2. Mathematical models

In this section, the mathematical models for terrestrial hydrology, which accounts for the maximum number of physical and meteorological factors and can be described by the shallow water equations, are considered. The SWE or the Saint-Venant equations are fairly simple model describing the free surface dynamics of incompressible fluid (see the details in the review of [23]). The model assumes that vertical equilibrium in a medium exists at every moment of time. The conservation laws of mass and momentum in the integral form for a thin layer of moving substance with additional sources and forces are:

$$\frac{d}{dt} \iint\limits\_{S(t)} H(\mathbf{r}, t) \, dS = Q(\mathbf{r}, t) \, \frac{d}{dt} \iint\limits\_{S(t)} H \mathbf{v} \, dS = -\text{g} \iint\limits\_{S(t)} H \nabla \eta \, dS + \iint\limits\_{S(t)} H \mathbf{f} \, dS,\tag{1}$$

where Q is the sources function, ∇ = ex∂/∂x + ey∂/∂y is the nabla operator, S(t) is the crosssectional area of the "liquid particle", r = xe<sup>x</sup> + ye<sup>y</sup> is the radius vector, v = ue<sup>x</sup> + ve<sup>y</sup> is the velocity vector, η(r, t) = H(r, t) + b(r) is the free surface elevation, b(r) is the bottom profile, f(r, t) = f<sup>b</sup> + ffr + fCor + f<sup>w</sup> + f<sup>s</sup> is the sum of the external forces, accounting for the bottom friction fb, the internal friction (viscosity) ffr, the Coriolis force fCor, the effect of atmospheric wind fw, and the force f<sup>s</sup> determined by the liquid momentum due to the action of the sources Q. The surface density of sources is σ(r, t) = σ(+) + σ(-) = dQ/dS, where σ(+) and σ(�) are the liquid sources (rain, melting snow, flows through the hydro-constructions, groundwater, etc.) and sinks (infiltration, evaporation, etc.), respectively.

The digital elevation model (DEM) provides the quality of numerical simulations of real hydrological objects to a significant extent. The DEM is determined by the height matrix bij = b(xi, yj) on numerical grid {xi, yj} (i = 1, …, Nx, j = 1, …, Ny). The DEM elaboration utilizes diverse geoinformation methods for the processing of spatial data obtained from various sources. A matrix of heights has been built in several stages. At the beginning stage, the remote sensing data are accounted by the function b(xi, yj). The river sailing directions and the actual water depth measurements allow to construct the DEM for large river beds (for example, for the Volga River and the Akhtuba River). To improve our DEM, the data on various small topography objects such as small waterways, roads, small dams, etc., should be included into the consideration. The numerical simulation results of the shallow water dynamics reveal flooding areas which may be compared with real observational data (both from the remote methods and our own GPS measurements). Such approach qualitatively improves the DEM as a result of the iterative topography refinement.
