## 1 Introduction  
The subject of this article is relevant to the integration of vertical movements of the Earth's crust determined from various geodetic survey techniques, both terrestrial and satellite. The aim of this paper is to study the possibility of integrating the vertical crustal movements developed from two independent sets (leveling and GNSS PPP--Precise Point Positioning data) by using a hybrid network adjustment (nonlinear method) and mathematical transformation combined. This paper hypothesizes that the combined adjustment of vertical motion networks from hybrid data is comparable to the use of yield transformations. The leveling network consists of the benchmarks between which the elevation measurements are made. It forms a leveling network. The leveling network is subject to adjustment with respect to a point or points of known elevation. Re-measurements of elevations are carried out at intervals (measurement campaigns) form a double-leveling network [1]. This allows for the calculation of the movement of the benchmarks identified with the vertical crustal movements. Two approaches are implemented: static--based on the difference in the height of the benchmarks--and non-linear--based on the adjustment of the double-leveling network (adjustment of vertical movement between the benchmarks). Each method gives good results under certain conditions, i.e., it depends on the existing leveling material; for example, both campaigns were carried out along the same leveling lines; successive campaigns were carried out in different parts of the area [2]. The use of nonlinear methods makes it possible to determine vertical movements associated with earthquakes and movements of a local nature. It also allows for the incorporation of tide gauges data and data from other surveying methods into this study. The incorporation is carried out as vertical movement at a point (sea level change--tide gauges data) or movements between points (e.g., GNSS permanent stations) [1]. The advantage of nonlinear methods is the greater ability to influence the \"observation\" data through observation errors. The disadvantage is the transformation of the hybrid network (triangulation) each time when GNSS stations are added or subtracted. The advantage of transformation is the rapid preparation of new sets in case of GNSS station additions or subtractions. The disadvantage is that it is difficult to evaluate the choice of location and number of adjustment points and the reliability of vertical movements determined on them [3,4]. These solutions are not commonly used to combine vertical crustal motions (deformations) determined from dual leveling and GNSS coordinate time series. The Polish area is not tectonically active, and the GNSS time series do not always reflect the vertical motion of the crustal surface [5]. If GNSS stations reflecting the station's own movements and those indicating the influence of underground mining are eliminated, it can be presumed that the resulting vertical movement of the station is compatible with the deformation of its crust. In the case of repeated leveling measurements, dozens of papers published (even from before the era of satellite measurements) clearly indicated consistency between leveling measurements and crustal movements. In areas with low tectonic activity (e.g., Poland), vertical movements determined from GNSS data are linear [5]; so, despite the low temporal resolution of leveling measurements, it can be assumed that the same relationship exists for leveling data. Models of the Earth's vertical crustal movements (VCM) or vertical land movements (VLM) are developed as absolute models and relative models. Absolute models refer to a rotational ellipsoid or quasi-geoid, while relative models refer to a specific level, e.g., the sea level, defined by a point or points (tide gauges). Other Global Navigation Satellite Systems (GNSS) references or stations, whose vertical motion is considered constant over time or have a known'velocity', can also be adopted as reference points [6].