit", gpu ( 2 2 1 gpu = 10 m s − ). Thus, 1 mgpu corresponds to approximately 1 mm in height.
For the same reason, formulas are presented to the precision of 0.01 mgpu, which is a usual computation precision in precise levelling.
Section 2 presents general results on the permanent tide and aims to clarify its relationship with the reference potential 0 W . After reviewing different determinations of the time-average of the tide-generating potential, I propose to use for the IHRF the function of the IERS Conventions but with a different and more transparent normalisation. Section 2.4 discusses tide-free coordinates from the ITRF, and tide-free GGMs, which play a central role in gravity field modelling. IHRF requires zero-tide potentials as a stepping-stone to the final mean-tide potentials and mean-tide coordinates: I provide specific formulas to correct for the tide-free quantities at different phases of the modelling.
Section 3 points out that the permanent tide was treated inconsistently in the empirical estimation of 0 W that was the basis for the IAG adoption of the IHRS conventional 0 W . The consistent estimate differs by 2 2 0.0943 m s − + from the estimate preferred by Sánchez et al. (2015Sánchez et al. ( , 2016. However, after the rounding off to 2 2 0.1 m s − precision, the consistent estimate agrees with the IAG conventional value. Section 4 treats the practical and theoretical difficulties that the (minor) dependence of the permanent tide-generating potential T W on height could cause for the IHRF mean-tide geopotential numbers. In section 5, I then propose a solution: use the mean-tide geoid as a reference surface for the IHRF geopotential numbers but eliminate the height dependence of T W from them by convention.
This amounts to the way the permanent tide is treated when national and regional mean-tide height systems are created using levelling networks.
2 General results on permanent tide

Basic relations
In the spectral decomposition of the tide-generating potential, only the evendegree zonal tides have non-zero time averages (Zadro and Marussi, 1973 Here, is the sum of the time-averages of the tide-generating potential for Sun, Moon, and planets • ) , ( φ r are the geocentric radius and latitude, respectively where the coefficients have been derived from the KSM03 tidal expansion (Kudryavtsev 2004(Kudryavtsev , 2007 1 . They agree with the digits shown with the HW95 tidal expansion Wenzel 1995a, 1995b). The coefficient of the second-degree term changes by 2 -2 0.00063 m s − per century, due to the change in the inclination of the ecliptic. The second-degree term is generated by the Moon, the Sun, and the planets -the contributions of the planets sum to one unit in the last decimal shown. The fourth-degree term is generated by the Moon; the contribution of the Sun and the planets are negligible. In tidal spectroscopy the second-degree term with the Sun and the Moon (sometimes also including the planets) is usually called M0S0. I will return to the numerical values later in this paper.
In what follows, I drop the fourth-degree term, although it has the size of the last digit (0.01 mgpu) which is traditionally carried over in e.g., precise levelling calculations. That is, however, done in order to decrease round-off errors (when the objective is 0.1 mgpu precision), which in the present case is not relevant.
Thus, in the sequel I identify ) , ( If V is the potential of the Newtonian attraction of the masses of the Earth including the permanent tidal deformation, and Ω W is the potential of the centrifugal acceleration of the Earth's rotation, the potential ZT W in the zero-tide case is . (4) Using the same notation, the potential MT W in the mean-tide case is .
The generation of the conventional tide-free potential can be illustrated using the deformation response of a spherical non-rotating elastic and isotropic Earth. The Newtonian potential field of the model Earth deformed by the potential of Eq. (3) is where k is a Love number. If the tidal response of the Earth is modelled by forcing the Earth with the full tide-generating potential, including the time-independent part, the response will also include the contribution T kV with the same value of k as for the time dependent tides, nominally 0.3 k = . Removing the total response (time-dependent and time-independent parts) gives us the conventional tide-free In modern geodetic practice, the time-dependent tidal response of a realistic rotating Earth is modelled using a large spectrum of spherical harmonics, frequency-dependent Love numbers, and taking into account the anelasticity of the mantle (e.g., Petit and Luzum 2010). Nevertheless, if there is a modelling step where the complete tide-generating potential (including the time average) is used, the corrected geopotential will have a tide-free second-degree zonal harmonic, just as in the simplified case above (Eqs. [6] and [7]), with the Love number k used at that particular step. In addition, depending on the computation scheme, there might be a tide-free fourth-degree zonal term. This is treated in section 2.4, along with the restoration of the zero-tide values.
The conventional tide-free coordinates are generated analogously: by forcing the Earth with a tide-generating potential that also contains the time-independent part.
The correction then removes the time-independent part with the same Love number h and Shida number ℓ with which the time-dependent tide is corrected for.
In the "secular tide-free concept" a "secular" or "fluid limit" Love number of about 0.93 s k = is found. The secular tide-free concept was never considered viable as a reference for geodetic quantities. It would create a reference very much different from physical reality. Compared with this, the traditional argument against it (Groten 1980(Groten , 1981Angermann et al. 2016) "that the s k is poorly known and in practice unknowable" is insignificant.
Henceforth, I use "tide-free" without attributes as being synonymous with "conventional tide-free". This is in line with Chapter 2 ff. of IERS Conventions 2010 (Petit and Luzum 2010). Although Fig. 1.1 and Fig. 1.2 of Section 1.1 (op cit) appear to suggest that "tide-free" without attributes should point to secular tide-free quantities, it would be quite impractical to reserve the concise expression "tide-free" to the secular concept that is never used in geodetic referencing, and to always have to use the long "conventional tide-free" for the concept that is actually used.
Taking the zero-tide potential ZT W as reference we have the difference of the tidefree potential NT W and of the mean-tide potential MT W relative to it In the spherical-harmonic expansion of a zero-tide GGM using the same scale parameter a, the second-degree zonal term is 2 20 Here GM is the geocentric gravitational constant and 20 ZT C is the zero-tide seconddegree zonal coefficient. We can merge the geopotential and the permanent tidegenerating potential in the surface spherical harmonic on a sphere. On the sphere r a = 20 ,2 20 2 2 But we cannot replace 20 claim that the result would represent "the second-degree zonal term in a mean tide GGM". If we nevertheless try to use such a construct for computations, we will get erroneous results. For instance, at the poles ( 2 (sin ) 1 , Thus, while 2-D displays and spherical formulas of the permanent-tide quantities, such as in Fig. 1 of Mäkinen and Ihde (2009)

