The key issue with interpolating features for which orientation is important (e.g. channels) is the ability to incorporate direction into the method used to develop them from observations. Kriging – a method of interpolation for which the interpolated values are modelled by a Gaussian process – is often employed to create continuous surfaces from point data e.g.. The approach accounts for the statistical properties of observations within a local search neighbourhood using a variogram function . Using this, it is possible to incorporate various types of anisotropy within the basic framework . However, the method only holds when applied over regions sharing the same overall statistical properties whether that be, for example, the same geologic rock type or the same directional bias. When anisotropy is defined relative to a fixed Cartesian coordinate system, and where data are sparse, kriging is impractical for sinuous features with constantly varying direction such as channels see also. Specifically, dividing a region into areas of shared anisotropy (thus satisfying the assumption of stationarity within a search window) that are data sparse prevents the adequate population of the variogram with which to statistically model the region.

To enable interpolation across channel widths, one approach uses cross-sectional profiles, but to do so, typical channel sinuosities present a problem. As an intermediary step to interpolating sinuous channels in DEMs, several approaches have been developed to transform the coordinate system of a given channel – moving from Cartesian coordinate space to channel coordinate space – enabling removal of complex sinuosity and the creation of an artificially straight channel cf.. Channel space (sometimes denoted as s,n in the literature) differs from Cartesian space in that locations are transformed relative to their distance along the channel (s) and perpendicular to the centreline (n). Observations within channel space – a now-straightened channel – can be locally interpolated by considering a single direction as opposed to a continuously changing one. The interpolated channel can then be transformed back to Cartesian space. The approach breaks down, however, where multiple channels merge together at confluences. Furthermore, in the absence of sufficient observations, such an approach cannot be used alone to predict along-channel geometry without additional interpolation. For example, manual digitisation has been applied to individual channels to assist in the development of a realistic bed topography for Thwaites Glacier, West Antarctica . Additionally, channel straightening through coordinate transformation becomes difficult where channels manifest high levels of sinuosity or sharp changes in direction .

Further issues with regard to maintaining morphological characteristics of channels, in particular ensuring known depths are honoured, are apparent in low-resolution datasets particularly where interpolation methods are applied . Where resultant data products are to be used in modelling studies, honouring known maximum depths is key as incorrect values can adversely affect results – especially with regard to maximum and minimum elevations . To ensure true morphology is maintained, proposed a routine which initially interpolates glacial channels along a mean direction vector. Connectivity between points along the trough is then established, and the locations of gridded points are adjusted to be within the vicinity of the now-defined channel. Elevations are then mapped with minimum elevations being applied to adjusted points now in the channel. This “mathematical morphological” approach is effective in regions where observations (gridded or not) covering features of interest are available. The adjustment of gridded points to follow channel directions provides a succinct approach to avoid the constraints of regular gridding, which mask channel structures especially at lower resolutions. However, observations are required to identify channels, and application of the mathematical morphology approach becomes complicated in the case of multiple interconnected dendritic type networks.

Subglacial channels, which occur beneath grounded ice, are significantly easier to interpolate into DEMs than fjords as a mass conservation optimisation scheme can be applied . This approach is independent of traditional geostatistical interpolation methods. Bed elevation values are calculated from ice thickness values, which are derived from a combination of radar sounding measurements and surface velocity observations and of course using the assumption that mass is conserved along flow. Despite such an approach being useful for subglacial channels covered by grounded ice, this approach cannot be applied for regions of open ocean or non-grounded ice as is the case for fjords and cross-shelf troughs on formerly glaciated continental shelves.

