# Subsidence calculation

The subsidence model calculates the gradual downward settling of the ground's surface where a peat layer is present. The considered subsidence takes place due to oxidation and compaction of the peat layer. The total subsidence is currently calculated as the result of oxidation and compaction over the years. Oxidation is dependent on the ground water level and the clay thickness. Compaction is dependent on the toplayer thickness, the peat fraction and raised surface terrain.

The ground water level is initialized by either raster or surface area data and optionally adjusted each year (indexation). When indexation is active, the surface water level is adjusted for the settling each year and the ground water levels will change accordingly.

The calculation model is configured by the subsidence overlay wizard and it's result are visualized with the subsidence overlay.

## Calculations

During a calculation step, the following aspects are calculated in order:

• The temperature at the start of the year is calculated
• Based on that, parameter a of the oxidation formula is calculated
• The oxidation subsidence is calculated
• The compaction subsidence is calculated
• The surface water levels are lowered by the amount of subsidence times the indexation
• The ground water level is lowered based on subsidence and surface water level indexation
• The new ground water level serves as input for the next calculation step

### Yearly recalculation of Parameter A

The a parameter is recalculated yearly, to account for a changing climate. The new value for the a parameter is obtained by multiplying the original Parameter a is with a factor. This factor is calculated based on an interpolated average temperature for that year, a soil temperature factor, micro activity and an oxidation factor.

The value of the a parameter is updated each year as follows:

Delta temperature is calculated as:

${\displaystyle \Delta {T}=(T_{final}-T_{start})\cdot {\frac {y_{y}-y_{start}}{y_{final}-y_{start}}}\cdot f_{st}}$

Delta micro activity is calculated as:

${\displaystyle \Delta {MA}=Q_{10}^{\Delta {T}/10.0}-1.0}$

Delta subsidence is calculated as:

${\displaystyle \Delta {s_{sub}}=\Delta {MA}\cdot f_{ox}}$

The new parameter a value for year y is calculated as:

${\displaystyle a_{y}=a_{0}\cdot (1.0+\Delta {s_{sub}})}$

where

${\displaystyle a_{y}}$ = The a parameter for the current year.
${\displaystyle a_{0}}$ = The a parameter value at the start of the calculation, defined by the model attribute A.
${\displaystyle y_{y}}$ = The current year. The first year is defined by START_YEAR.
${\displaystyle y_{start}}$ = The start year of the simulation, defined by CLIMATE_START_YEAR.
${\displaystyle y_{end}}$ = The final year of the simulation, defined by the addition of the start year and YEARS.
${\displaystyle T_{start}}$ = The average temperature of the start year, defined by CLIMATE_START_TEMP.
${\displaystyle T_{final}}$ = The average temperature of the final year, defined by CLIMATE_FINAL_TEMP.
${\displaystyle Q_{10}}$ = The Q10 factor, configured as 3. This value is currently not adjustable.
${\displaystyle f_{st}}$ = Soil temperature factor, defined by CLIMATE_SOIL_TEMP_FACTOR.
${\displaystyle f_{ox}}$ = Oxidation factor, defined by CLIMATE_OXIDATION.

### Ground water depth

At the start of a simulation, the ground water depth is initialized with the ground water depth GeoTIFF (if provided) and is optionally overwritten by (managed) water areas' water level.

Furthermore, the ground water level can be managed with drainages (provided as underground buildings), either actively or passively.

Additionally the terrain height can change due to subsidence that occurred in previous years and due to actions taken that raised the terrain. Managed water areas can react to these changes when indexation is configured. For indexation, see indexation formula.

The following formulas describe how the yearly adjusted ground water depths are obtained.

#### Ground water level managed by drainages

${\displaystyle d_{gw,y}={\begin{cases}d_{d,active},&{\text{if }}n{\text{ drainage is larger than 0}}\\max(0.0,d_{a}-\Delta {d_{a,y}}+d_{d,passive},&{\text{if }}n{\text{ drainage is smaller than 0}}\\max(0.0,d_{gw}+\Delta {d_{a,s,y}})&{\text{ otherwise }}\end{cases}}}$

where

${\displaystyle d_{gw,y}}$ is the calculated ground water depth of a grid cell at year y.
${\displaystyle d_{d,active}}$ is the ground water depth actively maintained by a drainage
${\displaystyle d_{a}}$ is the ground water depth of the managed water area.
${\displaystyle \Delta {d_{a,y}}}$ is the area water level adjustment for year y.
${\displaystyle d_{d,passive}}$ is the ground water level passively maintained by drainages.
${\displaystyle \Delta {d_{a,s,y}}}$ is the ground water level increase calculated based on the are water level adjustment due to indexation and the subsidence in the previous year.