Geoids in different concepts of the permanent tide
It is frequently stated that the potential 0 W at the geoid is "free of zero-frequency tidal distortion", "independent of the tidal concept used", i.e., that the same 0 W is appropriate for the secular tide-free, conventional tide-free, the zero-tide, and the mean-tide geoid. This statement is sometimes (e.g., Burša 1995c) treated as a theorem that requires proof, for instance using Bruns' formula or an explicit form of the permanent tide-generating potential. However, the statement is better considered as the definition of the tidally-different geoids 3 . Suppose we take for instance the zero-tide geoid with a given value 0 W as a starting surface. We then "distort" it by adding the permanent tide-generating potential T W . Which of the new equipotential surfaces with various values of the constant 1 W can be regarded as the tidally-distorted version of the original surface of Eq. (12)? Surely the answer is The same logic applies between all of the tidal geoids. The distances between the geoids can then be calculated using Bruns' formula. The height of the mean-tide geoid above the zero-tide geoid is and the height of the tide-free geoid above the zero-tide geoid is where g is gravity.
When we consider a range of geoids with different tidal definitions but all with the same 0 W , it is important to keep in mind that the potential function W , by which the equipotential surface is defined, is different in each case.

Numerical values for permanent-tide quantities
where 6378137 m a = is the semi-major axis of the GRS80 ellipsoid, and compare the coefficients A. Where can we obtain good estimates for T W , i.e., for the coefficient A? That would be from the time-independent terms (M0S0) of the time-harmonic expansion of the tide-generating potential (Table 1). Except for items Nº 8 and Nº 9, the values in the column "A" in Table 1 were derived not from the original papers (column "Reference") but from Hartmann and Wenzel (1995b) and Wenzel (1996), where they have been renormalised to the same format ("HW95") as item Nº 6 , and when necessary also updated with new astronomical constants. Where the original paper has less digits than those given in Table 1, I have put the extra digit in parentheses. The coefficient in Eq. (16) enters Table 1 through both items Nº 2 and Nº 9.
Item Nº 9 uses the renormalisation advice of Petit and Luzum (2010); the values are given under Eq. (16). The advice appears to ignore the original parameters of Cartwright-Tayler-Edden. Wenzel (1996) states that they have been taken into account (item Nº 2). This apparently leads to the minor difference between items Nº 2 and Nº 9, column "A".
Starting with item Nº 2, the differences in the coefficient A (the last column of  Wenzel (1995aWenzel ( , 1995b, Roosbeek (1996) and Kudryavtsev (2004Kudryavtsev ( , 2007. It is proposed to adopt the ( , ) T W r φ of IERS Conventions also for the IHRF Conventions. However, the Cartwright-Taylor-Edden normalisation is unwieldy and opaque for analysts outside of the tide community. Thus, for the IHRF we should adopt Here 2 ( ) P ⋅ is the second-degree Legendre polynomial and 2 ( ) P ⋅ is the seconddegree fully normalised Legendre polynomial.

