A methodological framework for reconstructing historical delta front morphology: case study at Macquarie rivulet delta within Lake Illawarra, Australia

Abstract

Reconstructing past delta front morphology is a challenging task, due to complex morphological formation processes. We have developed a methodological framework to reconstruct the delta front morphology by integrating the information from historical shorelines, spatial distribution of depositional environments, relative sea-level changes and a modern Digital Elevation Model (DEM). The delta front morphology was reconstructed based on spatial connections between mud basin morphology, subaerial DEM and the historical shoreline. In addition, available sedimentation data at the delta front was utilized to aid in reflecting complex morphological formation processes. Taking Macquarie Rivulet delta within Lake IlIawarra, Australia, as an example, we generated the historical delta morphology for 1892. The modelled sedimentation rate appears to be consistent with the measured ones. We have also applied this method to reconstruct the historical morphology in 1938 and 1981. The model results indicate a progressive infilling influenced by switching river mouth locations. The cautions and implications of this method are also discussed. An increased resolution of sedimentation data should be able to improve the accuracy of the model. The reconstructed morphologies, elucidating fundamental information about delta evolution and sediment mass volumes in the past, can be employed in management activities.

Appendix

The morphological variation is described by the semi-variogram that evaluates the spatial correlations of the residuals of the observations after detrending the mean surface (Olea 1991). For a set of N observations at locations x_i (i = 1, 2…., N), a regionalized variable h(x) represents the elevations. An estimator of experimental semi-variogram γ(d) is given below, where d is the distance between the pairs of observations:

$$ \gamma (d)=\frac{1}{2N}{\sum}_{i=1}^N{\left[h\left({x}_i\right)-h\left({x}_i+d\right)\right]}^2. $$

(4)

The semi-variogram is a function of distance and is independent of the location. If the mean of the regionalized variable is stationary, the covariance is C(d) = C(0) - γ(d), where C(0) is the sample variance in the experimental semi-variogram. When the distance of a pair of observations is greater than a certain range r, they are no longer correlated and the semi-variance stops increasing and remains stable, which is called the sill and can be used to approximate the population variance (Barnes 1991). Anisotropy means that r varies in different directions, resulting in different spatial correlations depending on both the directions and distances of geospatial data. Even at small distances there is a small difference in semi-variances and this is called the nugget and represents measurement errors and can be considered as white noise δ.

One of the common mathematical functions to model the experimental semi-variogram is called the spherical model (Shapiro and Botha 1991), which is defined as:

$$ {\gamma}^{\prime }(d)={\sigma}_0^2\left(\frac{3d}{2r}-\frac{d^3}{2{r}^3}\right), $$

(5)

where ${\sigma}_0^2$ is the sill and is equal to C(0) when the distance d is equal to or greater than the range r. The fitted semi-variogram model is then used as a distance-based weight of the kriging estimator. The kriging estimator is a family of generalized linear regression techniques (Webster and Oliver 2008) and the ordinary kriging estimator (Davis 2002) is given below:

$$ \widehat{Z}\left({x}_0\right)=u\left(1-{\sum}_{i=1}^k{\lambda}_i\right)+{\sum}_{i=1}^k{\lambda}_i{Z}^{\ast}\left({x}_i\right), $$

(6)

where $\widehat{Z}\left({x}_0\right)$ is an estimator at location x₀; Z^∗(x_i) is the observation of the regionalized variable Z at location x_i; u is the mean of the regionalized variable Z; and λ_i is the kriging weight between the values at location x₀ and the values at location x_i, which is derived from the semi-variogram model γ ′ (d). ${\sum}_{i=1}^k{\lambda}_i$ is forced to be 1 to ensure unbiased estimates for all weights obtained when the error variances are as small as possible, and the mean error is zero. In the ordinary kriging, there is no need to know the mean value u or the trend, as ${\sum}_{i=1}^k{\lambda}_i$is equal to 1.0.