In this study, we consider a coupled lake-catchment model that consists of three modules: the lake, the adjacent groundwater reservoir and the remaining catchment area. The basis for the modelling is to maintain the water mass balance between input and output over the lake and its catchment area on an annual time step:

where

$\mathsf{\Delta}{V}_{L}$ and

$\mathsf{\Delta}{V}_{r}$ are changes in lake and groundwater storage volumes (L

^{3}) respectively;

$\mathsf{\Delta}t$ the time step (T). We have assumed that the change of the groundwater volume in the upstream part of the catchment is equal to zero (∆

V_{C} = 0). The reason is that ∆

V_{C} is expected to be small relative to ∆

V_{L} and ∆

V_{r}, since the upstream part of the catchment is located further away from the lake, and will therefore be much less influenced by its (considerable) level variations.

P is precipitation (L/T) estimated as being spatially uniform in this study for Lake Babati region from available rainfall records (

Figure 2).

${A}_{l}$,

${A}_{r}$ and

${A}_{c}$ are the surface areas (L

^{2}) of the lake, the groundwater reservoir adjacent to the lake, and the remaining upstream part of the catchment, respectively. These were determined from the lake bathymetry and catchment topography (

Figure 4).

${E}_{l}$,

${E}_{r}$ and

${E}_{c}$ are the evaporation over the lake, the actual evapotranspiration over the groundwater reservoir adjacent to the lake and the actual evapotranspiration over the catchment (L/T), respectively.

$G{W}_{f}$ is the groundwater flux (L

^{3}/T) draining out of the lake and

${R}_{f}$ the possible runoff volume (L

^{3}/T) due to lake overflow. These evaporative and water flux components were estimated as described in the following sections.

#### 2.2.1. Catchment Topography and Lake Bathymetry

The area–volume relationship of the Lake Babati was derived from a combination of catchment topographic and lake bathymetric data (

Figure 4). Catchment topography is based on a 90 m resolution Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM) ([

31]) and paper topographical maps (1:50,000 scale and 50 feet contour spacing) produced by the Directorate of Overseas Surveys for the Tanganyika Government in 1964 from aerial photographs taken in 1958–1960. The paper maps were georeferenced and important features (contours, river channels, lake shore, extent of alluvial deposits) were digitized. Hence, a topo-to-raster interpolation with drainage enforcement [

32] within ArcGIS 10.1 spatial analyst package was applied to the digitized contours to generate a hydrologically correct 50 m resolution DEM, based on which (together with the SRTM DEM) the catchment boundary was delineated and its area calculated.

Subsequently, the contour line corresponding to the lake overflow level was used to determine the maximum lake surface area. Then, lake bathymetry was derived by spatial interpolation from lake vertical profile measurements along five transects acquired during a 2010 field visit (

Figure 4). From the bathymetric map, the area and volume corresponding to different lake depths (from the lowest point to the maximum lake level, at 1 cm step) were calculated using the “surface volume” tool in ArcGIS 3D analyst. Finally, the area of the groundwater reservoir around the lake was determined based on the extent of the alluvial deposits as digitised from the topographic and geological maps.

#### 2.2.2. Evaporation over the Lake

Lake surface evaporation is one of the main factors controlling the water balance of closed basin lakes. There are several methods to estimate lake surface evaporation, including direct measurements, empirical methods, water budget, energy budget, mass transfer and combination methods [

33,

34,

35,

36] or from pan evaporation records [

37]. The Penman [

35] and the energy balance methods have been extensively applied in the study of East African lakes [

3,

4,

5].

In this study, the energy budget method was adopted. The approach is among the best methods for estimating open water evaporation [

38,

39,

40,

41]. Comparatively speaking, the energy budget method is relatively simple and less data intensive [

42] and considers the effect of cloud cover, which is often ignored in other methods, despite its large impact on evaporation estimates. The approach is based on the assumption that, under steady-state conditions, the energy gained by the lake water surface from net radiation is compensated by the energy loss through latent and sensible heat transfer, such that:

where

${R}_{n}$ is the net radiation (W/m

^{2}),

$H$ is the sensible heat flux (W/m

^{2}) and

${H}_{L}$ the latent heat flux to evaporate water from the lake (W/m

^{2}). In our case, it can be assumed that heat exchange between the lake surface and the deeper part of the lake is negligible, since the time step used (one year) is large enough to exclude the influence of diurnal cycles [

33].

Equation (2) can be rewritten by introducing the Bowen ratio (B), which represents the fraction of sensible to the latent heat flux, as

The evaporation rate (

${E}_{L}$) can be introduced and represents the mass of water that needs to be evaporated to consume the latent heat energy of the latent heat and can be expressed as:

where

${L}_{e}$ is the latent heat of water vaporisation (

$2.46\times {10}^{6}\mathrm{J}/\mathrm{kg}$).

The net radiation (

${R}_{n}$) can be expressed in terms of the net short-wave radiation (

${R}_{s}$) and long-wave radiation (

${R}_{L}$) as:

Since there were no direct measurements,

${R}_{S}$ and

${R}_{L}$ were estimated in this study using empirical equations. The net shortwave radiation, which is the radiant flux resulting directly from solar radiation was estimated from [

43] as:

$\mathsf{\alpha}$ being the albedo of the water surface (

$\mathsf{\alpha}\approx 0.66$ [

33]) and

${R}_{TOA}$ is the solar radiation at the top of the atmosphere, which is a function of latitude [

43].

${R}_{TOA}$ was estimated to be 416.4 W/m

^{2} at Lake Babati.

The parameter

C, the cloud cover, explains the portion of the day during which clouds are not obstructing the sun.