Derived expressions
It is useful to derive from Eq. (18) where the overbar in ) , ( h W T ϕ is not normalisation-related, but is used to avoid possible confusion due to change of variables compared with earlier notation. Equation (19) The dependence of ) (ϕ T g on h is negligible and is not shown. Eq. (20) shows the value that should be added to zero-tide gravity in order to obtain mean-tide gravity (The IAG definition of gravity is zero-tide).
The ratio ) , where ) ( 0 ϕ γ is the GRS80 normal gravity at the ellipsoid, can for instance be used to get an idea about the difference between metric zero-tide heights and metric mean-tide heights.

Tide-free quantities
In the IHRF both coordinates and potential shall be mean-tide. Gravity field modelling, however, cannot be done with the mean-tide potential, as it contains the permanent tide-generating potential T W , generated by masses outside the Earth. The potential T W can only be added at the end. It is straightforward to do the modelling in zero-tide. However, key inputs are tide-free: the published ITRF coordinates, and many GGMs. Therefore, many analysts prefer to work with tidefree quantities and reduce to the zero-tide at the end. This reduction is often done by using the generic formulas of Ekman (1989).
The purpose of this section is to recount how the zero-tide quantities can be restored before the computation. If the computation is, nevertheless, performed with tide-free quantities, formulas are provided to restore zero-tide at the end.
They are specific to tide-free ITRF coordinates and to tide-free GGMs that are generated by applying the IERS Conventions.

ITRF coordinates
Here, r is the unit vector from the origin to the station, n is the unit vector at right angles to it in the northward direction, φ is the geocentric latitude, and 2 ( ) P ⋅ is the second-degree Legendre polynomial.
It is useful to express Eq. (22) Thus, ( ) T v ϕ is the correction from the tide-free north coordinate to the mean-tide north coordinate in a local (north, east, up) coordinate system at station height.
Eq. (23) and Eq. (24) are valid wherever Eq. (22) is. The correction to the geodetic latitude corresponding to Eq. (22) is It decreases insignificantly (in absolute value) with increasing height of the station above the ellipsoid: at 10 km the coefficient of the sin 2ϕ term is -0.813.
When the potential NT W or ZT W is to be evaluated using a geopotential model and tide-free 3-D coordinates from ITRF, firstly, the mean-tide position should be restored using Eq. (22). If the GGM is, nevertheless, evaluated at the ITRF tidefree position, the correction W ∆ to the GGM + the centrifugal potential, corresponding to Eq. (22), can be calculated from Eq. (23), multiplying it by (-g) where g is gravity. We can get a good estimate of the correction by replacing g with the normal gravity at ellipsoid 2 4 2 2 0 ( ) ( Restoring the zero tide to GGMs and consequences for the potential Petit and Luzum (2010) provide a formula (6-14 in Section 6.2) for restoring the zero tide to the tide-free fully-normalised zonal coefficient 20 NT C , obtained by processing the solid Earth tides with the IERS Conventions. I will look at the general formulas. The second-degree zonal term of a GGM is for fully normalised spherical harmonics 2 0 20 20 2 where GM and the scale parameter 0 r are specific to the model. Normally, we are free to re-scale GGMs but here 0 r should be the scale that was originally used in the processing of satellite gravity observations, i.e., to scale the effects of the solid Earth tide on the geopotential. If 20 C in Eq. (27) has been provided tide-free, it means that in the model the permanent tide-generating potential was part of the forcing by zonal tides. Thus, a part of the Earth's actual contribution to zero-tide 20 C (there is no mean-tide 20 C ) was already removed together with the timedependent part. As in Eq.
Obviously, the formulas would look simpler if we would also have the same scaler 0 r in Eq. (18). In view of the accuracy that is needed, in Eq. (29c) and Eq. (30) we can consider 0 ( ) 1 r a ≈ in any case.
The IERS Conventions starting with McCarthy (1996)  It appears that most of the recent GGMs were calculated using these values or values that are sufficiently close to them. The correction term of Eq. (29c) agrees with the formula (6-14, Section 6.2) by Petit and Luzum (2010) within the number of significant digits that are involved in their computation.
In a tide-free GGM that is produced by observing the IERS Conventions (Petit and Luzum 2010), there is also a tide-free fourth-degree zonal coefficient 40 C . It comes from the correction to fourth-degree geopotential coefficients due to second-degree tides and has nothing to do with fourth-degree tides. From Eq.
Using Eq. (29c) and putting 0 ( ) 1 r a ≈ The fourth-degree zero-tide and tide-free zonal components of the GGM are then related by 5 0 40 40 20 40 The coefficient of ( ) where the function of geocentric latitude in parenthesis (= 40 (sin ) P φ ) is maximally one in absolute value. Thus, the correction term in Eq. (34) is maximally 0.23 mgpu in absolute value. It is not clear in which tide-free GGMs the tidal correction to 40 C according to Eq. (6.7) of Petit and Luzum (2010) was in fact applied. Therefore, the correction term of Eq. (34) is not included here in corrections to tide-free GGMs. Now, suppose that the potential values of the GGM were evaluated using the tidefree version and we want to restore the zero-tide potential a-posteriori. Denote by  Combine the corrections for tide-free ITRF coordinates and for tide-free

