**6. A revisited linear inverse model to estimate sea level trends**

The linear inverse model proposed by Kuo et al. [48, 49], and then by Wöppelmann and Marcos [50], assumes that the absolute sea level change rates are similar at all the tide gauge (TG) sites. This assumption is particularly important for the successful inversion. The explanation will be provided in this section.

The difference between the absolute sea level rise (ASLR) and the relative sea level rise (RSLR) rates, i.e. the velocities at which the sea level vertical motion is observed by satellite altimeters and TGs, denoted respectively with *g*\_ and *s*\_, is an estimate of the vertical velocity at which the land beneath TGs is moving. Such vertical crustal velocity, as previously stated, is named vertical land motion (VLM) and indicated by *u*\_. A subscript *i* is added to denote that the quantities *g*\_, *s*\_ and *u*\_ refer to the *i*-th TG of a group of *N*:

$$
\dot{u}\_i = \dot{\mathbf{g}}\_i - \dot{\mathbf{s}}\_i \qquad i = \mathbf{1}, \ldots, N \tag{1}
$$

Eq. (1) is sufficient to obtain good estimates of the VLM rates at each TG, provided that all the variables in the equation refer to the same period and to coherent geophysical processes and have negligible inherent drifts and errors. Eq. (1) can be expressed in vector–matrix notation:

$$G\dot{u} = d; G = I\_N \tag{2}$$

where *u*\_ is the column vector of the unknown VLMs *u*\_ ¼ *u*\_ ð Þ 1, … , *u*\_ *<sup>N</sup>* , and *d* is the column vector whose elements are formed by the right-hand side of Eq. (1): *d* ¼ *g*\_ <sup>1</sup> � *s*\_1, … , *g*\_*<sup>N</sup>* � *s*\_*<sup>N</sup>* . In this picture all the unknown VLMs are mutually independent, and the linear system is easily inverted, offering the solution component by component. However, the solution is affected by large errors, as the period during which Eq. (1) is valid corresponds to the overlap period of TG and satellite altimetry observations, and thus no more back in time than 1992. In fact, the current time span of satellite altimetry data is less than 30 years. Such a short time span hinders the derivation of accurate trends from altimeter-gauge time series, as they are affected by inter-annual and decadal sea level signals, in particular by the 18.6-year lunar nodal tide, leading to uncertainties of the order of 1–2 mm yr�<sup>1</sup> [47, 54, 55]. For this reason, Kuo et al. [48] proposed a more elaborate linear system, in which constraints formed by the differenced time series of TGs over longer time periods (>40 yr) pose strong limits to the magnitude of the final errors thanks to the length

project, started in 2017, is extending the processing to the coastal zone, and an experimental coastal sea level product is going to be released to the public, in six selected regions: Northern Europe, Mediterranean Sea, Western Africa, North Indian Ocean, Southeast Asia and Australia [37]. This product is along-track and combines the enhanced spatial resolution provided by high-rate data (20-Hz), the post-processing strategy of X-TRACK and the advantage of the ALES retracker [38]. The product relies on the GPD+ wet tropospheric correction [39] and the FES2014 tidal corrections [40]. The X-TRACK/ALES SLCCI 20 Hz along-track

The trend is an indicator describing how sea level has changed over long time. It provides a simple predictive scenario if what observed in the past might be representative in the near future. The classical approach is to calculate a straight line through sea level data using a linear regression. The most used method for fitting data is least squares. However, other methods based on more complex models exist to estimate trends from sea level time series [41]. The trend estimation is sensitive to the length of the record and start/end periods. There might be variability at different interannual to decadal timescales occurring within the data. Moreover, in addition to the

A single tide gauge cannot explain to what extent the observed trend is related to ocean and/or land changes, without any nearby GPS. With the advent of satellite radar altimetry and the possibility to use altimeter passages nearby tide gauges a new method was proposed by Cazenave et al. [43]. It assumes that both the tide gauge and altimetry system measure the same ocean signal and the difference is a measure of VLM at the gauge: hereinafter we refer to this method as the "direct" or "classical" method. Another assumption is that there are no instrumental errors introducing significant drifts. This direct method provides VLM at the selected tide

Different implementations of the basic idea were successively proposed involving more tide gauges, more rigorous error analysis with mitigation of the uncertainties introduced by the assumptions and taking advantage of longer and improved altimeter-derived time series available at that time (e.g., [44–47] and others).

