We generated flow-routing surfaces using FlowFill from DEMs of two study regions. The first study region includes a reach of the Sangamon River in Illinois, USA, located at 39.97∘ N, 88.72∘ W. The low-relief plains left behind in this postglacial landscape contain closed depressions that may impact hydrologic connectivity as a function of runoff . We resampled the 2.5 ft (0.76 m) resolution lidar DEM to 15 m resolution for our analysis. At several locations, bridges and other man-made structures cross the river channel at this study site. These are clearly visible on the lidar and artificially elevate the topography, creating blockages to water flow. These were manually removed using GRASS GIS before running FlowFill by digitizing the problematic bridges, converting these to null cells, and then performing a bilinear interpolation to populate these cells with more realistic values. The second study region was the Río Toro basin, located mainly in Salta Province, Argentina, around 24.5∘ S, 65.8∘ W. Its steep topography was shaped primarily by fluvial processes but has also been impacted by mountain glaciation, landslide dams, and tectonically driven isolation of much of the basin from the foreland (though it has since re-incised) . The DEM of this region was resampled to 120 m resolution from 12 m TanDEM-X data. The two landscapes differ in their topographic setting in terms of tectonics, glaciation, drainage integration, average slope, and spatial scale.

In these examples, we prescribed uniform runoff across each DEM in order to easily compare depression filling and hydrologic connectivity (i.e. drainage integration). We tested runoff inputs of 1 mm to 15 m in the Río Toro basin and 1 mm to 20 cm in the Sangamon River basin (Table ). For comparison, a typical storm event in the Río Toro basin drops ∼1 cm of rain the Supplement, which equals the minimum runoff value discussed here. This minimum amount of runoff was not capable of filling many of the depressions in the landscape, and therefore our calculations indicate that some significant segmentation of the Río Toro catchment remains regardless of the size of the rainfall event. The median daily rainfall on a day with rain near the Sangamon River is 3.3 mm, and the maximum recorded single-day rainfall is 99 cm (USGS 05590050: data from 1 October 2005 to 19 February 2019). This large range suggests that hydrologic connectivity in the Sangamon River basin depends on storm intensity.

We used FlowFill to fill depressions both at the Sangamon River reach (Figs.  and ) and in the Río Toro basin (Figs.  and ). We applied varying amounts of runoff to demonstrate differing levels of depression filling (Fig. ) and hydrologic connectivity (Fig. ). Both study sites contain persistent depressions that are unlikely to be permanently filled and connected via surface water, as well as smaller depressions that may be filled during modest rainfall–runoff events.

We varied the input runoff depth at the two sites, with a maximum value selected based on how much runoff was required to fill all depressions in each DEM, giving a result comparable to existing flood-fill algorithms. This maximum runoff depth was 0.2 m for the Sangamon River site (Fig. ), significantly lower than the 15 m runoff required for the Río Toro site (Fig. ). However, at the Río Toro site, most depressions were filled by significantly shallower runoff (0.1 m or less), and only a few large depressions persisted as we dramatically increased the initial runoff depth.

At both sites, deeper runoff fills more depressions, thus increasing hydrologic connectivity across the landscape (Fig. ). We define both drainage integration and hydrologic connectivity based on Strahler stream order (Fig.  and Table  ), which we calculated using the “r.stream.order” extension to GRASS GIS . The effect of variable runoff on drainage integration is more prominent in the Sangamon River landscape due to its larger proportion of depressions that can, if unfilled, significantly break up the drainage network. The deep runoff required to completely fill all depressions (Fig. ) or produce higher-order drainage networks (Fig. ) implies that either heavy rainfall or a long period of rainfall – which may or may not be plausible, depending on the climate and the hydraulic conductivity of the substrate – is necessary for the landscape to become fully hydrologically connected. Both landscapes contain a few larger depressions that persist once most other depressions have been filled, though based on their locations, they have somewhat less importance in setting overall hydrologic connectivity, which saturates at runoff values below the maximum required to flood all depressions (Fig.  and Table ).

The processing time for FlowFill varies depending on the selected starting runoff depth, the number of cells in the domain, and the topographic structure of the site. Runtimes for our test cases varied from 0.97 to 32.42 min (Table ). We performed each calculation using eight processors on an Intel i7-5820K CPU (3.30 GHz) on a desktop computer running Ubuntu Linux with 64 GB DDR3 RAM and a solid-state hard drive.

Due to the slight overfilling of some depressions in the outputs from FlowFill, a correction was performed, as discussed in Sect. . In the two sample study regions discussed in this paper, the volume of the adjustment for overfilling was insignificant, ranging from 0.003 % to 0.29 % of the total volume of water stored on the landscape. Cases in which the supplied initial runoff was deeper tended to have a slightly higher proportion of overfilling.

