The problem of filling a depression up to a pre-defined level can be envisioned as filling a hole in a true–false mask (whether or not the local topography is below the a priori chosen water level of the lake or ocean. There are several algorithms for solving this problem, generally known as flood-fill algorithms (Zhu et al., 2006). They are commonly known for their use in the “bucket” tool of several paint programs. A thorough description of the stack-based flood-fill algorithm is given by Zhu et al. (2006). The default algorithm as used by Zhu et al. (2006) starts with a “seed”, defined as a designated element from where neighbouring elements are flooded. In many hydrological applications, the seed will be a local minimum in a DEM. Beside the seed element, the algorithm builds a “stack”, which is an array listing the indices of the map elements. During each iteration of the algorithm, all stack elements are checked. If the elevation of a stack element is below the water level, the element is filled with water, removed from the stack and consequently all its neighbours are added to the stack, thus expanding the filled area outwards. If the element does not lie below the water level it is removed from the stack without any further action. The iterations are continued until the stack is empty and the elements below the water level are identified.

Once the horizontal extent of the lake has thus been determined, its volume is calculated by integrating the water depth over the calculated area.

A pseudo-code example of this algorithm is illustrated below. A step-by-step illustration of one iteration of the algorithm being applied to a simple true–false mask is shown in Fig. 1.

An example of a MATLAB implementation of this algorithm is provided in the Supplement, being script “fill_1km.m”.

The number of operations required for filling a hole with this algorithm is approximately proportional to the number of elements of the depression. This means that application to a very large area with a high resolution results in a long computation time, since doubling the resolution in the horizontal plane will quadruple the number of operations.

Determining the land–ocean mask of a DEM using this algorithm is straightforward, since the water level (the elevation of the water surface with respect to the Earth's centre) that determines the fill criterion for all elements is, by definition, at sea level. For a lake where the water inflow is not balanced by evaporation, the water level depends on the topography surrounding the depression. The water level is then equal to the elevation of the “spillover point” – the origin of the river that transports spillover from the lake to the open ocean. The only way to determine the location of this point, and therefore the water level of the lake, is to start at the chosen local minimum in the DEM and to iteratively increase the water level until this spillover point is reached.

In the case where evaporation balances inflow, such as for several large present-day inland seas, the water level should be increased iteratively until the integrated evaporation over the lake matches the inflowing water flux.

A pseudo-code example of this algorithm is illustrated below.

The threshold condition can be the overflowing of the lake into the sea, the total evaporation over the lake balancing total water influx, or any other logical condition.

Depending on the vertical resolution of the DEM and the accuracy required to determine the lake depth, such a calculation requires dozens to hundreds of iterations. This is not always a restriction for practical applications where one is interested in the water runoff pathways and drainage basins for a given DEM. However, when performing an ensemble of simulations, or a simulation where the topography changes over time, reducing the computation time can be crucial for the performance and feasibility of the application and optimizations are required. In the next paragraph we present several optimizations to the standard flood-fill algorithm as described above, which reduce computation time considerably for maps with a large area or a high resolution.

As the first example, we use different versions of the flood-fill algorithm to determine the land–ocean mask for a DEM of North America and Greenland (Amante and Eakins, 2009), using an oblique stereographic projection at 1 km resolution. The region covers an area of roughly 7000 by 5000 km, resulting in approximately 17 million ocean elements to be filled. This extremely large size is useful for testing the efficiency of different setups and is needed in order to model the pressure exerted by lakes in gravitationally consistent calculations during the evolution of ice sheets in North America.

The chosen horizontal resolution of 1 km is very high for such models. However, a lower resolution would overlook topographical features, such as small river valleys, that would limit the water level of the lake through drainage. This would lead to a systematic overestimation of the total water volume unrelated to the uncertainty of the modelled location of the ice sheet margin, which is usually not as high as 1 km. A comparison between the calculated water volume of a lake at 1 km resolution versus the calculated volume of the same lake at 40 km resolution showed that the difference can be as large as 20 %.

Creating the land–ocean mask depicted in Fig. 3 with the standard version of the flood-fill algorithm at a 1 km resolution takes approximately 82 s. This serves as benchmark to which all subsequent experiments will be compared.

We performed two series of experiments with block inspection at different resolutions. The first series considers a sequence of 40, 8, 4, 2 and 1 km resolutions. The second series considers a sequence of 40, 20, 10, 5 and 1 km resolutions. The results in terms of computational efficiency are presented in Tables 1 and 2, respectively.

Both the conversion and filling parts of the tables can be read as “coming from [row] resolution, going to [column] resolution”. For example (Table 1; highlighted in yellow), converting a 40 km map and stack to their 1 km equivalent takes 0.20 s. A 1 km fill starting with that map and stack will then take 27.54 s. The diagonal elements in the filling parts of the tables indicate the computation time of a fill at that resolution starting from a single seed element.

An overview of several possible resolution schemes is given in Table 3. It shows that the most efficient scheme (40, 5 and 1 km) is almost an order of magnitude faster than the default algorithm at 1 km resolution.

The total gain in efficiency from the improvements to the algorithm will of course depend on the details of the problem it is applied to. In the optimized algorithm, the number of elements that can be filled at the low resolution roughly scales with the area of the depression, whereas the number of elements that need to be filled at the high resolution roughly scales with the depression's circumference. The gain in computational efficiency therefore depends largely on the shape of the depression, becoming larger when the depression becomes more circular.