GGM: a levelling analogy
We have seen that the corrections for tide-free ITRF coordinates and for tide-free GGM are independent, both theoretically and practically. The pseudo-obligatory binding of "tide-free crust" and "tide-free potential" to a single tide-free concept originated with precise levelling and ceased to be valid when GGMs became the method with which to evaluate potential values at a large scale.
If we work with tide-free ITRF coordinates and with a tide-free GGM but, nevertheless, skip both of the two corrections going into Eq. (40) then we have evaluated the tide-free geopotential at the tide-free coordinates. This is an analogy to tide-free geopotential numbers from levelling. They are interesting for us because many analysts who work with ITRF tide-free coordinates and tide-free geopotential verify their results by comparing them with tide-free levelling results.
If we keep the coordinate representation at mean-tide, as we normally should, the tide-free geopotential number NT C is (from Eq. [7]) 0 ( ) The last form of Eq. (41) refers specifically to Eq. (37). In the levelling analogy we define Now, a tide-free geopotential number arising from a tide-free correction to levelling is not necessarily influenced by the permanent tide-generating potential exactly in the same way as NT C ɶ is. Details will depend on the correction. It is unlikely to contain the complicated tidal response of the shape and potential of the can be compared with tide-free potential modelling.
Many of the formulas in this section update and supersede the popular formulas by Ekman (1989). The context, however, is different because Ekman (1989) presented formulas for transforming between different tidal concepts. Here, the perspective is strictly that of correcting tide-free quantities to obtain zerotide/mean-tide quantities. As to the numerical differences between the formulas, Ekman worked in a spherical approximation evaluating potentials on the surface of a sphere, and his tide-free model is a generic model. Here, there are two specific tide-free models: ITRF coordinates and tide-free GGMs generated by IERS computation schemes were addressed. Ekman (1989) was only concerned with differences and one-mm accuracy. Comparisons show that the differences of his formulas to mine are of the order of one millimetre only. Since the flattening of the Earth corresponds in proportion to 1 mgpu of the permanent tide-generating potential, one millimetre is the best that one can achieve in a spherical approximation.