In addition to these two study sites, we used FlowFill on a subset of the Sangamon study site at four different resolutions in order to assess how the resolution of the input data affects the results. The results of this analysis can be seen in Figs.  and . A small subset of the Sangamon study site was selected in order to keep runtimes manageable at higher resolutions. The resolutions selected were 0.762 m (2.5 ft, the original resolution at which we obtained these data), 3, 5, and 15 m to match the resolution used for the entire Sangamon study site.

The results show that resampling data to a different resolution has an impact both on the number and morphology of depressions in the unfilled DEMs and on the results obtained from FlowFill. When resampling to coarser resolutions at this site, the number of large depressions in the study area visually appears to increase, as seen in Fig. . However, Table  shows us that the total number of depressions actually drastically decreases. This is due to the abundance of small depressions in the higher-resolution data. Instead, the total area of depressions present increases in lower-resolution data: the total area of depressions present in the unfilled DEM at 15 m resolution is almost 70 % more than the area of depressions at 0.762 m resolution. This trend is also reflected in the intermediate 3 and 5 m resolutions. While these coarser resolutions have resulted in higher depression areas, the smoothing effect of resampling has also resulted in depressions becoming shallower, and hence total depression volumes are smaller at coarser resolutions. This effect is less consistent: 15 m resolution exhibits the lowest total depression volume and 0.762 m has the highest, but the 5 m resolution DEM has a higher depression volume than the 3 m resolution DEM. Systematic changes in the shape of the landscape occur as resolution changes, but not all of these changes relate linearly to the change in resolution. All of these differences in depression number and morphology at different resolutions are an important reminder that the results of any landscape study based on remotely sensed data such as this are limited by the accuracy of the input data, and any preprocessing steps may change this accuracy. The effects of different resolutions on depression storage are discussed in more detail by , and discuss the effects of different resolutions on watershed delineation and streamflow prediction.

We completed 12 runs of FlowFill using DEMs with each of the four resolutions and with starting runoff depths of 0.2, 0.01, and 0.005 m (Figs.  and ; Table ). Regardless of the resolution, 0.2 m of starting runoff filled all of the depressions, while lower amounts of runoff left some depressions unfilled. Overall patterns in depressions filled appear visually similar at all resolutions, with 15 m resolution showing the greatest difference from other resolutions. At 15 m resolution with 0.01 m of runoff, several of the larger depressions on the southern edge have been filled, while these remained unfilled at finer resolutions. Drainage patterns also follow similar patterns at different resolutions, with the dominant river channels visible at all four resolutions. Channel widths are inflated at coarser resolutions due to the larger cell size (Fig. ). Flow-routing pathways also differ between the 15 m resolution DEMs and those at finer resolution. When depressions are fully filled in the finer-resolution DEMs, a channel in the northeast corner of the DEM flows off the southern edge of the map. At 15 m resolution, however, the head of this channel is diverted and flows off the eastern edge of the map.

Using data at the highest available resolution prevents the loss or distortion of the data and ensures that analyses are not introducing any additional errors due to downsampling. Unfortunately, runtimes for FlowFill for large datasets can become prohibitively long. This is a limitation of FlowFill, and more computationally efficient methods for dealing with depressions in flow-routing surfaces are needed. We begin to address this problem in a companion paper .

Runtimes for this subset of the Sangamon site (Table ) ranged from 2 s to 13 min for 15, 5, and 3 m resolutions with all input runoff depths, but they escalated drastically to over 33 h for 0.762 m resolution with 20 cm of initial water depth. The reason for this nonlinearity lies in the fact that FlowFill moves water from cell to cell (Fig. ). Increasing the resolution increases the total number of cells that must be calculated and requires more iterations of cell-to-cell water exchange for the water to move the same real-world distance. Runtimes times per cell are given in Fig. .

We present FlowFill results at two locations with differing landscape characteristics (Table ). The results include flow-routing surfaces with selectively filled depressions and the associated changes in the drainage integration of the landscapes. A visual inspection of stream networks in cases in which lower runoff values are used tells us that hydrologic connectivity is lower in these cases. This is quantitatively supported by the Strahler stream-order data shown in Table . More higher-order streams occur when deeper runoff is used to create the flow-routing surfaces, hence filling more depressions. Channels were extracted using the FlowAccumulation functionality in RichDEM with a threshold of 100 units of accumulation.

The impact of using different amounts of runoff within FlowFill on the hydrologic connectivity of the landscape was more apparent in the Sangamon River basin study site, where more depressions were present. The presence of these depressions significantly reduced connectivity between stream segments. It is likely that FlowFill will be most useful in cases in which the geologic and geomorphic history of a landscape produces a surface with many depressions, such as this postglacial landscape. The high number of depressions seen may also be partially due to the finer resolution of these data relative to the Río Toro study site – the density of depressions has been shown to relate to grid spacing as an inverse power law – but this does not belie the finding that significant real depressions exist and impact hydrologic connectivity.