Despite the approaches that have been developed to interpolate channels in DEMs, there are a number of recurring problems in applying these methods in the next generation of the Greenland DEM. In particular, all of the methods assume that there are at least some data from which to extend and predict the structure of a given feature. Furthermore, no method is explicitly designed to include or represent the known physical characteristics, in particular the cross-sectional geometry, of the particular type of channel system (e.g. u-shape of glacial; v-shape of fluvial), with morphological information only being extended from available observations. Thus, there remains a disconnect between the presented frameworks and cases where features (1) are known to exist; (2) are assumed to conform to a structure related to the processes by which they were created (e.g. an assumed u-shape in the case of fjords where no other data are available); and (3) have no observations available to provide geometric constraints. A framework for fjord channel systems which addresses these issues, and can be applied to a large area such as the Greenland coast, must be able to

impose morphological geometry to features of known process origin;

Stochastic models of bathymetry have long been employed to abyssal-hill features in the deep ocean e.g.. In such places, stochastic models are appropriate for use because the frequency power spectra of deep-ocean bathymetry follow well-defined parametric relationships . Specifically, the high-frequency tail of the power spectra is characterised by power law relationships (i.e. the Brownian regime, which can be stochastically modelled), with lower-frequency behaviour characterised by a flat region of the power spectrum (i.e. the white regime, which cannot be stochastically modelled) . This spectral behaviour is common across other types of natural terrain, and consequently spectral analysis of natural terrain often focuses upon establishing the transition between high- and low-frequency behaviour, and the characterisation of the high-frequency power law relationships . Whilst the spectral properties of mid-ocean bathymetry and subglacial channels have been assessed, to the best of our knowledge this has not been done for fjord bathymetry. As part of this study, we use data that are available from surveyed fjords to constrain the stochastic models of the bathymetry of many Greenland fjords.

The ability to map a given fjord where no observations are available requires the provision of a skeleton mesh, which hinges on the presence of a centreline – an imaginary line that is equidistant from the two fjord edges. Consequently, the first step in the synthesis of a given fjord's geometry requires a centreline to be defined. Approaches exist for automatic centreline identification for glacier surfaces e.g. as a means of avoiding manual digitisation. Such applications are, however, informed by the availability of a glacier surface elevation DEM. An equivalent non-geomorphologically based method includes the definition of the medial axis cf. or topological skeleton and is frequently used in image processing and computer graphics applications see. Various packages are available to calculate topological skeletons e.g.. However, these algorithms are based purely on an input image and are sensitive to image pixel resolution. For our intended application, this can result in the development of a centreline (or skeleton) with multiple branches along a single channel feature.

The centreline mapping method that we introduce allows fjord systems with multiple branches to be accounted for. Each centreline extends from a predefined seed point (or points) at the head of the fjord, ending at a predefined end zone (e.g. the fjord mouth). The centreline itself is defined by a series of points or vertices, each with a unique identifier. Fjord confluences and the implementation of network structure are described in Sect. . We define the centreline as being any path between a seed and the end zone which minimises the path length whilst maximising the distance of the path from the fjord walls. This removes the issues of multiple side branches that arise using existing skeletonisation approaches. The algorithm incorporates direction and thus an aspect of evolutionary landscape process knowledge, which ensures that the centreline captures a leading-order feature from the landscape it represents. Furthermore, this approach ensures that the paths and vertices are given unique identifiers enabling them to be specifically referenced, which is important when defining the channel mesh (see Sect. ).

The generation of centrelines would be straightforward if we knew in advance the start and end points for each. In that case, we would simply compute a path that minimises the line integral J=∫CL(x,y)dc, where C is the entire centreline and L is some function that grows towards the channel edge. However, there are a large set of potential start points for every centreline, because we do not know ahead of time where any given channel should start. There are also a large set of potential end points, for the same reason and also because there may be more than one centreline originating at any given start point if the channel is forked.

Fjords were identified as channels between areas of land and ice leading towards the open ocean, identified here using the modified GIMP land classification product (see Fig. a). At the head of each fjord, multiple seeds were manually created from which to initiate a path. The end target was defined as a broad region rather than a specific point (in this case, the edge of the land classification mask). The following algorithmic steps were then undertaken:

Using the land classification mask, we calculate the distance of all locations between land/ice and ocean within the channel (d), from which the shortest distance of any location within the ocean from regions of land or ice can be identified (see Fig. ).