Permanent-tide in the empirical estimation of W0
Thus, the GGMs with other tidal components, and their surfaces, bring nothing new, it is the same estimate repeated three times over.
What will happen if the potential models and the surfaces are paired in another way? Suppose that, for instance, we average the zero-tide potential model where I have denoted by avg ( ) S ⋅ the average taken over the region S of the ellipsoidal surface corresponding to the sea region in question. Sánchez et al. (2016) take a weighted average, with 2 1 P γ as weights (Eq. [12], op cit); my notation and the discussion below cover both a weighted and a non-weighted average. I have argued above that the potential and the surface must be matched. Thus, when the point P is on the MSL, we must include the permanent tide-generating potential T W in the averaging as I just did above in Eqs. (45) and (46) If we would be able to integrate over the whole ellipsoid, the bias (whether from a weighted or a non-weighted average) would nearly disappear. (It would not be exactly zero.) As things are, the Legendre polynomial 2 (sin( )) P φ in T W means that the bias will decisively depend on the latitude range and ocean mask used.
In their tide-free alternative, Sánchez et al. (2015Sánchez et al. ( , 2016 (Ihde et al. 2008, Eq. [5-7]) is the same formula as Eq. (23) To first order The ratio of the biases is 02 0 given that 0.30 k ≈ . The numerical values by Sánchez et al. (2015Sánchez et al. ( , 2016 produce the corresponding ratio as -0.0665/(-0.0943)=0.705, which is close enough to Eq. It seems to me that the problem with the approach by Sánchez et al. (2015Sánchez et al. ( , 2016 is that they try to use the systems of permanent tide as if they were, well, systems.
That is, they perform formal transformations between them as one would do between, say, coordinate reference systems. But tidal systems are not like that.
One must always keep in mind the physical significance of the operations. I

Mean-tide heights in a rigorous definition?
Using the notation of section 2, the mean-tide geopotential number ( )

Technique-related issues
The first question is whether the dependence of ( ) T W X on the height h in Eq. (19) will naturally show up in some of our observations or data. Then, it would be more difficult to dismiss the height dependence in our conventions and practice.
Global geopotential models (GGMs) do not include the time average of the tidegenerating potential, let alone its height dependence. Both are parts of the total tide-generating potential in modelling observations in satellite gravity, but they do not show up in the end-product, the GGM. The formula of Eq. (18) can be added as an extra member to a GGM (but it cannot be merged with its second-degree zonal term). The augmented GGM would then include the dependence on h. In the conventional proposal for IHRF geopotential numbers (section 5), if Eq. (18) is used for the IHRF stations, it would need to be evaluated at the ellipsoid, not at the observation point.
The possibility of determining potential differences using the redshift effect of the frequency of clocks (Bjerhammar 1975;Vermeer 1983) is progressing rapidly (e.g., Wu and Müller 2020). The frequency stability of the best clocks is now around 1 × 10 −18 (McGrew et al. 2018;Oelker et al. 2019). A frequency shift of 1 × 10 −18 corresponds to a potential difference of 0.09 m 2 s -2 , which is approximately 9 mm. For geodesy, high accuracy in frequency comparisons at a distance are needed (Lisdat et al. 2016). The clocks sense the total potential including the permanent tide-generating potential, but only potential differences are accessible through clock comparisons. The accuracy needed for clock pairs situated at different elevations on topography, to detect the effect of the elevation h on the permanent tide-generating potential (Eq. [19]), does not seem attainable in the near future.
In precise levelling, an observed height difference 2 , 1 h ∆ between two bench marks (1 and 2) is converted to a geopotential difference 2 , 1 C ∆ by multiplying it with the average gravity on the interval. In practice this is usually the average gravity of the two bench marks 2 , 1 2 1 2 1 2 , 1 Now, in a rigorous mean-tide height system, the gravity g in Eq. (57) shall obviously represent the gradient of the total potential field, i.e., include ) (ϕ T g , the contribution of T W to gravity (Eq. 20). Note that this theoretical issue is separate and independent from the tidal correction to . Whether or not we should do it is a different question that will be discussed in the next sections.

