For each meander bend a lithogenetic cross section was constructed based on multiple hand corings from meander base to apex (Fig. 1b, c), following a line perpendicular to the scroll bars, with cores obtained at high and low points in the scroll bar and swale topography. In a separate paper, Candel et al. (2018) provide details of these cross sections (including detailed facies descriptions and an explanation of the lithogenetic interpretation) and use the information to reconstruct palaeo-discharges. Here, we concentrate on those aspects that are directly relevant to the reconstruction of channel position throughout the lifetime of the meander. Most of the cores reached a depth of 4 to 6 m, penetrating the full point-bar deposits (extremely fine to coarse sand) including the channel lag (medium fine to coarse sand with gravel), which occurred at depths varying from 2.5 to 4 m below the surface. Some aeolian reworking occurred locally, as is evident from the DEM (Fig. 1). The lithological and geomorphological information was used to select the suitable depths to collect OSL samples (see below).

Maps show the position of the river channel at the time of the map survey; however, the use of this information for quantitative reconstruction of channel position over time is not straightforward. First of all, historical maps are likely less accurate than their modern counterparts. Hence there is an uncertainty in the exact position of the channel. In this section we outline how we quantify this spatial uncertainty. Next, to allow Bayesian analysis of the sequence, this spatial uncertainty must be translated into a temporal uncertainty. The necessity of this step and the approach we take are outlined in Sect. 3.4.

OSL dating makes use of a low-intensity light signal emitted by mineral grains (e.g. quartz) upon optical stimulation. The intensity of the signal can be linked to the age of the sediment. The (latent) OSL signal builds up as the grains are exposed to natural ionizing radiation in the environment (from uranium, thorium, potassium, and cosmic rays; known as the environmental dose), which causes the accumulation of trapped charge in the crystal structure. When the grains are exposed to daylight during transport, the trapped charge is released and the OSL signal is reduced (bleached) to a low level, often close to zero. When the grains are deposited and buried, they are no longer exposed to light and the signal can build up again.

Through a comparison of the natural luminescence signal with that induced by laboratory irradiation, the environmental dose received by the grains since burial can be estimated (the palaeo-dose). Combined with information on the environmental dose rate experienced by the grains, the age can be calculated by age (ka) = palaeo-dose (Gy) / dose rate (Gy ka-1) (see, e.g., Rhodes, 2011 for more information on OSL dating).

With dating of fluvial sediments, the assumption in luminescence dating that the OSL signal was completely zeroed prior to sediment burial is questionable (Murray et al., 1995) due to the limited exposure of grains to light when transported in turbid water (e.g. Wallinga, 2002a). This means that at the time of sediment burial, some trapped charge that was built up since the previous deposition may remain, which may consequently lead to a remnant OSL signal upon deposition and an overestimation of the age. Poorly bleached sediments mostly consist of a mixture of poorly bleached and well-bleached grains (Olley et al., 1998), which can be detected by broad and skewed equivalent dose distributions (e.g. Wallinga, 2002b). Heterogeneous bleaching in channel site samples may result in apparent doses ranging over several orders of magnitude and may limit the accuracy of dates (Murray et al., 1995; Olley et al., 1998), although methods have been developed to largely overcome these challenges (e.g. Galbraith et al., 1999; Cunningham and Wallinga, 2012).

Bayesian statistical approaches were used to combine the numerical age data obtained from the historical maps and OSL dating, with auxiliary data on the stratigraphic order. This analysis was performed using the freely available OxCal software (version 3.4; main introduction to the programme described by Bronk Ramsey, 1995; see also Bronk Ramsey, 2009). Bayesian analyses of deposition sequences are normally based on age versus depth profiles (Bronk Ramsey, 2008). Here, we analyse age as a function of lateral migration distance, where the location of the oldest swale (meander base) is taken as the origin and distance is measured along the central axis of the scroll bar deposits. For each of the age constraints (OSL sample age or survey age of a historical map), we have both an age and a location along the central axis. In addition, we know that the meander grew from its base to the final position, providing an order for all information. In the Bayesian approach, all this information is combined to obtain a robust reconstruction of the channel position through time.