For each fjord centreline, points normal to each centreline vertex were defined up to the fjord edge taken from the GIMP land mask (Fig. ). The angle of the normal vector along which these new points were defined was calculated from its orthogonal relationship with the vector joining the neighbours of a given centreline vertex. To avoid an irregular distribution of new points in the interpolated profile at the channel edges, the points used to define the vector from which the normal was calculated were sometimes selected from more distant neighbours. This was particularly pertinent at more sinuous sections of a centreline. This smoothing of the profile is adapted from . Vertices normal to the centreline were calculated up to the mouth of the channel, at which point the fjord centreline was manually clipped.

Elevations were attributed to the point mesh by first constraining the seed and fjord-edge bed elevations through association with the nearest bedrock/bathymetric observation (from ice-penetrating radar where ice-covered see; from altimetry where bedrock was exposed ; or from bathymetric observations, OBS1516). This method ensured that at the head of the fjord three elevations were available – the two edges of the fjords (taken as the elevations at the first land locations encountered at the fjord edges) and a centreline elevation. For future applications along the Greenland coast where seed data are sparse, modelled estimates from the mass conservation optimisation scheme could be used.

Two approaches were taken to assign bed elevation values along the centreline of a given fjord. In both approaches, bed elevations were linearly interpolated between a known bed elevation at the head of the fjord (taken from OBS1516, Fig. b) and a known bed elevation at the mouth of the fjord – the mouth having been manually located and consistently used for all model runs. For the first run (SynthIBCAO), the bed elevation at the mouth was taken from the nearest IBCAO observation (20 km from the mouth of the fjord system depicted in Fig. ) and was set at -803 m. This was chosen for the first simulation as until recently IBCAO provided the most extensive bathymetric dataset for Greenland, and the distance from a fjord head to the nearest observation is often ∼ 10 s of kilometres. For the second run (SynthOBS), the gridded bathymetric observation from OBS1516 at the same position was used (-920 m; see Fig. b). Should high-frequency stochastic perturbations wish to be added along the profile (see Sects.  and ), they would be applied at this stage. Bed elevations up to the termini of most glaciers in Greenland, albeit predominantly modelled, are now available . We justify the use of the OBS1516 data for defining the elevations at the head of each fjord in the presented simulations as it enables a comparison of synthetic and observational data directly, removing the need to consider uncertainties inherent of modelled elevations.

In the absence of large-scale studies on fjord bathymetric geometry, we base our cross-sectional fjord geometry on the prior analysis of over 8000 glacially eroded valleys now exposed by interglacial ice sheet retreat . In their study, profiles were acquired from different glacial and geological environments, including valleys from the Southern Alps (New Zealand), the Pyrenees, and north and south Patagonia. For the valleys the elevation, Vd, was fitted to a power law relationship for the form Vd=α|w|β, where α is the form ratio (valley depth / valley top width), w is the distance along the cross section from the centreline (the position of which corresponds to w=0) and β is the power law exponent. Best fit parameters of α=0.20 and β=1.38 were obtained . A value of β=2 (i.e. a parabolic relation) follows .

Equation () assumes that a given fjord's cross section is symmetrical about the centreline, with the centreline as the deepest point – an assumption which usually does not hold exactly. Additionally, the fjords are often seeded with edge elevation data that are significantly higher on one side of the fjord than the other. To define the cross-sectional fjord geometry, we define a parabola of the form ax2+bx+c , where a, b and c are calculated based on the known elevations and relative locations of the edge and centreline points. This enables us to relax the constraint that the centreline must be the deepest point. Thus, the parabola used to define across-fjord geometry in this study is inspired by the analyses of but not a direct application of it.