An advanced method to estimate VLM that includes supplementary constraints from adjacent tide gauges has been proposed by Kuo et al. [48]. Its solution is based on the inversion of a linear system, formed mixing differences of altimetry- and tide gauge-derived trends, and differences of trends from neighboring tide gauges only, introduced in the linear system through Lagrange multipliers. As the solution of such a system requires its inversion, the method is referred to as Linear Inverse Problem with Constraints (LIPWC), or shortly "inverse" method. The new method optimally combines short-term altimetry records with long-term tide gauge observations. It assumes that absolute sea level change at tide gauges over a long time span is the same. The advantage of the method is that long (>40 years) tide gauge records contribute to reduce the error in the final VLM solution, and random and systematic errors in one or more time series trend are shared among all the other, cutting down the impact on the originating one. The disadvantage is that the method cannot be applied if the absolute sea level change is different from place to place. Nevertheless, this method can be useful in closed and semi-enclosed basins and could be adapted to

Kuo et al. [48] applied the inverse method within a semi-enclosed sea (Baltic Sea region of Fennoscandia). The results showed a significant reduction of uncertainties

work also in case a GPS at the coast is used instead of a tide gauge.

linear trend, there might be autocorrelation of the noise in the data [42].

dataset will be indicated with SLCCI-AT hereinafter.

*Geodetic Sciences - Theory, Applications and Recent Developments*

**5. Methods of estimating sea level trends**

gauge station only.

**100**

of the time series. Such constraints are formed imposing that the rate of relative vertical motion between two TGs must equal the difference of their VLMs:

$$
\dot{r}\dot{u}\_{i\dot{\jmath}} = \left(\dot{\mathbf{g}}\_i - \dot{\mathbf{s}}\_i\right) - \left(\dot{\mathbf{g}}\_j - \dot{\mathbf{s}}\_j\right). \tag{3}
$$

A unique linear system incorporating the constraints (7) in the system (6) is formed recurring to the Lagrange multipliers technique: the inverse linear problem with constraints [56] (LIPWC). It stems from the minimization of the expression <sup>Φ</sup>ð Þ¼ *<sup>u</sup>*\_ *<sup>e</sup>Te* <sup>þ</sup> <sup>2</sup>*λT*ð Þ *Fu*\_ � *<sup>h</sup>* , obtained as the sum of the *L2* norm of the prediction error *e* ¼ *Gu*\_ � *d* and the inner product of the constraint equations *Fu*\_ � *h* ¼ 0 by the Lagrange multipliers *λT*. The resulting ordinary least squares (OLS) equation in

*Coastal Sea Level Trends from a Joint Use of Satellite Radar Altimetry, GPS and Tide…*

λ � � <sup>¼</sup> *<sup>d</sup> h*

*<sup>Y</sup>*; *<sup>Y</sup>* <sup>¼</sup> *<sup>d</sup>*

*h*

1

CCCA

� � (8)

*F* 0 !

� � (9)

. Such

(10)

(11)

� � �

� �, then the reference to the ASLR

*I F<sup>T</sup> F* 0 ! *u*\_

which is a linear system of the form *<sup>A</sup>* � *<sup>X</sup>* <sup>¼</sup> *<sup>Y</sup>*, with *<sup>A</sup>* <sup>¼</sup> *I F<sup>T</sup>*

*<sup>X</sup>* <sup>¼</sup> *<sup>u</sup>*\_ λ � �

Ω *A*�<sup>1</sup> � �*<sup>T</sup>*

*<sup>X</sup>* <sup>¼</sup> *<sup>A</sup>*�<sup>1</sup>

� � can be rewritten as *ru*\_ *ij* ¼ �*s*\_

posed by De Biasio et al. [58]:

Ω ¼

system is solved by direct inversion, provided the inverse of matrix *A* exists:

<sup>¼</sup> *<sup>A</sup>*�<sup>1</sup>

The standard errors of the *u*\_ are estimated as the diagonal elements of the

, with Ω given by

<sup>1</sup> 0 … 0

where N and L are respectively the number of parameters *u*\_ *ij* and constraints *ru*\_ *ij*. The previous expression for Ω holds assuming no autocorrelation and heteroscedasticity of the regression residuals. The resultant errors of the OLS estimators are generally referred to as heteroscedasticity-consistent standard errors or White– Huber robust standard errors [57]. Finally, the estimated parameters are given by