C values are highly variable and may have a great influence on lake evaporation estimates. It can be estimated from Equation (8) as:

where

n and

N are actual number of hours of bright sunshine and the number of daylight length, respectively [

33]. The optimised value of

C after model calibration was 0.4 (see following sections).

The net longwave radiation (

${R}_{L}$), the radiant flux resulting from emissions of the atmospheric gases and the land and water surfaces on earth, was estimated following [

44] as:

where

${\mathsf{\epsilon}}_{w}$ is the emissivity of water (

${\mathsf{\epsilon}}_{w}\approx 0.97$);

$\mathsf{\delta}$ is the Stephan–Boltzmann’s constant (

$\mathsf{\delta}=5.6697\times {10}^{-8}{\mathrm{Wm}}^{-2}\cdot {\mathrm{K}}^{-4}$),

T_{s} is the lake surface temperature (K), and

${e}_{sat}$ is the saturation vapor pressure (hPa), which can be calculated with Clausius–Clapeyron’s relation as:

where

$\mathsf{\epsilon}$ = 0.622 is the molar mass fraction of water vapour in dry air and

${R}_{d}$ is the gas constant of dry air (287 J/kg/K).

The Bowen Ratio (

B) is based on the eddy diffusivity hypothesis and is given by the relationship:

where

T_{s} and

T_{r} are the temperature (K) at lake surface and at a reference height respectively. An average value of 19.4 °C (292.55 K) for the reference temperature was used. The lake surface water temperature was considered to vary with the air temperature (

T_{a}) by a constant quantity Δ

T, using the relation:

This simplification holds, since for such a shallow lake, a low heat storage value in the lake and thus a similar evolution of the air and water temperatures can be expected [

4].

${Q}_{r}$ and

${Q}_{s}$ are the specific humidity (%) at a reference height and close to the water surface respectively. The latter is approximated, assuming the near surface air temperature and the water surface temperature are the same and the relative humidity close to the surface is 100% (saturated air) using the relation:

where

${e}_{sat}$ is the saturation vapour pressure (Pa) close to the water surface (Equation (10));

${p}_{o}$ is the atmospheric pressure at the lake surface, calculated by the relation:

where

${p}_{sea}$ = 101,325 Pa is the sea level pressure,

g = 9.81 m/s

^{2} is the acceleration of gravity,

R = 287 is the gas constant of dry air,

${T}_{c}$ is the average catchment temperature, and

$\mathsf{\Delta}h$ is the altitude difference between the lake level and sea level.

${M}_{v}$ and

${M}_{d}$ are the molar masses of water vapour and dry air respectively (g/mol), assuming that the masses of the water vapor

${m}_{v}$ and dry air

${m}_{d}$ are such that

${m}_{v}\ll {m}_{d}$. The optimum value of 50% for relative humidity was determined from model calibration.

#### 2.2.3. Evapotranspiration over the Groundwater Reservoir and the Catchment

Evapotranspiration over the ground is governed by several parameters including (but not necessarily limited to) the amount of water available in the surface layer, atmospheric conditions, the soil hydraulic conductivity of the soil, and plant type. Due to limited information about these variables in the study area, we used the empirical equations by [

45,

46] to estimate ground evapotranspiration. These relate evapotranspiration with precipitation and ground temperature as:

where

${E}_{a}$ and

${E}_{p}$ are the actual and potential evapotranspiration rate respectively (kg/y/m

^{2} or mm/year), P the precipitation (mm/year), and

T_{g} is average ground surface temperature (K). Since no measurements of catchment temperature were available,

T_{g} was determined relative to the reference temperature at Babati town (1350 m a.s.l.), by assuming the standard atmosphere with a temperature increase of 0.6 K for every 100 m decrease in altitude. Therefore,

${E}_{r}$ and

${E}_{c}$ in the water balance Equation (1) correspond to the actual evapotranspiration calculated at the mean altitude of the groundwater reservoir adjacent to the lake (1400 m a.s.l.) and at the mean altitude of the remaining upstream part of the catchment (1800 m a.s.l.) respectively, and adjusted by a calibration factor (

${X}_{c}$) to account for catchment dependency uncertainties [

47], such that:

with

E_{a} calculated at 1400 m a.s.l. and

with

E_{a} calculated at 1800 m a.s.l.

The factor

X_{c} was determined through calibration to be 1.15 (see

Section 2.3).

#### 2.2.4. Groundwater-Lake Water Interaction

Lake Babati water interaction with the surrounding groundwater reservoir also plays an important role in the water budget. The extent of the groundwater reservoir adjacent to the lake was determined from available topographical and geological information. The lake-groundwater interaction was modelled by assuming the change in groundwater storage depth

$(\mathsf{\Delta}{D}_{r})$ equals the change in lake level

$(\mathsf{\Delta}{D}_{l})$ from the net change in volume (

$\sum \mathsf{\Delta}V$) after each time step, such that:

with

$\mathsf{\phi}$ being the mean porosity of the aquifer (-).

The groundwater flux leaving the lake (

$G{W}_{f}$) was approximated by a Darcy flow, as

where

${k}_{h}$ is the aquifer horizontal hydraulic conductivity (m/s) and

${s}_{g}$ the groundwater gradient (m/m), assumed equal to the topographical slope downstream of the lake.

${A}_{LW}$ is the cross-sectional area perpendicular to the flow (m

^{2}), expressed by the product of the water depth in the lake (

${D}_{L}$) and the average width of the downstream edge of the lake. Based on hydrogeological maps showing sandy aquifer conditions in the Lake Babati vicinity [

48], the hydraulic conductivity (

k_{h}) and porosity (

$\mathsf{\phi}$) were set accordingly [

49,

50] to 50 m/day and 40% respectively.