The ability to use FlowFill with varying user-selected starting runoff values makes it ideal for comparing network connectivity in wet vs. dry seasons (Fig.  and Table ) or for analysing the effects of storms of different sizes. Shallow runoff inputs to FlowFill imitate real-world conditions with low amounts of rainfall (e.g. during the dry season). During these times, hydrologic connectivity is significantly reduced, and routing flow across a completely depression-filled landscape becomes unrealistic. Deep runoff inputs simulate wet seasons or flood conditions. Due to the associated greater hydrologic connectivity, more of the region contributes water to basin outlets.

FlowFill is a cellular automaton that models water flow across landscapes. While we designed FlowFill to fill depressions on digital elevation models in a way that conserves water mass, rather than to reproduce a physics-based transient flow response, we propose that its mechanism of moving flow between cells may be useful for modelling applications. The amount of water moved between cells at each iteration is gradient based, meaning that FlowFill can approximate real transient flow in the landscape as a result of both topography (body forces) and water depth (pressure forces). The outputs of FlowFill diverge significantly from a true flow solution, for example the backwater equation, in that a parcel of water in FlowFill can move at most one cell per iteration, regardless of the underlying slope. Furthermore, FlowFill cannot accommodate different roughness values that would modulate flow velocity in one region versus another. With these limitations in mind, it could still provide a useful approach for simulations with approximately constant roughness and in which differences in elevation are consistently less than the flow depth. Fortunately, such examples are common in geomorphology and include reduced-complexity approaches towards simulating the dynamics of braided rivers and river deltas . With these uses in mind, users can view intermediate (pre-equilibrium) result outputs from FlowFill after a set number of iterations or at frequent intervals. The compute time for an individual iteration of FlowFill ranged from 10-3 to 10-6 s for the regions discussed in this article.

While we have created a way to handle the problem of real-world depressions in a DEM, FlowFill does have some limitations. Firstly, FlowFill requires significantly more compute time than flood-fill methods that fully fill DEM sinks . Secondly, it creates flat areas in DEMs, which present their own challenges for flow routing. Thirdly, the threshold value for |Δhmax| is distinct in different landscapes and as such is a user-defined parameter. Finally, if an input topography contains three-dimensional structures such as bridges, FlowFill can cause water to artificially dam behind them.

FlowFill can be time-consuming to run, especially for large DEMs, high starting runoff depths, or relatively flat study sites. Runtimes for each of the results shown here are shown in Table , with the compute time per cell shown graphically in Fig. . This may make it an unappealing choice in cases with large study areas or very-high-resolution data. Therefore, while FlowFill can route runoff to create flow-routing surfaces, a more computationally efficient solution to the problem outlined in this paper will permit faster analyses of a wider range of DEMs.

Like some other depression-filling algorithms, FlowFill produces flat areas where it fills depressions. Post-processing these into a gentle slope may be required in order to create reasonable or visually appealing flow networks. Fortunately, tools to do so efficiently already exist. To produce our drainage networks for the stream-order calculations (Fig. ), we used RichDEM to impose a gradient on flat areas as the final step in constructing each flow-routing surface.

While a single criterion for convergence on a final flow-routing surface would be ideal, we were unable to find one and instead have left this as a user-selected parameter. We attempted to use the maximum amount of water moved between two cells in an iteration, the total amount of water moved in an iteration, the rate of change in the maximum amount of water moved between two cells in an iteration (averaged over various time windows), and the root mean square error of the linear regression that created the aforementioned slope. We also attempted each of these methods while normalizing for the initial amount of applied runoff. None of these approaches were able to collapse the response curves of flow over the landscape. However, we do observe that the maximum amount of flow between two cells in a single iteration asymptotes to a consistent value in each landscape. We therefore selected the exit criteria based on a time when the change in the maximum amount of flow between cells from one iteration to the next is below some small, user-selected threshold. It is necessary for a user to select this threshold value since the amount of noise after the plateau has been reached varies from one landscape to the next. As a result, we suggest that users who want to test multiple initial runoff depths first run FlowFill with a modest amount of runoff (to speed compute time: Fig. ) in order to create a maximum water depth moved vs. iteration curve like those shown in Fig. . From this, users may pick a threshold value for |Δhmax|. Selecting a value that is too large will cause FlowFill to exit before reaching this plateau, while a value that is too small will cause FlowFill to continue running for its maximum limit of 1 000 000 iterations, resulting in a long compute time. Based on our two study areas, suitable threshold amounts are landscape specific but appear to be agnostic to the amount of runoff selected for a given landscape. Following this approach, we chose thresholds of 0.01 mm (Sangamon) and 1 mm (Río Toro).

All depression-filling algorithms can produce “lakes” as artefacts. In these cases, the two-dimensional topography does not represent efficient three-dimensional flow paths – such as flow under bridges or through culverts . FlowFill is especially sensitive to these, as their damming effect can also create bottlenecks that significantly increase the number of iterations required to evacuate the water behind them, even when a narrow flow path exists to bypass them. Even after convergence, these areas often require the additional step to reduce overfilling discussed in Sect. . This common problem further motivates work to remove these artificial blockages from rivers in DEMs for flow routing .