*<sup>Y</sup>*; *Xi* <sup>¼</sup> *<sup>u</sup>*\_ *<sup>i</sup>* if 0<sup>≤</sup> *<sup>i</sup>*<sup>≤</sup> *<sup>N</sup>*

A possible attenuation of the condition that *g*\_*<sup>i</sup>* � *g*\_ *<sup>j</sup>* ¼ 0, for each pairs of TGs

0 *i* � � � �*s*\_

has disappeared. Such situation can be achieved by a change of variable, as pro-

(

*<sup>δ</sup>Xi* <sup>¼</sup> *diag A*�<sup>1</sup>

involved in a constrain, arises from the observation that if *ru*\_ *ij* ¼ *g*\_*<sup>i</sup>* � *s*\_*<sup>i</sup>*

*g*\_*<sup>i</sup>* ! *g*\_ 0

> \_ *<sup>ζ</sup><sup>i</sup>* ! \_ *ζ* 0 *<sup>i</sup>* <sup>¼</sup> \_ *ζi*–*g*\_*<sup>i</sup>*

*s*\_*<sup>i</sup>* ! *s*\_ 0 *<sup>i</sup>* ¼ *s*\_*<sup>i</sup>* � *g*\_*<sup>i</sup>*

<sup>2</sup> … 0 ⋮ ⋮⋱ ⋮ 0 0 … σ<sup>2</sup>

*N*þ*L*

λ*<sup>i</sup>* if *N* þ 1≤ *i*≤ *N* þ *L*

*i*

<sup>Ω</sup> *<sup>A</sup>*�<sup>1</sup> � �*<sup>T</sup>* � �

0 *j*

*<sup>i</sup>* ¼ *g*\_*<sup>i</sup>* � *g*\_*<sup>i</sup>* ¼ 0

σ2

0

BBB@

0 σ<sup>2</sup>

matrix form is:

*DOI: http://dx.doi.org/10.5772/intechopen.98243*

covariance matrix *A*�<sup>1</sup>

*Xi* � *δXi*:

*g*\_ *<sup>j</sup>* � *s*\_ *<sup>j</sup>*

**103**

At this stage, the constraints still contain explicitly the ASLR at sites *i*, *j*, which are not known back in time beyond the beginning of the altimetry era. But if each couple of TGs in Eq. (3) are observing the same ASLR for some reason, for example they can be inside a lake or a semi-enclosed basin, Eq. (3) simplifies to:

$$
\dot{r}\dot{u}\_{ij} = \begin{pmatrix} \dot{s}\_j - \dot{s}\_i \end{pmatrix},\tag{4}
$$

leaving out any reference to the ASLRs. Containing only the differences of the RSLR, Eq. (4) can be extended to the whole period of overlapping observations of the two TGs, which usually are longer and affected by lower errors. To distinguish the RSLR observed at the TG in the altimetry era from that observed in the common, longer period of observations of TGs *i*, *j*, we rewrite Eq. (4) as:

$$
\dot{\boldsymbol{r}}\dot{\boldsymbol{u}}\_{ij} = \dot{\boldsymbol{\zeta}}\_{j} - \dot{\boldsymbol{\zeta}}\_{i} \tag{5}
$$

where we used the Greek letter ζ to indicate that the TG RSLR difference is calculated over the complete overlapping time span of the two tide gauges, even before the altimetry era.

The two linear systems for the Eqs. (2) and (5) are written in vector–matrix form as:

$$G\dot{u} = d; \ \dot{u} = \begin{pmatrix} \dot{u}\_1 \\ \vdots \\ \dot{u}\_N \end{pmatrix}; \ d = \begin{pmatrix} \dot{g}\_1 - \dot{s}\_1 \\ \vdots \\ \dot{g}\_N - \dot{s}\_N \end{pmatrix}; \ G = I\_N,\tag{6}$$

$$F\dot{u} = h; \ h = -F\begin{pmatrix} \dot{\zeta}\_1 \\ \vdots \\ \dot{\zeta}\_N \end{pmatrix}; \ F = \begin{pmatrix} 1 & -1 & 0 & 0 & \dots & 0 \\ 0 & 1 & -1 & 0 & \dots & 0 \\ & \dots & & & \dots \\ 0 & 0 & 0 & \dots & 1 & -1 \end{pmatrix}.\tag{7}$$

The matrix *F* is the design matrix by which the constraints are formed and introduced in the linear system. The constraints can be chosen arbitrarily, but they must be linearly independent so that the rank of matrix *F* is L ≤ N – 1. For the system to admit an OLS solution, L, the rank of the matrix *F*, must be < N, so that one degree of freedom is left in the linear system for the OLS procedure to perform the unknowns estimate. Without such degree of freedom, the system would become even-determined, and the N constraints *ru*\_ *ij* <sup>¼</sup> \_ <sup>ζ</sup> *<sup>j</sup>* � \_ ζ*<sup>i</sup>* would automatically determine the solutions for the unknown *u*\_ *<sup>i</sup>*, and there would be no need for an OLS estimation.

