1.1. Hydraulic Geometry Concept Applied to Deltas
 Hydraulic geometry (HG) is a set of empirically derived power law relations between the channels' width, mean depth, and mean flow velocity, and the discharge conveyed by the channel [Leopold and Maddock, 1953]. In river deltas, channel geometry scales according to the downstream HG relation logA ∼ β logQ, where Ais channel cross-sectional area,Q is water discharge conveyed by the channel, and β is an exponent typically lying in between 0.8 and 1.2 [Edmonds and Slingerland, 2007]. In tidal systems, the exponent β often shows the same range of variation, but the tidal prism or peak tidal discharge is used instead of a discharge with a constant frequency of exceedance [Friedrichs, 1995; Rinaldo et al., 1999; D'Alpaos et al., 2010]. Channel geometry in tidally influenced river deltas can show a mixed scaling behavior between that of river and tidal channel networks, as the channel forming discharge is both of river and tidal origin. As a consequence, tidal processes play a prominent role in the morphological evolution of tidally influenced river deltas [Geleynse et al., 2011], which may also have an impact on the response of the delta to permanent changes in river discharge [Edmonds et al., 2010] and ultimately on its evolutionary structure [Wolinsky et al., 2010]. Although tidal effects on delta morphology can be studied by adopting a process-based morphodynamic modeling approach [van der Wegen et al., 2011; Geleynse et al., 2011], studies on HG relations may help to acquire a synoptic insight into the morphology of delta channel networks affected by tides, and provide the basis for idealized models of delta evolution [e.g., Kim et al., 2009].
 In this paper we show that the traditional tool of hydraulic geometry can be used to map tidal hydrodynamic processes onto a delta network, which can bridge the gap between physical oceanographic research on tides and the geological literature on river deltas. Traditionally, researchers use tidal amplitude or tidal prism to quantify the tidal influence on a delta [e.g., Syvitski and Saito, 2007]. However, to asses the degree in which channel morphology is influenced by the tidal motion throughout a delta, which may be used to improve the common classification based on the tripartite division between river-, wave-, and tidally dominated deltas [Galloway, 1975], a more objective and reliable set of metrics is required. Hydraulic geometry provides a set of relations for describing the dominant controls over delta channels synoptically, linking explicitly process and form.
 Inside a delta channel network, bifurcations are key elements controlling the division of water and sediment discharge over downstream channels. Flow division at river bifurcations has been investigated intensively with theoretical models [Wang et al., 1995; Bolla-Pitaluga et al., 2003], with numerical models [Lane and Richards, 1998; Dargahi, 2004; Zanichelli et al., 2004; Kleinhans et al., 2008], and on the basis of flume experiments [Bertoldi and Tubino, 2007]. In tidally influenced deltas, however, tides intruding at the mouths of distributaries complicate significantly the processes governing flow division at tidal junctions [Buschman et al., 2010; Wu et al., 2010; Sassi et al., 2011b]. Frings and Kleinhans  presented a comprehensive data set on sediment transport and hydrodynamics at three tidal junctions in the River Rhine, showing complex variations in sediment transport during a flood wave. They observed a poor correlation between sediment fluxes and river discharge. In a tidal junction of the Sacramento River, Dinehart and Burau observed that velocity patterns during ebb and flood can be highly asymmetrical. Asymmetry of the tides and tide-induced residual circulations may exert a significant impact on the division of river discharge over the distributary channels in tidally influenced deltas [Buschman et al., 2010; Sassi et al., 2011b]. The inclusion of tides in the HG concept and the application to delta channel networks will allow quantification of the degree to which tides influence the variation of river discharge throughout the network [Singh et al., 2003; Dodov and Foufoula-Georgiou, 2004; Eaton and Church, 2007].
1.2. Tidal Processes in River Deltas
 Tidal rivers are intrinsically complex, as tidal propagation is influenced by river discharge and vice versa. Tidal waves propagating upstream become distorted and damped, which is caused both by bottom friction and by the river flow [Godin, 1999; Horrevoets et al., 2004; Buschman et al., 2009]. Adopting the hydrological perspective, tides impact the river flow by inducing fortnightly variations, which are generated by variation of the tidally averaged friction over a spring-neap cycle. At spring tide, high levels of tidally averaged friction act to block the river discharge, increasing water depth and allowing discharge waves to be admitted during neap tide. These effects of the spring-neap cycle can extend far upstream from the estuary [LeBlond, 1979; Godin, 1991].