To jointly analyse the OSL priors and the historical map data, both were added as entries in a Bayesian deposition model using the P_Sequence (Fig. 3). This command fixes the chronology of the sampling points using the known order of events (based on the known meander expansion direction) and makes use of a Poisson distribution. With this depositional model, channel migration is assumed to be random (i.e. varying in rate but always positive) giving approximate proportionality to distance (factor z) (Bronk Ramsey, 2008). The P_Sequence model needs information on the size of the units (deposition events) within the sequence. Here, we chose to use the average dimension of a scroll bar as unit, i.e. the mean distance between two swales. Based on meander amplitude and the visual counting of scroll bars from the DEM, this distance was estimated to be 77 and 42 m for Junner Koeland and Prathoek respectively. This provides an event spacing (k0 value) of 0.013 events m-1 for Junner Koeland and 0.024 events m-1 for Prathoek. By applying the regular boundary function, the entire distributions were allowed to be used in the calculation.

Likelihood distributions of estimated burial age (see Sect. 3.3.4) were uploaded to the OxCal directory as priors (*.prior file), with the unit in years CE. For this OSL age data, the position is relatively well known (∼5 m), while there is considerable uncertainty in the time domain (age estimate).

For the historical map data we assume the age is known exactly (survey date; see Table 1), while reconstructed locations are uncertain (up to 107 m; see RMSEs in Table 1). Incorporating this data into the Bayesian model requires uncertainty to be expressed in the time domain rather than spatial domain. Moreover, a slight correction is needed as the OSL data provide age information on the scroll bar formation (i.e. inner bank), while the channel centre line was reconstructed from the maps. Channel width varied significantly between the historical maps and may not represent the true low-water channel width of past times. The historical river channel was therefore deduced from each map as a low-water channel centre line. To make sure that the age of the map is assigned to the inner bank instead of the channel centre in the chronological analysis, we corrected the distance (z value) for each centre line using half the channel width. For this calculation, we assumed that channel width has been relatively stable throughout meander formation and is represented by the width of the abandoned channel (∼30 m) as derived from the DEM (Fig. 1).

Uncertainty in the spatial domain was calculated for each map based on σ (see Sect. 3.2.2) and assuming a normal distribution. To transpose this spatial uncertainty to an age uncertainty, the migration rate of the channel is required. This provides a problem, as the procedure aims at a reconstruction of channel position through time, which is needed to determine the migration rate. Nevertheless, a crude estimation of migration rate is possible based on the available data, and an iterative approach can be used to improve this estimation with the model output. For the first iteration error bands were converted to years using an estimated migration rate based on the bsMAM age of the OSL sample closest to the meander base and historically known maximum channelization date (1894 and 1906 CE for Prathoek and Junner Koeland respectively), when the channel reached its final position. For subsequent iterations, migration speed was calculated for each map individually by dividing the difference between the modelled mean ages for its error boundaries (in years) by the error width (in m). This migration rate was then used to convert RMSE (available in Table 1) to years. The converted RMSE (i.e. σ) was used together with the map age (i.e. the mean) as input for the next iteration. This process was repeated until the model outcome converged and remained stable. This required four iterations of the Bayesian model for both investigated meander bends.

Queries were built into the model to generate output for the requested distances, including the meander base (calculated by OxCal using extrapolation of the model; consequently, the uncertainty for this point will be larger) and the spatial boundaries of each error band to allow the calculation of a migration speed for each historical map as described above.

Upon convergence and stabilization of the model outcome, the deposition model of the final (i.e. fourth) iteration was used to interpolate ages throughout the scroll bar deposits. Using the graphical options in OxCal, interpolation results were plotted versus distance from the meander base as the origin toward the meander apex using linear interpolation in between the data points. Plots of the input data were created with the ggplot2 package (Wickham and Chang, 2016) employed in R (version 3.4.0).

We performed sensitivity tests to check model sensitivity to (1) the initial migration rate that was used to convert error bands to years, (2) the correction for channel width, and (3) a rigid versus variable k0 value. Test outcomes indicated that (1) either increasing or decreasing the initial migration rate by a factor of 2 yielded similar results after several iterations, (2) model outcome with and without using the channel correction differed only by a couple of years, and (3) a rigid or variable k0 value resulted in age differences of no more than 1 year. Based on these tests we concluded that model sensitivity to these factors is minor.