Not always assumption *g*\_*<sup>i</sup>* � *g*\_ *<sup>j</sup>* ¼ 0 is valid for every couple of TGs paired in a constraint. In such case independent linear systems are to be considered for each group of homogeneous TGs. Wöppelmann and Marcos [50] for example, applying this method to the seas of the southern Europe, treated separately the sites inside the Mediterranean Sea from those on the Atlantic coast of the Iberian Peninsula, as the oceanographic behaviors of the two sets of TG sites were observed to be markedly different. In this case the two linear system are totally independent and there cannot be a connection between the two groups.

*Coastal Sea Level Trends from a Joint Use of Satellite Radar Altimetry, GPS and Tide… DOI: http://dx.doi.org/10.5772/intechopen.98243*

A unique linear system incorporating the constraints (7) in the system (6) is formed recurring to the Lagrange multipliers technique: the inverse linear problem with constraints [56] (LIPWC). It stems from the minimization of the expression <sup>Φ</sup>ð Þ¼ *<sup>u</sup>*\_ *<sup>e</sup>Te* <sup>þ</sup> <sup>2</sup>*λT*ð Þ *Fu*\_ � *<sup>h</sup>* , obtained as the sum of the *L2* norm of the prediction error *e* ¼ *Gu*\_ � *d* and the inner product of the constraint equations *Fu*\_ � *h* ¼ 0 by the Lagrange multipliers *λT*. The resulting ordinary least squares (OLS) equation in matrix form is:

$$
\begin{pmatrix} I & F^T \\ F & \mathbf{0} \end{pmatrix} \begin{pmatrix} \dot{u} \\ \lambda \end{pmatrix} = \begin{pmatrix} d \\ h \end{pmatrix} \tag{8}
$$

which is a linear system of the form *<sup>A</sup>* � *<sup>X</sup>* <sup>¼</sup> *<sup>Y</sup>*, with *<sup>A</sup>* <sup>¼</sup> *I F<sup>T</sup> F* 0 !. Such system is solved by direct inversion, provided the inverse of matrix *A* exists:

$$X = \begin{pmatrix} \dot{u} \\ \lambda \end{pmatrix} = A^{-1}Y; \ Y = \begin{pmatrix} d \\ h \end{pmatrix} \tag{9}$$

The standard errors of the *u*\_ are estimated as the diagonal elements of the covariance matrix *A*�<sup>1</sup> Ω *A*�<sup>1</sup> � �*<sup>T</sup>* , with Ω given by

$$
\boldsymbol{\Omega} = \begin{pmatrix}
\sigma\_1^2 & \mathbf{0} & \dots & \mathbf{0} \\
\mathbf{0} & \sigma\_2^2 & \dots & \mathbf{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{0} & \mathbf{0} & \dots & \sigma\_{N+L}^2
\end{pmatrix} \tag{10}
$$

where N and L are respectively the number of parameters *u*\_ *ij* and constraints *ru*\_ *ij*. The previous expression for Ω holds assuming no autocorrelation and heteroscedasticity of the regression residuals. The resultant errors of the OLS estimators are generally referred to as heteroscedasticity-consistent standard errors or White– Huber robust standard errors [57]. Finally, the estimated parameters are given by *Xi* � *δXi*:

$$X = A^{-1}Y; \ X\_i = \begin{cases} \dot{u}\_i \text{ if } 0 \le i \le N \\\ \lambda\_i \text{ if } N+1 \le i \le N+L \end{cases}$$

$$\delta \mathbf{X}\_i = \text{diag}\left(A^{-1}\Omega \left(A^{-1}\right)^T\right)\_i \tag{11}$$