Orthometric heights
The orthometric height of a point P is the distance of P from the geoid, measured along the plumb line. In this definition, the tidal type of geopotential only appears through the geoid definition (tide-free, zero-tide, mean-tide) and the plumb line, not through what quantities are contained in the geopotential number that gives the potential difference between the geoid and P. The geopotential number becomes just a computational means to an end. Thus, it would seem that there are no theoretical problems involved, when we convert a mean-tide geopotential number to an orthometric height above the mean-tide geoid by dividing it with average gravity along the plumb line. If the geopotential number contains the small height-dependent part and we want to be rigorous, then we divide it by mean-tide average gravity (i.e., zero-tide gravity augmented by ) (ϕ T g from Eq. [20]). If not, we divide it by zero-tide gravity and still are rigorous. If we do not care and would rather use a mix, that would also present no problem. The situation would be comparable to an error in the gravity value at the levelling benchmark. The value often comes from interpolation and might have an uncertainty larger than the error of less than 0.1 mGal that would come from using the wrong kind of gravity in the conversion. This line of thought is equally valid for Helmert heights and rigorous orthometric heights. In section 5, however, I will propose a definition of IHRF geopotential numbers that circumvents the issues just discussed.

Normal heights
Normal height is the height above the ellipsoid that produces the same potential difference in the normal gravity field as the geopotential number gives in the actual gravity field. The key expression here is "normal gravity field". If the conventions are rigorously mean-tide, then we must have a normal gravity field that includes the permanent tide-generating potential. The concept of normal height is not just some pretext to conveniently allow us to divide the geopotential number with average normal gravity instead of the troublesome (if done rigorously) calculation that is needed for orthometric heights. Instead, the normal height is a building block in a rigorous theory to solve for the shape of the Earth.
Making mean-tide normal heights rigorous would force us to include the permanent tide-generating potential in the normal potential field, in the same way that the potential field of the centrifugal force is already included, i.e., to enlarge upon the Somigliana-Pizzetti theory. It can be done as has been demonstrated by Vermeer and Poutanen (1997). Obviously, it would mean a complete disruption of all or of a part of the current ellipsoid-based reference system. (Vermeer and Poutanen [1997] demonstrated how the enlargement can be performed without changing the geometric ellipsoid.) The level ellipsoid is then an equipotential surface of the attraction of the Earth's masses + centrifugal force + permanent tide. Observed and normal gravity include the contribution of the permanent tidegenerating potential which is eliminated from gravity anomalies, just as the centrifugal force already is. Stokes' formula would be valid. Such a model might be worthwhile in connection with a total overhaul of the ellipsoidal reference, if ever undertaken. Note that such an overhaul would not in any way interfere with the ITRS or ITRF.
In section 5, I propose a definition of IHRF geopotential numbers that excludes the height dependence of the permanent tide-generating potential.

A conventional definition for geopotential numbers in IHRF
Recall the generic relation (Eq. [56]) between mean-tide and zero-tide geopotential numbers (P) (P) (P) where P is the field point. The dependence of (P) = is the mean-tide geoid instead of the zero-tide geoid, but the potential differences on the plumb line through Q are measured the same way for both zero-tide geopotential numbers and IHRF geopotential numbers. Eq. (59) is in fact the definition of mean-tide geopotential numbers that is used in national and regional levelling networks.
Since the distance between the geoid and ellipsoid is maximally around 100 m, in practice 0 (Q) with an error of, maximally, one unit in the last decimal given or 0.01 mgpu.
Obviously, one can also evaluate Eq. (18) at the ellipsoid.
The definition of Eq. (59) means that the IHRF geopotential number will differ (very slightly) from the "natural" or "generic" mean-tide geopotential number of Eq. (58) with the same datum at the same location. The difference on the Earth's topography will always be less than 0.3 mgpu. ). This argumentation with respect to Stokes' formula that was decisive at the IAG 1979 Assembly has lost its weight as the relevant wavelengths in modern geoid computations are taken from GGMs. Sánchez and Sideris (2017) show that the residual effects from disparate national height systems that are implicit in the gravity anomalies are negligible. IHRF mean-tide metric heights embedded in gravity anomalies will be harmless in the same way. Thus, there is no reason to discourage the use of the IHRF metric heights in producing gravity anomalies. On the contrary, the IHRF heights could also provide a long-overdue unification in this respect.
Eq. (21) can be used to get an idea about the differences of IHRF metric heights and corresponding zero-tide metric heights.