 In tidal rivers, the along-channel tidally averaged friction is mainly balanced by a subtidal pressure gradient [Buschman et al., 2009]. As a consequence, river-tide interaction induces a water level setup, which becomes progressively larger in the upstream direction [LeBlond, 1979; Godin and Martinez, 1994]. At a bifurcation, a mismatch may occur between the water level setups that would develop in the two channels if they would have been disconnected. The water level setup in the channel where river-tide interaction and the associated propensity for water level setup is largest will promote the allocation of river discharge to the other channel.
 Tides can affect river discharge division in one other way. The Stokes transport, which is the drift associated with a traveling wave that can be calculated as the Lagrangian mass transport minus the Eulerian mean, can be different in two adjacent channels that join at a bifurcation. Based on an idealized model, Buschman et al. found that asymmetries in subtidal flow division at the apex of a tidal river splitting over two sea-connected branches were enhanced when one of the sea-connected branches is deeper or shorter. In their study, bed roughness differences resulted in the opposing effect.Sassi et al. [2011b] elaborated on the work by Buschman et al. , showing that differences in water level setup may play a key role in the division of discharge at tidal junctions. The division of river discharge at tidal junctions leads to variation of the HG exponents, and helps to explain the complexity in HG relations of mixed river-tide dominated deltas.
 Studies of estuarine morphology tend to focus on overtide generation to describe the unique geomorphological characteristics of tidal environments [e.g., Wang et al., 1999]. This stems from the fact that landscape-forming discharges in tidal systems are highly influenced by peak discharges, which are typically controlled by tidal asymmetry [Lanzoni and Seminara, 2002; Fagherazzi et al., 2004]. Tidal asymmetry is generally attributed to non-linear interaction of the main semidiurnal tide with itself [Friedrichs and Aubrey, 1988]. Persistent asymmetrical tides may also be produced by the interaction of diurnal and semidiurnal constituents in tidal regimes where both semidiurnal and diurnal tides contribute significantly to the tidal motion [Hoitink et al., 2003]. Friedrichs  indicated that in stable channels the minimum shear stress τsnecessary to maintain a net zero gradient in the along-channel sediment transport leads to a convergence point, from whichτs decreases both seaward and landward. Downstream HG relations of the area of such channels will then exhibit exponents (β) greater than unity in flood-dominated systems, and values ofβsmaller than unity in ebb-dominated systems.Rinaldo et al.  suggested that ebb/flood transitions in tidal channels are marked by a break in the slope of the HG relation of the area, which may be interpreted as downstream variation in β. This notwithstanding, we argue that processes such as differential water level setup driven by river-tide interaction [Sassi et al., 2011b], which need weeks to develop rather than days, exert a strong control on the morphological evolution of tidally influenced river deltas, via the re-distribution of river discharge at bifurcations.
 Here we focus on the River Mahakam, which constitutes the major navigable river in East Kalimantan, Indonesia. The River Mahakam flows through a relatively flat basin characterized by a very mild slope. At about 150 km from the river mouth, an alluvial plain marks the transition to the upper reaches of the catchment, where a system of interconnected lakes with a total area of about 400 km2 is located. Water level fluctuation induced by the tide has been observed upstream of the lakes region [Hidayat et al., 2011]. It has been suggested that tidal processes may have a dampening effect on the fluvial dynamics of the delta region, causing a characteristic progradation pattern [Allen et al., 1977]. Recently, the absence of discharge peaks was also ascribed to the non-flooding discharge regime resulting from the buffering effect of the lakes [Storms et al., 2005; Hidayat et al., 2011]. While flood flows up to 5000 m3 s−1 can cause a rise in water level up to five meters in the upper reaches, flood surges are virtually damped by the buffering effect of the lakes [Storms et al., 2005; Hidayat et al., 2012], which eliminates sudden and large variations in river discharge in the lower reaches of the river. Indeed, the lack of channel migration [Allen et al., 1977] and the absence of channel avulsions [Storms et al., 2005] support these two hypotheses, rendering the Mahakam delta channel network virtually fixed. The relatively constant river discharge in the downstream reaches of the Mahakam allows us to investigate the subtle processes of river-tide interaction, which are often obscured by short-term and high-magnitude tidal processes, and/or fluvial instability. Since the Mahakam river discharge typically fluctuates at timescales longer than a fortnight, the system can adjust to subsequent quasi-equilibrium states (in the order of months), greatly simplifying the complex dynamics that river-tide interactions impose.