A possible attenuation of the condition that *g*\_*<sup>i</sup>* � *g*\_ *<sup>j</sup>* ¼ 0, for each pairs of TGs involved in a constrain, arises from the observation that if *ru*\_ *ij* ¼ *g*\_*<sup>i</sup>* � *s*\_*<sup>i</sup>* � � �

*g*\_ *<sup>j</sup>* � *s*\_ *<sup>j</sup>* � � can be rewritten as *ru*\_ *ij* ¼ �*s*\_ 0 *i* � � � �*s*\_ 0 *j* � �, then the reference to the ASLR has disappeared. Such situation can be achieved by a change of variable, as proposed by De Biasio et al. [58]:

$$
\dot{\mathbf{g}}\_i \to \dot{\mathbf{g}}\_i' = \dot{\mathbf{g}}\_i - \dot{\mathbf{g}}\_i = \mathbf{0}
$$

$$
\dot{\mathbf{s}}\_i \to \dot{\mathbf{s}}\_i' = \dot{\mathbf{s}}\_i - \dot{\mathbf{g}}\_i
$$

$$
\dot{\boldsymbol{\zeta}}\_i \to \dot{\boldsymbol{\zeta}}\_i' = \dot{\boldsymbol{\zeta}}\_i - \dot{\mathbf{g}}\_i
$$

of the time series. Such constraints are formed imposing that the rate of relative vertical motion between two TGs must equal the difference of their VLMs:

� � � *<sup>g</sup>*\_ *<sup>j</sup>* � *<sup>s</sup>*\_ *<sup>j</sup>*

At this stage, the constraints still contain explicitly the ASLR at sites *i*, *j*, which are not known back in time beyond the beginning of the altimetry era. But if each couple of TGs in Eq. (3) are observing the same ASLR for some reason, for example

*ru*\_ *ij* ¼ *s*\_ *<sup>j</sup>* � *s*\_*<sup>i</sup>*

leaving out any reference to the ASLRs. Containing only the differences of the RSLR, Eq. (4) can be extended to the whole period of overlapping observations of the two TGs, which usually are longer and affected by lower errors. To distinguish the RSLR observed at the TG in the altimetry era from that observed in the com-

<sup>ζ</sup> *<sup>j</sup>* � \_

*g*\_ <sup>1</sup> � *s*\_ 1 ⋮ *g*\_ *<sup>N</sup>* � *sN*\_

<sup>ζ</sup> *<sup>j</sup>* � \_

1

1 �10 0 … 0 0 1 �1 0 … 0 … … 00 0 … 1 �1

0

B@

� �

*:* (3)

� �, (4)

ζ*<sup>i</sup>* (5)

CA; *<sup>G</sup>* <sup>¼</sup> *IN*, (6)

1

ζ*<sup>i</sup>* would automatically determine

CCCA*:* (7)

*ru*\_ *ij* ¼ *g*\_*<sup>i</sup>* � *s*\_*<sup>i</sup>*

*Geodetic Sciences - Theory, Applications and Recent Developments*

they can be inside a lake or a semi-enclosed basin, Eq. (3) simplifies to:

mon, longer period of observations of TGs *i*, *j*, we rewrite Eq. (4) as:

*u*\_1 ⋮ *u*\_ *N* 1

CA; *<sup>d</sup>* <sup>¼</sup>

0

BBB@

The matrix *F* is the design matrix by which the constraints are formed and introduced in the linear system. The constraints can be chosen arbitrarily, but they must be linearly independent so that the rank of matrix *F* is L ≤ N – 1. For the system to admit an OLS solution, L, the rank of the matrix *F*, must be < N, so that one degree

the solutions for the unknown *u*\_ *<sup>i</sup>*, and there would be no need for an OLS estimation. Not always assumption *g*\_*<sup>i</sup>* � *g*\_ *<sup>j</sup>* ¼ 0 is valid for every couple of TGs paired in a constraint. In such case independent linear systems are to be considered for each group of homogeneous TGs. Wöppelmann and Marcos [50] for example, applying this method to the seas of the southern Europe, treated separately the sites inside the Mediterranean Sea from those on the Atlantic coast of the Iberian Peninsula, as the oceanographic behaviors of the two sets of TG sites were observed to be markedly different. In this case the two linear system are totally independent and there

of freedom is left in the linear system for the OLS procedure to perform the unknowns estimate. Without such degree of freedom, the system would become

0

B@

1

CA; *<sup>F</sup>* <sup>¼</sup>

\_ζ1 ⋮ \_ ζ*N*

0

B@

even-determined, and the N constraints *ru*\_ *ij* <sup>¼</sup> \_

cannot be a connection between the two groups.

before the altimetry era.