Dissemination issues
By the IHRS definition, the IHRF geopotential ( ) IH RF M T C X numbers shall be related to mean-tide coordinates X . Section 2.3 provided formulas on how to deal with the tide-free ITRF coordinates in geopotential modelling. The issue will emerge again when IHRF heights are disseminated. The users only have immediate access to tide-free ITRF coordinates ' X . It might be too awkward and error-prone to have them perform their own conversion from ' X to X before accessing IHRF. Instead, the IHRF models that are distributed to them should contain the conversion, i.e., be expressed as a function of ' X .
Example. Sánchez et al. (2021, Eq. [4]) present the results of regional geopotential modelling in the form 0 0 (P) (P) (P) where the coordinates of P must be mean-tide. Here ζ is the height anomaly from the solution of the Geodetic Boundary Value Problem using the GRS80 normal gravity field as a reference, and (P) γ is normal gravity at P . Then

Summary and discussion
I have written the previous sections while consistently using "concepts" instead of "systems" of permanent tide, as I consider the term "systems" to be misleading. In this section, I will discuss the current practices where the use of "systems" is firmly embedded. Because of that I will also be using "systems".
In section 3, I demonstrated the tidal inconsistencies in the empirical estimation of 0 W , which was the basis for the IAG adoption of the IHRS conventional 0 W . My point is not "what should be the first digit behind the decimal point in the 0 W ".
My point is that the errors that were made in the estimation of 0 W were a consequence of the current "systems" approach to permanent tide.
The standard way of operating with these systems, which by now is deeply ingrained in the minds of geodesists', is like this: (1) Learn the mantra given by the three bullet points in the Introduction (replacing "concept" by "system"); (2) Locate where your quantity is in the taxonomy of the three entities of the permanent tide; (3) This provides an identifier for your quantity in the same way that coordinate reference systems have identifiers in transformation libraries; (4) Transform all of your quantities to the same tidal system using the identifiers thus established and formulas from the literature; (5) Now you are alright.
Except that you are not alright, as Chapter 3 shows. The problem comes from implicitly assuming that the three systems of permanent tide are equally valid for any purpose and that it is sufficient to consider formal operations between them.  Ekman (1989Ekman ( , 1996: "concepts" or "cases"? One can indeed argue that the confusion might be less today if we had stayed with "concepts" or "cases". The word "systems" prompted the creation of geodetic quantities in all the "system" varieties as it was taken to imply that all such quantities are needed or useful, or they at least make sense. But some of those quantities were inconsistent, such as the "mean-tide ellipsoid" of Burša (1995aBurša ( , 1995b as explained in section 2.1. Others are of questionable physical validity, for instance, many tide-free quantities related to the physical properties of the Earth (see the discussion by Ekman [1981;).
More importantly, the illusory clarity of the "systems" removed the urgency from harmonising the treatment of the permanent tide in geodesy. After the initial embarrassment at the accidental adoption of the tide-free quantities, it was possible to take the standpoint of: "Well, the situation is not ideal, but if there is a system for these things, it is OK".
Would it help to return to "concepts" now? I do not think so. Just changing names would not make a difference anymore. We would continue to operate with "concepts" in the same way as we now operate with "systems".
Does all of this matter? After all, the error in the case discussed was the equivalent of +10 mm of sea level only. However, one might point out that this was the fortuitous result of much larger biases averaging out in the integration.
The bias at the equator is +99 mm (Eq. [21]). The bias at the North Pole would be -196 mm. Further, obviously, the pitfalls in the current paradigm of treating the permanent tide are not limited to the problem of estimating 0 W , or more generally to questions concerning the MSL. Is a paradigm that facilitates (one is tempted to write "invites") such confusion in the treatment of the permanent tide sustainable in the long run? What could be done to remedy this?
The "systems" and the quantities already produced in them are not going away.
But, below, I aim to sketch one possible line of development on how to use them.
1. There are too many systems in use. The zero-tide and the mean-tide are nature's systems. The tide-free is a human system: an assemblage of mostly non-connected missteps, and of decisions with unintended consequences. In the long run, discontinue this system as a data environment.
2. In the short run, work with zero-tide quantities. As long as the normal gravity field does not contain the permanent tide (see section 4), the zerotide system is the only set of quantities that both describes the actual Earth and facilitates the solution of the geodetic boundary value problem. To emphasise this, it might be useful to re-introduce zero-tide coordinates as a synonym for mean-tide coordinates.