Following the development of a complete fjord elevation mesh, a surface was then made for each fjord, with mesh point elevations being mapped to a regular grid, thus creating a continuous surface. The resultant grid was then masked using the GIMP mask, thus removing any values outside the extent of the point mesh which arose as a result of the regular gridding process. The individual fjord grids were then combined, from which a final grid of the minimum values (or maximum depths) was created. Thus, the lowest value at any location where two fjords overlap was retained . As a result, deeper grid values took precedence over those that were shallower. This approach coupled with the aforementioned setting of edge elevations at confluence locations avoids the creation of ridge artefacts in the final DEM. The fjord DEM was then integrated into the wider landscape DEM (Bed2013), which includes non-fjord regions. Prior to the merge, Bed2013 was masked, removing any values in the area occupied by the synthetic fjord(s).

In this section we describe spectral analysis methods used to constrain the fjord's statistical features and the inverse methods that can be used to generate a synthetic profile. Our analysis is based upon analogous analysis of abyssal-hill features in the mid-ocean , although it is simpler in the respect that fjord bathymetry is approximated as a one-dimensional problem. Using the centreline mapping approach presented in Sect. , centreline points were established, with vertices on a 150 m interval for nine fjords along the west Greenland coast, selected where mean gridded observations were contiguous along fjord centrelines according to OBS1516 (Fig. ).

The lengths of the nine fjord sections in our example are constrained by the length of the shortest fjord section (∼ 30 km). Elevations were extracted for each centreline vertex, providing one-dimensional centreline elevation profiles. Where a centreline contained missing data at a level no greater than 20 %, a cubic interpolation routine was applied to give a continuous elevation profile (Fig. ). Prior to performing the spectral analysis, each elevation profile was linearly detrended, which acts to emphasise the overall variation of the small-scale trends . Each elevation profile was then transformed using the numerical fast Fourier transform algorithm, converting it to the frequency domain . Power spectra for each fjord were then obtained from the square of the complex modulus, and arithmetically averaged over the nine selected fjord profiles, to create a composite power spectra. This arithmetic averaging approach is as described in for mid-ocean bathymetry and enables longer wavelength features to be statistically constrained along with the higher-frequency features that are repeatedly sampled.

Of interest in this study is demonstrating, in a proof-of-concept manner, how the composite power spectra for the fjords can be used to generate different one-dimensional realisations of synthetic bathymetry that are consistent with the overall statistical properties. In order to generate the different realisations of bathymetry, we use the inverse Fourier transform method outlined by (the sinusoidal approximation method described in Sect. 4 of their study). Their method was introduced in the context of generating one-dimensional random road profiles, which is a mathematical analogue of fjord profiles. In their formulation the Fourier amplitudes of each harmonic are determined by the power spectra of the profile, with stochasticity present via the random relative phase of each harmonic. Our only modification to their method is to use a different parametric form for the power spectra, which is motivated by our observed results described in Sect.  and is consistent with the generality of their method.

We consider areas of maximum over- and underestimation of bed elevation that are present in Bed2013 within the region covered by OBS1516. Bed2013 is a continuous DEM extending from the bed beneath the contemporary ice sheet out to the continental shelf in the ocean, with all bathymetric information derived from IBCAO. IBCAO was combined with the bed elevation component of Bed2013, with triangulation used as an interpolator to provide values where IBCAO was unconstrained by data . Triangulated portions of the resultant DEM were then smoothed using a 2 km window . Where there was an unrealistic offset between the two surface datasets (e.g. bathymetry was higher than the glacier bed), some areas were manually dropped to force them to adhere to a subjectively more realistic profile (i.e. a fjord would be lower than the glacier bed upstream of it). The result of differencing Bed2013 from the OBS1516 dataset is presented in Fig. a, relative to Greenland (Fig. b), with the frequency distribution of the differences presented in Fig. c. On average, Bed2013 underestimated the depth of OBS1516 by 115 m, for which a skewness of -0.7 from the difference frequency density distribution was identified. These overall dataset statistics obscure the regions of maximum depth underestimation which are focused within the fjords themselves. Absolute maximum under- and overestimates of OBS1516 by Bed2013 reached -1329 and 1077 m respectively. Regions containing these extreme values can be directly associated with portions of the IBCAO dataset that were unconstrained by observations and were themselves the result of triangulation and spline interpolation .

