## 2 Materials and Methods
### Methodology
As Equation (4) characterizes the state of the system as defined by local variables, it can be used to determine the entropy of the system. Therefore, the state of equilibrium at the beginning and end of the period is determined, not the inner processes, because entropy is a property of the system. When Equation (3) is balanced, the delta study aggrades vertically with the MSL rise, no horizontal shoreline movement occurs, and the system is in a stationary state. The disorder in the system and entropy generation are low. System deviations from this balance result in delta regression (progradation) or transgression (retrogradation). Inner processes occur and bring the system back to a stationary state. Therefore, any deviation from the stationary state increases the speed of entropy production. In the case of an MSL rise, delta submergence results in additional space for sediment to be disposed of on the topset (storage increase), which enhances sediment aggregation and reduces shoreline regression. If the sediment supply is low, the available space on the delta plain generated by the MSL rise cannot be filled, and the delta will be transgressive (\(\mathrm{Q_{river}\cdot f_{r}<R\cdot A_{delta}}\)) to sustain the dynamic cycles necessary for activities to be coupled. If the sediment supply is sufficient, the delta can maintain its size and equilibrium (\(\mathrm{Q_{river}\cdot f_{r}=R\cdot A_{delta}}\)) or prograde seaward (\(\mathrm{Q_{river}\cdot f_{r}>R\cdot A_{delta}}\)). Rises that change the delta area can be explored based on the deviation from the equilibrium (Equation (3)) and used for sustainability assessment, assuming that the entropy or disorder (\(\mathrm{S_{delta}}\)) is the absolute difference between the sediment supply and the effects of MSL rise on the delta's natural system. \[\mathrm{S_{delta}}=\mid\mathrm{Q_{river}\cdot f_{r}-R\cdot A_{delta}}\mid. \tag{4}\]  
This is a state function, imbalance, or risk that arises because of disturbances caused by global and local human activities or natural causes. Theoretically, \(\mathrm{S_{delta}}\) can be expressed as follows:  
\[0<\mathrm{S_{delta}}<\infty. \tag{5}\]When the value of S\({}_{\text{delta}}\) approaches 0, the system is in near-perfect order, its processes are in a stationary state, and the inner irreversible changes maintain the stability of the equilibrium state. As the deviation or disorder increases, the speed of entropy production and the value of S\({}_{\text{delta}}\) increase, and the entropy in the system increases. Therefore, any deviation from the stationary or equilibrium state causes entropy generation, which reduces sustainability. Finally, it can theoretically approach a large value (maximum disorder), which indicates that the delta dry land area increases when the amount of sediment load in the delta is constantly high, or the sea area or bay increases when the sediment supply reaches very low values. In both cases, the sum of entropy changes in the system is always positive. The speed and duration of the process determine the outcome or level of change. The entropy change for a delta system during a process in a certain period is as follows:  
\[\text{DS}_{\text{delta}}=\text{S}_{\text{delta},\text{ final}}-\text{S}_{\text{delta},\text{ initial}}. \tag{6}\]  
Therefore, when the entropy change of a delta system has a minimal positive value or equals zero if the state of the system does not change during the process (study conditions), the delta hydomorphology is sustainable, and the system maintains its organization. However, when it is positive, the delta system is unsustainable and changes the biophysical system. In the proposed approach, only (historic) qualitative assessment is applied because it is only possible to assess the direction of change (transgressive or regressive) and not the speed of change. The direction of change can be assessed by assuming an equilibrium change between the sediment supply and the rate of MSL increase (Equation (3)).