#### Ground water level managed by water areas

The effect of changes of managed surface water levels (by water areas) on the ground water level. The x-axis indicates the relative depth of the managed surface water level. The y-axis indicates the meters of change to the ground water level, per meter change in surface water level. The smaller the relative depth is, the less the ground water level equalizes with changes in the managed surface water level.

The ground water level can also be managed by water areas, which control the surface water levels. To make sure ground water levels do not rise (too much), surface water levels in the water areas can be lowered automatically with respect to the subsidence. This process is called indexation. The new ground water level can then be estimated based on the indexation on the managed water level of surface water and the calculated subsidence of the previous year. The increment in ground water level is rarely exactly the adjustment of the managed surface water level and is often less, depending on the relative depth of the surface water level. The following formulas have been developed to estimate the increase of the ground water level.

The new relative surface water level depth is calculated as:

${\displaystyle \Delta {d_{inc}}=-\Delta {d_{a,y-1}}-s_{y-1}}$

${\displaystyle d_{a,adj}=d_{a}+\Delta {d_{inc}}}$

Next, based on the old and new depths, the ground water depth change is divided into three sections, each which its own rate of contribution.

${\displaystyle \Delta {d_{low}}=0.6\cdot min(0.6,d_{a,adj})-min(0.6,d_{a})}$

${\displaystyle \Delta {d_{mid}}={\frac {\max(0.6,min(1.0,d_{a,adj}))+max(0.6,min(1.0,d_{a}))}{2.0}}\cdot (\max(0.6,min(1.0,d_{a,adj}))-max(0.6,min(1.0,d_{a})))}$

${\displaystyle \Delta {d_{high}}=1.0\cdot (\max(0.0,d_{a,adj}-max(0.0,d_{a})}$

Finally, the ground water depth change is calculated and the new ground water depth is obtained:

${\displaystyle \Delta {d_{gw,y}}=\Delta {d_{low}}+\Delta {d_{mid}}+\Delta {d_{high}}}$

${\displaystyle d_{gw,y}=max(0.0,d_{gw,0}+\Delta {d_{gw,y}})}$

where

${\displaystyle d_{a}}$ is the current relative water depth of water area a.
${\displaystyle d_{a,adj}}$ is the adjusted relative water depth of water area a.
${\displaystyle \Delta {d_{a,y-1}}>}$ is the water level adjustment for indexation of current year y.
${\displaystyle s_{y-1}}$ is the subsidence that occurred up to year y.
${\displaystyle d_{gw,y}}$ is the calculated adjusted ground water depth for year y, based on the occurred subsidence and indexation of previous years.
${\displaystyle d_{gw,0}}$ is the initial ground water depth at the start of the simulation.
${\displaystyle \Delta {d_{gw,y}}}$ is the estimated change of the ground water level based on occurred subsidence and indexation.

### Oxidation calculation

The amount of subsidence due to oxidation is calculated by the following formula:

${\displaystyle S_{O}=a_{y}\cdot d_{gw,y,c}-b\cdot d_{ct,c}-c}$

where

${\displaystyle S_{O}}$ is the subsidence due to oxidation.
${\displaystyle d_{gw,y,c}}$ is the calculated ground water depth for year y in cell c.
${\displaystyle a_{y}}$ is the calculated a parameter value for the current year ${\displaystyle y}$.
${\displaystyle d_{ct,c}}$ is the clay thickness of the terrain in cell c, defined by CLAY_THICKNESS
${\displaystyle b}$ is the b parameter value, defined by the model attribute B
${\displaystyle c}$ is the c parameter value, defined by the model attribute C

This formula was provided by experts, who have derived this formula empirically.

### Compaction calculation

The amount of subsidence due to compaction is calculated by the following formula:

${\displaystyle S_{C}={\begin{cases}(a_{peat}\cdot p_{c}+a_{tl}\cdot d{tl,c})\cdot log10(t_{days})+a_{h}\cdot \Delta {h}+b_{p}\cdot p_{c}+b_{tl}\cdot d_{tl,c}&{\text{if }}\Delta {h}{\text{ is larger than 0}}\\0&{\text{ otherwise}}\end{cases}}}$

where

${\displaystyle S_{C}}$ is the calculated subsidence due to compaction.
${\displaystyle p_{c}}$ is the peat fraction of the respective grid cell c.
${\displaystyle d_{tl,c}}$ is the top layer thickness of the respective grid cell c.
${\displaystyle \Delta {h}}$ The terrain height increase as a result of the actions taken during a session, such as the creation of levees.
${\displaystyle t_{days}}$ is the number of days in a year times the current year being calculated.
${\displaystyle a_{peat}}$ is the first regression constant (0.015853041) for the effect of the peat fraction and time in days.
${\displaystyle a_{tl}}$ is the first regression constant (0.006617643) for the top layer thickness and time in days.
${\displaystyle a_{h}}$ is a regression constant (0.200468677) for the change in surface height, for example caused by added materials.
${\displaystyle b_{peat}}$ is the second regression constant (0.02348519) for the effect of the peat fraction.
${\displaystyle b_{tl}}$ is the second regression constant (-0.010061616) for the top layer thickness.