The first implementation of the synthetic fjord routine, SynthIBCAO, defines the elevation at the mouth of the fjord based on the nearest IBCAO bathymetric observation, with the elevation of the point at the head of the fjord taken from the OBS1516 dataset (Fig. a). Points normal to each centreline vertex were then calculated as described in Sects.  and . The resultant combined surface DEM with the inclusion of synthetically created fjord bathymetry, providing a new realisation of the bathymetry as in Bed2013 (Fig. b), is displayed in Fig. c.

The SynthIBCAO channel geometry is both deeper and more concave than Bed2013 (Fig. b), particularly with regard to the narrower fjord regions. Based on the contour pattern (Fig. c), these narrower fjord regions now display a deeper and more concave cross-sectional profile than was rendered in Bed2013. For the wider confluence region centred south from (-705, -1340) in Fig. c, there is a clear change in the overall depth profile, with SynthIBCAO reaching -731 m compared to -391 m in Bed2013. SynthIBCAO reaches a minimum bed elevation different to the defined elevation at the mouth of the fjord (-803 m) as a result of the regular gridding of the fjord mesh elevations described in Sect. .

Comparing the difference between Bed2013 and SynthIBCAO (Fig. a), the latter dataset has elevations consistently lower than the former. The mean offset between the two datasets was 274 m. The changes along the narrower portions of the fjords – up to ∼ 3 km from each respective fjord head (see Fig. b for mapped channel centrelines) – are relatively small (∼ 0–50 m). Larger offsets are apparent where fjords enter the confluence region centred south from (-705, -1340) in Fig. a, with a maximum offset of 547 m. The increased concavity of SynthIBCAO is well illustrated with a mean increase in depth along the confluence zone centreline of ∼ 370 m.

Subtracting SynthIBCAO from OBS1516 (Fig. b) reveals a mean offset between the two datasets of around 50 m. Relatively good agreement along the first ∼ 3 km of each fjord (see Fig. b for mapped channel centrelines) is displayed, and indeed portions of the main confluence region, with differences centred at 0 m. The main region of synthetic elevation overestimation (i.e. lower than the observations) is focused at the confluence point of fjord 1 (refer back to Fig. b for fjord numbers), with the region of overestimation focused around (-709, -1344) in Fig. b up to a value of ∼ 580 m. This overestimate is likely indicative of the presence of a sill-like feature in OBS1516. The two main regions of depth underestimation using SynthIBCAO are centred at (-704, -1338) and (-704, -1355) in Fig. b, with maximum underestimates of ∼ 385 and ∼ 358 m respectively. These underestimates possibly relate to the presence of overdeepening type features present in OBS1516.

For reference, a comparison with OBS1516 subtracted from Bed2013 for the same area of interest is drawn (Fig. e). As described in Sect. , Bed2013 consistently underestimates bed elevation. However, as with SynthIBCAO, the main areas of underestimation are focused at the same locations – namely (704, -1338) and (-704, -1355) in Fig. e for which overdeepenings are likely present.

The second implementation of the synthetic fjord routine, SynthOBS, defines the elevation of points at both the head and mouth of the fjord based on gridded elevations from OBS1516 at the same location. The resultant combined surface DEM with the inclusion of synthetically created fjord bathymetry is displayed in Fig. d. SynthOBS demonstrates deeper concave geometry across the fjords than Bed2013. The changing relief of the banks of the synthetic fjords are steeper than those rendered in the original DEM (Fig. b). Between fjords, there are also changes in the elevations of the ridges such as at (-706, -1330) in Fig. d. The differences between the synthetic and the original DEMs are further quantified by the difference plot illustrated in Fig. c.

The SynthOBS surface is generally lower than Bed2013 (Fig. c), with a mean offset between the two datasets of 316 m. The only locations where SynthOBS was higher than Bed2013 were at the edges of the fjords within ∼ 3 km from each respective fjord head (see Fig. b for mapped channel centrelines). This possibly highlights overly smoothed sections of Bed2013 (where it was combined with the IBCAO dataset – see ). As with the SynthIBCAO approach, the largest offsets are apparent as fjords enter the confluence region centred south from (-705, -1340) in Fig. c, with a mean offset in this region of ∼ 400 m.

