## 1. Introduction

[2] For the past several decades, hydrologists have widely used digital elevation model (DEM) data for delineating watersheds, extracting stream channel networks [e.g., *Band*, 1986], and computing topographic parameters such as slope [e.g., *Jones*, 1998; *Zhang et al.*, 1999], flow direction [e.g., *O'Callaghan and Mark*, 1984; *Tarboton*, 1997; *Quinn et al.*, 1991], flow accumulation area, and topographic index [e.g., *Quinn et al.*, 1995; *Wolock and McCabe*, 1995; *Pan et al.*, 2004]. Except for slope, all other parameters can be computed only after the flow direction at each pixel is determined. In hydrologic applications, flow direction is defined as the direction(s) in which water flows out of a pixel. Most flow direction algorithms published in the literature can be classified into three types [*Pan et al.*, 2004]: single flow direction (SFD) [*O'Callaghan and Mark*, 1984], bi-flow direction (BFD) [*Tarboton*, 1997], and multiple flow direction (MFD) [*Quinn et al.*, 1995] methods. All of these algorithms determine flow direction based on elevation gradient measured outwards from a grid cell. Downward elevation gradients are defined as positive. Virtually all watersheds contain flat or depression (sink) pixels that may be actual or artifactual, where the maximum outward elevation gradient is zero or negative (uphill). The flow direction for such pixels cannot be computed based on local elevation information alone and a method for handling these flat or sink pixels is required. In addition to the flow direction problem, flat or sink pixels (hereafter, FS pixels) can produce problems in other calculated topographic characteristics: water flowing into FS pixels cannot flow out, resulting in underestimates of the catchment drainage area and the flow accumulation area for downslope pixels; if FS pixels are inside the channel network, the extracted channel network will be discontinuous; and zero slope for FS pixels will prevent calculation of some topographic characteristics, e.g., the topographic index [*Beven and Kirkby*, 1979]. Therefore, before flow direction and other important topographic parameters can be computed for hydrologic applications, all artifactual FS pixels in a DEM data set must be rectified [e.g., *Lindsay and Creed*, 2006].

[3] The fundamental objective of treating flat areas and depressions in DEMs is to enforce an outward flow direction for every FS pixel in the DEM spatial domain. The many methods that have been developed for treating FS pixels in DEM data sets can be classified into two general types. Methods of the first type determine flow direction in flat areas without altering pixel elevations. Methods of the second type recalculate the elevation at every flat or sink pixel. An early example of the first type of method to treat FS pixels was suggested by *Jenson and Domingue* [1988] (hereafter, JD method). The JD method raises elevations in flat areas to the level of the lowest boundary pixel with an outflow direction, and then assigns a flow direction for each FS pixel based on the shortest flow path to this lowest boundary pixel. However, the JD method only redefines flow directions at flat pixels without recomputing elevations (because after sinks are filled, they remain as flat areas). Because the JD method can efficiently determine flow directions for FS pixels, it has been widely used in GIS tools. For example, the “fill” and “flow-direction” functions in ArcGIS are based on the JD algorithm [*Ormsby et al.*, 2010]. However, the JD algorithm tends to produce unrealistic parallel patterns in the computed flow accumulation area (see Figure 11c, as an example), and thus in the extracted channel networks (see Figure 12c, as an example).

[4] Another example of the first type of method to treat FS pixels is the so-called “river burning” [e.g., *Hutchinson*, 1989; *Ehlschaeger*, 1989; *Soille et al.*, 2003; *Kenny et al.*, 2008; *Getirana et al.*, 2009a, 2009b]. In this method, a channel network GIS data layer is overlaid on the DEM, and then all flat and sink pixels are forced to flow to the nearest channel pixels without altering the elevation of any flat or sink pixel. The problem associated with this “river burning” method is that some flat and sink pixels can be located on hillslopes rather than near channels. If an algorithm forces flow in such pixels to the nearest channel, unrealistic linear patterns can occur in the derived channel network. Another limitation of this approach is that channel network GIS data are not always available. In such cases, channel networks must be extracted from DEM data.

[5] In addition to the commonly used JD algorithm and the “river burning” method, there are some other published methods belonging to the first type of method; e.g., *Chou et al.* [2004] applied the preference ranking organization method for enrichment evaluations theory to determine flow direction in depressions; *Wang and Liu* [2006] proposed the least cost search algorithm to treat depressions and determine flow direction; and *Zhu et al.* [2006] developed a neighbor-grouping scan method to assign flow direction over flat areas. However, because methods of the first type do not adjust elevations of flat areas, and the topographic index [*Beven and Kirkby*, 1979] is not defined for locations with zero slope (because slope is in the denominator), methods of this type are less useful for hydrological applications than methods of the second type.

[6] Methods of the second type adjust the elevation at every flat pixel. For example, the drainage enforcement algorithm proposed by *Hutchinson* [1989] can remove sinks or pits from DEM data using an iterative finite difference interpolation approach based on minimizing a terrain-specific, rotation invariant roughness penalty. The Topographic Parameterization (TOPAZ) method [*Garbrecht and Martz*, 1997; *Martz and Garbrecht*, 1998] adds or subtracts an arbitrary small number from the elevation at each FS pixel. This method is effective, but this approach for adjusting the elevation of FS pixels can introduce uncertainty. Recently, many alternative algorithms have been developed, e.g., *Soille* [2004] suggested an optimal method to treat spurious depressions in grid-based DEMs. *Grimaldi et al.* [2007] and *Temme et al.* [2006] developed algorithms for treating flat areas and depressions in DEMs using dynamic landscape evolution models.

[7] The objective of this paper is to recompute and replace elevation values inside each flat or sink area such that all grid cells have at least one positive downward slope in the outward direction. This approach uses linear interpolation between lower-elevation grid cells on the edge of each flat or sink area defined as outlets and higher-elevation values on the opposite side. Implementation requires an iterative solution to accommodate the irregular geometry of flat or sink areas and exceptions that arise. Linear interpolation across flat or sink areas provides a natural way to scale elevation adjustments based on the vertical scale of the surrounding topography, thereby avoiding the addition or subtraction of arbitrary small numbers that we regard as a disadvantage in some prior techniques (e.g., TOPAZ). The arrangement of this paper is as follows. Section 2 introduces the methodology and algorithms. Section 3 describes the application of our algorithm to treat flat areas and depressions in two artificially created DEMs and one real DEM. Comparisons among our algorithm, ArcGIS, and the TOPAZ tool [*Garbrecht and Martz*, 1997; *Martz and Garbrecht*, 1998] are also given in this section. Section 4 is a summary.