To illustrate this, the improved algorithm with a 40, 5 and 1 km resolution scheme was applied to several versions of the same topography field, passed through increasingly strong high-pass filters with a cut-off wavelength of 30 km, so that the remaining topographical features that determine the shape of the depression are all too small to be visible on the low-resolution map in Fig. 4.

The results of the experiment are given in Table 4. The area of the ocean basin and therefore the number of elements that require filling does not change much. However, the compactness factor C of the ocean basin, which is defined as C=A4π∅2 with ∅ the circumference and A the area of the basin, decreases sharply when the small-scale topographical features become more prominent.

This illustrates that the gain in computational efficiency from the block inspection depends strongly on the shape of the topography in which a depression must be filled. Natural topographical features generally have a vertical scale proportional to their horizontal scale, which leads to the relatively circular shapes of most natural lakes. This means that, for most natural topographies, the block inspection algorithm will substantially decrease the computation time required for filling depressions.

Similar to the previous experiment, we consider the lake formation in the North American region 30 000 years ago as a second example. At this time, large parts of the North American continent were covered by the Laurentide ice sheet. The depression left in the bedrock by the weight of the ice, combined with the mass of ice damming off the Hudson Strait lead to the formation of a massive proglacial lake over the area of what is now known as the Hudson Bay. The situation is shown in Fig. 5.

We apply the flood-fill algorithm to the same 1 km resolution topographic map, combined with an ice-thickness distribution, a bedrock deformation field and a geoid perturbation field (all generated by the ANICE–SELEN model – de Boer et al., 2014) to determine the size of the lake. These data fields have a 40 km resolution and are used to perturb the 1 km resolution DEM, such that the small topographic features are preserved. The 40 km resolution perturbation fields are mapped onto the 1 km resolution data fields using bilinear interpolation. The lake is found to cover an area of 3.5×106 km2, with a volume of 6.0×1014 m3 or about 1.7 m eustatic sea-level equivalent of water.

As a benchmark, we apply the standard flood-fill algorithm without any optimizations, using a 5 m depth increment. This means that the algorithm fills the depression up to a certain water level, increases the depth by 5 m and fills the depression up to the new water level from the same seed, until the lake overflows into the sea. This run costs 351 s (almost 6 min) to complete. This implies that for a 120 000 year period which is resolved every 5 years, which is a time step typically used to resolve the last glacial cycle in an ice-sheet model, computational time is excessive relative to the approximately 50 h such a simulation would otherwise require.

Performing the same test with a depth increment of 10 m takes 179 s. This is far less time because for each depth interval the complete lake has to be filled, the number of fills increases more or less linearly with the requested accuracy. A 5 m depth interval leads to an accuracy of about 3 % in the lake's volume, which is considered to be sufficiently accurate.

By implementing the “shoreline memory” improvement described in Sect. 2.2.2 to the algorithm we improve the performance in terms of computation time. As explained earlier, the improved shoreline memory implies that every element will only be checked once, yielding a drastic reduction in computation time, which is particularly relevant for large lakes. Filling the same lake at 1 km resolution only takes 23 s in the optimized case, a reduction in computation time of about 90 % with respect to the benchmark.

As a final improvement, we implement the depth estimation method. This method starts by filling the same lake on the 40 km resolution topography, in order to get a depth estimate. The resulting lake is depicted in Fig. 5. It has a volume of 6.6×1014 m3, overestimating the “true” 1 km volume by about 10 %. The 40 km estimate covers several river valleys visible in Fig. 6c, which should drain the lake. Since these valleys are only a few kilometres wide they do not appear on the smoothed 40 km resolution topography map. This causes the overestimation of the water level and the lake's extent and volume – hence a high resolution is needed. The resulting depth is reduced by an arbitrary amount of 20 % and a fill to this new depth is performed on the 40 km maximum topography field. The resulting lake, which will serve as a starting point for the final 1 km fill, is depicted in Fig. 6b.

The water level is then incrementally increased, while using the 1 km fill algorithm, until the lake overflows into the sea. The complete process takes 4.4 s to complete. This is about 5 times faster than the 1 km fill with shoreline memory and about 80 times faster than the default implementation, a reduction in computation time of about 99 % with respect to the benchmark experiment.

Note that the 20 % reduction in depth from the 40 km estimate is chosen somewhat arbitrarily. Depending on the smoothness of the local topography, it is possible that the 40 km estimate will be much more accurate. In such cases, it is justifiable to reduce the depth less, and hence improve efficiency even more. For this reason, it is advisable to check if the depth reduction from the 40 km estimate is enough to prevent overflow. If not, the depth needs to be increased even further before starting the 40 km maximum topography fill.

As a benchmark of our method, we compare the computational performance of our optimized algorithm to that of the commonly used drainage-pointer approach (Tarasov and Peltier, 2006). This method assigns to every map element a drainage pointer, indicating which one of its immediate neighbours receives run-off from that element. By following the run-off from an element until it reaches the sea, elements can be grouped together by the location where run-off reaches the sea, thus creating drainage basins.

We found that a MATLAB implementation of the first step of this approach, where drainage pointers are assigned to all high-resolution elements, required about five times more computation time than our optimized algorithm when used for determining the land–ocean mask as in Sect. 3.1. The reason for this is that the drainage-pointer approach needs to assign these pointers to every single high-resolution map element in order to work, whereas our method only needs to treat those elements that are beneath the water level and then only those near the shoreline at high resolution, thus limiting the number of operations.