When SynthOBS is subtracted from OBS1516 (Fig. d), the mean offset between the two datasets is ∼ -3 m. The spatial pattern is very similar to that described for SynthIBCAO, with the same regions of under- and overestimation being equally apparent. Specific values, however, differ. Maximum synthetic elevation overestimation focused around (-709, -1344) in Fig. d was ∼ 581 m. The two main regions of synthetic elevation underestimation (i.e. higher than the observations) using SynthOBS centred at (-704, -1338) and (-704, -1355) in Fig. d have maximum underestimates of ∼ 328 and ∼ 283 m respectively. Overall, the profile represented by SynthOBS is closer to OBS1516, as would be expected considering the elevation profile of each fjord was calculated between fjord head and mouth observations from the OBS1516 dataset.

Considering centreline profiles for all fjords, we illustrate the improvements made to the general elevation profile of each fjord (Fig. ) relative to those present in Bed2013, by considering the general agreement between the synthetic geometry and OBS1516. The synthetic realisations underestimate observed bathymetric elevation to a much lesser extent than Bed2013, capturing the generally sloping profile of OBS1516. The good agreement (approximately ±50 m) of synthetic–observed values along the first ∼ 3 km of each fjord – in particular for fjords 1, 2 and 5 – implies the presence of approximately linear profiles. Larger differences – indicative of where the synthetic approach performs less well – occur from ∼ 4 km along each centreline, which relates to the confluence region of the individual fjords (refer to Fig. ). Higher-frequency features (along-track peaks and troughs likely relating to sills and overdeepenings) are not captured using the presented synthetic fjord bathymetry generation approaches.

Following Sect. , we now consider the spectral characteristics of the fjord bathymetry along the centrelines of the nine fjords illustrated earlier in Fig. b, using the OBS1516 data. A log–log plot for the mean power spectra, S(k), where k is the wave number (linear spatial frequency), is presented in Fig.  (blue crosses). The power spectra exhibit an approximate power law relationship at higher frequencies (corresponding to a linear relationship in log–log space) and an approximate flattening at lower frequencies. A parametric model which captures this frequency transition is S(k)=F0kα+k0α, where k0 represents the approximate transition frequency between the high- and low-frequency regimes, α is the exponent for the high-frequency tail (for k≫k0,S(k)∝k-α) and F0 acts as a normalisation constant. The parametric model (Eq. ) is a generalisation of the model for the power spectra of ocean bathymetry in , which assumes α=2. In general, different types of natural terrain can exhibit a range of spectral exponents , and our parametric model is representative of this.

The parametric best-fit values were obtained using a non-linear least-squares solver and correspond to F0=17.6 m2 km-1, k0=0.069 km-1 and α=1.74 (Fig. , red solid line). The transition frequency, k0=0.069 km-1, corresponds to a transition wavelength of 14.5 km. This compares with a transition spatial frequency k0=0.025 km-1, and a transition wavelength of 40 km, for abyssal-hill features in the mid-ocean in .

Figure  shows two different realisations of synthetic fjord bathymetry using the parametric fit to the power spectra in Fig.  and the stochastic inverse Fourier transform procedure described in Sect. . The horizontal spacing of the synthetically generated profiles is set to be the same as the bathymetric data (0.2 km). If we draw a comparison between the stochastic model of synthetic fjord centreline profiles and the OBS1516 profiles (Fig. ), it is clear that the synthetic profiles do not contain the lowest-frequency oscillations (wavelengths ∼ 15 km or greater). This is consistent with the general flattening of the fjord power spectra at low frequencies. However, oscillations on a length scale ∼ 5 km (typical of sills and overdeepenings) are present in the synthetic profiles, although the specific locations of such features in these profiles are random.