*Fu*\_ ¼ *h*; *h* ¼ �*F*

*Gu*\_ ¼ *d*; *u*\_ ¼

form as:

**102**

*ru*\_ *ij* <sup>¼</sup> \_

where we used the Greek letter ζ to indicate that the TG RSLR difference is calculated over the complete overlapping time span of the two tide gauges, even

The two linear systems for the Eqs. (2) and (5) are written in vector–matrix

$$
\dot{\mu}\_i = \dot{\mathbf{g}}\_i - \dot{\mathbf{s}}\_i = \dot{\mathbf{g}}\_i - \dot{\mathbf{g}}\_i + \dot{\mathbf{g}}\_i - \dot{\mathbf{s}}\_i = \dot{\mathbf{g}}'\_i - \dot{\mathbf{s}}'\_i. \tag{12}
$$

principally the Permanent Service for Mean Sea Level (PSMSL) [61], the Venice Tide Forecast and Early Warning Center (Centro Previsioni e Segnalazioni Maree, CPSM) of Venice Municipality, the Istituto Superiore per la Protezione e la Ricerca Ambientale (Italian Institute for Environmental Protection and Research (ISPRA)) and the Institute of Marine Sciences of the National Research Council of Italy

*Coastal Sea Level Trends from a Joint Use of Satellite Radar Altimetry, GPS and Tide…*

**Figure 2** shows the position of the TGs on the map of the Adriatic Sea region. Some of the TG records have been formed by collating partial records from different sources. Such TGs are marked by an asterisk. The individual positions of the

C3S altimetry grid are shown in **Figure 3**: note that some of the grid nodes are represented over land. This is an artifact of the gridding procedure that partially

VEPTF is the shortest record in the set, as it started sea level recordings only in 1974. Nonetheless, its length is almost double that of the altimetry era, and abundantly double the period of the lunar nodal tide. To treat evenly all the TG records,

Plots of the in situ, as well as of the altimetry sea level anomaly monthly means

The altimetry dataset used to represent sea level anomaly in **Figure 4** is C3S. The

*Positions of six tide gauges in Adriatic Sea. Color bar indicates length of available time series of sea level at tide*

in situ and the remotely sensed sea level records are in good agreement, as the lowest Pearson's correlation coefficient between altimetry and TG sea level time series is 0.82 at the Rovinj station, while all the others reach values larger than 0.91. However, in some period a marked difference between in situ and altimetry SLA are seen, as for example in VENEZIA during 2012–2019 (TG sea level higher than altimetry), which is also confirmed by the nearby TG of VEPTF and seems to interest in a lesser extent also TRIESTE and DUBROVNIK, and for ROVINJ in

we consider in situ sea level data from 1974 up to 2018 for all the TGs.

observed at the six locations in the Adriatic Sea are reported in **Figure 4**: the seasonal and tidal signals have been removed from both the in situ and the altimetry datasets. The altimetry grid node associated to the TG time series has been chosen as the one whose time series has the higher correlation coefficient with the sea level time series of the TG, among the twelve grid nodes closest to the latter. All the sea level trend errors have been calculated considering serial correlation and are given

TGs with respect to the twelve closest nodes of the.

extrapolates over land the SLA field [36].

*DOI: http://dx.doi.org/10.5772/intechopen.98243*

with a 95% confidence interval.

**Figure 2.**

**105**

*gauges; shortest time series is about 50 years.*

(CNR-ISMAR).

In other words, we overcome the limitation of equal ASLR at all TGs by removing from both, the TG time series and the altimetry time series associated with the TG, a linear trend equal to that measured by the altimeter. Such change of variables (COV) does not alter the statistical properties of the TG and altimetry time series but eliminates any difference in relative sea level changes due to different absolute sea level changes. For it to work two assumptions are necessary:


While the first assumption can be easily verified by visual inspection or with more precise statistical methods, as the goodness-of-fit R<sup>2</sup> test [59], the second assumption, needed to permit the third change of variable in Eq. (12), can be more difficult to assess. In general, the linearity of a TG's RSLR trend can partly corroborate the validity of the second assumption, as the probability that two different, non-linear trends of the local ASLR and VLM perfectly combine by chance, to give an overall linear trend, is obviously low.
