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

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

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 gauge station only.

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

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

compared with those from conventional approaches, which are limited to the overlapping periods between altimetry and tide gauges. An extension of the method has been applied to Great Lakes and in open ocean regions, such as Alaskan coast [49]. It has been also extended along the coasts of southern Europe [50] with constraints between pairs of tide gauges based on correlation and overlapping periods. The same method has been extended to open ocean in New Zealand straddles, the Tasman Sea and Pacific Ocean [51]. All studies confirmed the superiority of the inverse method to the classical direct approach.

A new variant of the inverse method considers to difference sea level trends between pairs of tide gauge records and pairs of altimetry records [52]. Another study proposed different mathematical and statistical models, which enable simultaneous estimation of absolute and relative sea level trends and VLM at a tide gauge station merging altimetry and tide gauge records without the aid of geological information or GPS measurements [53].