This formula is based on provided expert data in the form of a reference table, indicating the amount of subsidence based on the parameters used in the formula above. The formula's results conform to the reference table to within an average of a tenth of the margin of error of the original table.

### Subsidence calculation

The total subsidence for year y is calculated as:

${\displaystyle S_{y,max}=d_{tl,c}\cdot p_{c}-S_{y-1}}$

${\displaystyle S_{O,y}=min(S_{y,max},S_{O,y})}$

${\displaystyle S_{y,max}=S_{y,max}-S_{O,y}}$

${\displaystyle S_{C,y}=min(S_{y,max},S_{C,y})}$

${\displaystyle S_{y}=S_{O,y}+S_{C,y}}$

where

${\displaystyle p_{c}}$ is the peat fraction of the respective grid cell c.
${\displaystyle d_{tl,c}}$ is the top layer thickness of the respective grid cell c.
${\displaystyle S_{y,max}}$ is the maximum possible subsidence in year y.
${\displaystyle S_{y}}$ is the total subsidence in year y
${\displaystyle S_{O,y}}$ is the subsidence due to oxidation in year y.
${\displaystyle S_{C,y}}$ is the subsidence due to compactions in year y.

### Indexation

Indexation is the policy of managing the surface water level such that it keeps the nearby ground water levels (more or less) at the same depth.

A water area which is fully indexed (1.0 = 100%) will have its surface water level lowered by the same amount as the terrain height has lowered due to subsidence. Since it lowers just as much as the terrain itself, the ground water level(s) relative to the surface of the land will remain the same.

In a water area which is not indexed (0%) the surface water level remains at the same level. Any subsidence taking place will lower the land, and thus reduce the relative water depth. An area indexed by 50% will have the surface water level lower by half of the amount of subsidence.

To make matters a bit more complex, the change in surface water level rarely changes the ground water level with the same amount. Instead, the amount the ground water level changes is estimated using the ground water depth formula for water areas.

The indexation formula is as followed:

${\displaystyle S_{a}={\frac {\sum (S_{c,y})}{n_{a}}}}$

${\displaystyle \Delta {d_{a,y}}=-S_{a}\cdot i_{a}}$

where:

${\displaystyle S_{a}}$ is the average subsidence for water area a.
${\displaystyle S_{c,y}}$ is the subsidence in grid cell c for year y.
${\displaystyle n_{a}}$ is the number of non-water grid cells within water area a.
${\displaystyle \Delta {d_{a,y}}}$ is the amount the water area is changed in meters.
${\displaystyle i_{a}}$ is the indexation fraction for area a.

### Terrain height

The height of the terrain can be manipulated during a session by stakeholders taking a land sculpting action, creating open water, or constructing levees. These actions will result in settlement, which can be found under the Settlement result type. Creating or demolishing constructions generally do not change the height of the land, and do not result in changes in the settlement results.

## Configuring overlays

The subsidence overlay has a number of ways to configure it. Both values which serve as input for the overlays directly, as references to attributes of areas which provide input for the calculations.

### Keys

The subsidence overlay has a "Keys" tab in the right panel in the editor. Most keys are attributes of areas. When the overlay calculates, it will look per grid cell for the existence of these attributes.

Besides these attributes, 2 more model parameters can be configured.

Years
The amount of years to simulate during the calculation, in 1-year steps. It is possible to set this value anywhere between 1 to 1000. This parameter can be configured with the YEARS model attribute of the Subsidence overlay.

Ground Water Tiff
If the option to use a Ground Water Tiff is checked, a GeoTIFF can be selected to use for the ground water levels. You can either use any of the provided default GeoTIFFs, or upload and use your own.

Three ground water GeoTIFFs covering the Netherlands are available by default. These GeoTIFFs are a combination between a high resolution GeoTIFF containing ground water levels for rural areas and a low resultion GeoTIFF containing ground water levels for city areas, where data from the low resolution GeoTIFF was only used to fill the gaps in the high resolution GeoTIFF. One of the available ground water GeoTIFFs, relevant for the Subsidence, is the Mean Lowest Watertable (MLW, or GLG in Dutch).

### Attributes

The subsidence overlay also has model attributes. All attributes have a default value, but can be changed to configure the subsidence calculation.