Figure 4 shows the distortion grid for the map of 1851 CE. As the grid appears virtually straight, geodetic distortions in the map section are minimal. Distortion grids for the other maps were similar (provided in the Supplement).

The channel centre lines that were deduced from the maps are shown in Fig. 5. On the map of 1894, Prathoek is already channelized (Fig. 2); therefore, this line is only drawn for Junner Koeland. Based on the current DEM and coring evidence, the georeferenced map dating from 1720 had to be rejected from the Bayesian modelling based on obviously erroneous representation of the river course. The 1720 map depicted the Junner Koeland meander where corings proved that river deposits were lacking, i.e. located too far west by up to 80 m. We tried to include the georeferenced map of 1786 in the Bayesian modelling, but this map resulted in poor model agreement (further discussed below). The correctness of this map was doubted because it displays the meander apex at Prathoek directed southwards instead of southwest like the other channel centre lines (Fig. 5). This would indicate that sedimentation towards the inner bend has taken place in order to move the apex westward, which seems unlikely. Based on this improbable meander planform and the statistical issue created by this map in the Bayesian modelling, we excluded it from further analyses.

The results of the OSL dating and bsMAM calculations are presented in Table 3. The spread in the equivalent doses of single aliquots for the scroll bar samples (example in Fig. 6a) shows that light exposure of the grains prior to deposition and burial was not sufficient to completely reset the quartz OSL signal of all grains. Strikingly, the equivalent dose distribution of many samples shows two distinct peaks: a younger population representing the age of recent transportation and deposition by the river and an older population with an age of about 11 ka, likely representing the depositional age of the grains prior to erosion and transport by the late Holocene Vecht river, as indicated by the OSL age obtained on a terrace remnant at Prathoek (circa 11 ka, Fig. 6b).

The Bayesian model was iteratively repeated until model results converged to similar output, which occurred after four iterations for both Junner Koeland and Prathoek; therefore, the deposition models that resulted from these iterations were used to interpolate ages throughout the scroll bar deposits. Table 4 lists both the un-modelled and modelled ages. The model agreement index (A value) should be above 60 % to indicate proper model functioning (Bronk Ramsey, 1995). The high A values in Table 4 show that the modelled ages fit the data well. The graphical output of the model as generated by OxCal is displayed on the right in Fig. 7b and d and demonstrates the improvement of the geochronology after the Bayesian modelling compared to the un-modelled data points shown on the left in Fig. 7a and c.

OSL sample NCL-2415156 suffered from very poor bleaching (Fig. 6a) and therefore had to be rejected from the Junner Koeland model; otherwise the model A value did not rise above 60 %. In addition, the historical map of 1894 had to excluded, as it resulted in poor agreement that could not be resolved by adjusting the model (e.g. k0 value). In the fourth (and final) iteration, the model had an A value of 143.8 % (Table 4a). For Prathoek, the poorly bleached sample NCL-2415160 resulted in poor agreement; after excluding this entry, the A value of the fourth iteration amounted to 112.2 %.

The Bayesian model also allows extrapolation of the results towards the meander base (Fig. 7). In our study area the positions of the two meander bases appear to be located close to the former valley side (Fig. 1); therefore, the modelled ages of the meander bases indicate the moment when the meanders sufficiently eroded the valley side to break free from lateral confinement. Model results indicate that this took place during the fourteenth century CE for Prathoek and about 65 years later during the fifteenth century for Junner Koeland. Subsequently, the meanders migrated outside the former river valley, reworking unconsolidated Pleistocene fluvial deposits and coversands, locally with Holocene drift sand on top. This occurred at an average rate of 2.6 and 0.9 m yr-1 for Junner Koeland and Prathoek respectively (calculated by dividing the distance from meander base to apex by the difference between their modelled mean ages). For both meanders, there seems to be an increase in lateral migration rate in the final stages prior to abandonment, roughly between 1825 and 1900 CE.