3. If one cannot work in the zero-tide system, it may be useful to at least think in it, as a reality checkpoint.
4. From this perspective, the tide-free quantities are just biased quantities, but they must be managed. There are tide-free coordinates and tide-free potential models. Section 2.4 contains exact formulas to correct for the two most common biases: in the ITRF coordinates and in the tide-free potential models in line with IERS Conventions.
5. The mean-tide system for the potential is an add-on after everything else has been done. Formulas for this can be found in section 2.3, and for the IHRF geopotential numbers in section 4.
6. Maintain realism. Whenever possible, talk about tide-free/zero-tide/meantide potential, tide-free/zero-tide=mean-tide coordinates, etc., rather than about potential/coordinates/etc., in the tide-free/zero-tide/meantide/system. The increased level of abstraction seldom illuminates anything and may imply a system that, possibly, is not even there. Almost always, two tidal systems are present and are independent: the coordinates and the potential. That is not scary. Specify both.
This physically realistic framework would be easier to explain to non-specialists than the present setup. Currently, most non-specialists seem to regard the tidal systems as an esoteric subject where they do not dare to venture, a subject that needs special erudition. On the other hand, any geodesist understands immediately the two fundamental features of tide-free quantities when they are explained in basic physics language: (1) ITRF coordinates of markers are not given at the timeaverage of their tidal variation in position, but off-average, and (2) many geodesists prefer to model the Earth with a part of the Earth's gravity field missing.
But, when they are told that these two facts are a part of a theoretical framework called "systems of permanent tide for geodetic quantities" and of its particular branch "conventional tide-free quantities", and somehow they must go together, many of them get confused and decide that the subject is not for them. Thus, in many cases the "systems of permanent tide" mystify things, instead of clarifying them. I believe that the protocol described in sections 1-6 would have an empowering effect on geodesists in general.
The supplementary material contains a compendium of legacy formulas related to the permanent tide.
Author Contribution. The study was in its entirety devised and written by JM.
Data availability. Only data in the published references were used.  Burša (1995aBurša ( , 1995bBurša ( , 1995c but do not seem to have been used in processing geodetic observations. Heikkinen's (1978) program of tidal corrections calculates the ephemerides of the sun and the moon using Newcomb, Brown and Eckert-Walker-Eckert, and from them directly the total tide-generating force in three dimensions. He then For the elevation of the mean-tide geoid above the zero geoid (obviously he was not using this terminology) Heikkinen Heikkinen's formulas did not however reach wide circulation until Ekman (1989) adapted them for use in his three systems of the permanent tide.
The Standard Earth Tide Committee was set up by the Permanent Commission of Earth Tides of the IAG following a recommendation of the IAG General Assembly in 1979. The committee's report (Rapp et al 1983) served as a basis for the resolution by the 1983 General Assembly. The committee recommended zero tide, the Cartwright-Tayler-Edden harmonic expansion with its time average for the tide-generating potential (M0S0), and the Wahr (1981) theory for timedependent tidal effects. The last-mentioned recommendation required an amendment to the Wahr tidal correction to observed gravity because it produced tide-free gravity. The correction to add to tide-free gravity to bring it to zero-tide where ψ is geocentric latitude.
Eq. (A-5) appeared in the International Absolute Gravity Basestation Network (IAGBN) Standards  where it apparently has baffled many gravimetrists over the years. It is not clear to which extent it was ever used by them. Wahr's definition of the gravimetric factor differed from that adopted by gravimetrists, and when the spectral approach became popular in applications requiring high accuracy, the M0S0 term could be dealt with from the beginning. (Melbourne et al 1983) are an early form of the IERS Standards/Conventions and were used extensively in dealing with the permanent tide before them. The Cartwright- Tayler Ekman 's (1989) formulas for the geoid and crust in different tidal systems come from a single quantity, his spherical approximation for T W g in the notation of this paper, W g in the notation by Ekman (1989 However, the differences between Ekman's formulas and mine do not only come from the approximation in Eq. (A-7). The spherical approximation influences different quantities in different ways. Of particular interest is the combined correction made to a geopotential number computed with the tide-free ITRF and with a tide-free GGM. It appears that Ekman's formulas are frequently used for this correction after potential modelling in the tide-free system. All Ekman's formulas are for metric heights.  in millimetres (green dotted line). Ekman's generic formula is performing very well even in this case, but its error could not be deduced from the error of his W g (the blue dashed line).