Turbidity currents are one of the main drivers of sediment transport from the continental shelf to the deep ocean. The resulting sediment deposits can reach hundreds of kilometres into the ocean. Computer models that simulate turbidity currents and the resulting sediment deposit can help us to understand their general behaviour. However, in order to recreate real-world scenarios, the challenge is to find the turbidity current parameters that reproduce the observations of sediment deposits.

This paper demonstrates a solution to the inverse sediment transportation problem: for a known sedimentary deposit, the developed model reconstructs details about the turbidity current that produced the deposit. The reconstruction is constrained here by a shallow water sediment-laden density current model, which is discretised by the finite-element method and an adaptive time-stepping scheme. The model is differentiated using the adjoint approach, and an efficient gradient-based optimisation method is applied to identify the turbidity parameters which minimise the misfit between the modelled and the observed field sediment deposits. The capabilities of this approach are demonstrated using measurements taken in the Miocene Marnoso-arenacea Formation (Italy). We find that whilst the model cannot match the deposit exactly due to limitations in the physical processes simulated, it provides valuable insights into the depositional processes and represents a significant advance in our toolset for interpreting turbidity current deposits.

Turbidity currents are density currents driven by sediment particles that are
suspended by turbulence in the containing fluid

This schematic representation of a dense gravity current (left) and a
corresponding depth-averaged shallow water approximation (right) shows
current height

The vast majority of the available data on turbidity currents are contained in
the sedimentary deposits that they leave behind. Significant effort is spent
on attempting to diagnose details about the turbidity current that produced
these deposits.

The task of obtaining a set of model input parameters based upon a desired model output represents an inverse problem. It can also be interpreted as an optimisation problem for which model parameters are sought to minimise the misfit between the deposit profile generated by the model and a target deposit profile, which is produced from measurements taken in the field.

In this paper, a shallow water model is used to simulate turbidity currents.
The shallow water equations are a set of partial differential equations
(PDEs). The optimisation of PDE-based models occurs throughout science and
engineering and is already applied, for instance, in ocean science

PDE models of turbidity currents require the definition of initial and boundary conditions. In the simplest case, this could involve the definition of a static lock-release laboratory configuration with uniform sediment depth and a single, uniform sediment grain size. Such a simple configuration would at least require the definition of the initial depth of the current, the concentration of sediment in the fluid, the ratio of initial depth to length, and the parameters controlling the particle settling velocity and flow front speed. More realistic initial conditions would be an inflow condition with time-varying depth, velocity, and concentrations of a wide range of sediment grain sizes along with information defining the topography of the bed, its composition, the parameterisations governing bed erosion rates, flow rheology, and bedload transport. As the model complexity and the choices of boundary and initial conditions increase, the range of deposit shapes that can be generated by the model also increases such that the model is capable of better recreating a range of deposits found in the field. However, with this added complexity, the parameter space grows and the manual tuning of the parameters becomes a greater challenge.

This paper presents a shallow water sediment-laden density current model,
released under the name AdjointTurbidity 1.0, that uses a novel finite-element mixed discontinuous Galerkin function space with adaptive
time stepping (Sect.

Shallow water models solve the Navier–Stokes equations in depth-averaged
form (Fig.

Shallow water sediment-laden density current models come in a variety of
forms.

The equations governing the current column height,

This single layer model ignores the effect of the motion of the overlying
fluid on the current. This approximation is valid for flows in which the maximum
column height is significantly less than the depth of the ambient fluid

The amount of deposited sediment,

The model is non-dimensionalised with the length, time, and velocity scales

Following

Applying this coordinate transformation, but keeping

Note that Eq. (

It is now shown that the boundary conditions (see
Eqs.

These are obtained using the method of characteristics, where

Due to the boundary conditions on momentum, the following is true:

As the cell size grows throughout the simulation, it is possible to use a much
larger time step at the end of the simulation than at the start of the
simulation. To exploit this property, an adaptive time-stepping scheme is used in
this model. A new time-dependent variable is introduced,

The system is discretised in time using a second-order explicit Runge–Kutta
time discretisation

The spatially weak form of the semi-discrete system (Eq.

Piecewise linear discontinuous Galerkin (DG) elements are used to discretise the
spatially varying state variables. Thus, the spatial and temporal discretisations
both have second-order accuracy. DG element types are known to be particularly
suitable for advection dominated problems

In order to construct a DG formulation, a regular partition

Notice that

The discretised DG formulation of Eq. (

Integrating the gradient terms by parts and slightly rearranging yields

Additionally, note that within domain boundary integrals, the

A choice of flux term must be made to handle the double-valued terms. This will
involve some coupling between the elements on either side of the interface. An
upwind flux is used for the advection term,

Using Eqs. (

This set of equations is solved for each time step of the simulation as a nonlinear variational problem using Newton's method with an LU decomposition solver for the linear problems.

Schematic diagram of the lock-release static initial
condition

Discontinuous Galerkin discretisations for convection dominated problems can
suffer from over- and undershoots at discontinuities that can cause
instability problems

The shallow water sediment-laden density current model described above was
built using the FEniCS framework

Many laboratory experiments and computer models are based around the
classical lock-release static initial condition (Fig.

Similarity convergence analysis. All variables are shown to converge
on the correct solution at the correct order.

The domain is unit length, as in all cases for this model. This solution is
valid for the model described in this paper so long as the settling velocity
of particles,

For the convergence test, the analytical solution is projected onto the model
function space forming the initial condition at

Similarity results for the finest resolution mesh (solid lines)
compared against the analytical results (dashed lines) at

Here, we describe the adjoint model and its derivation generally rather than specifically applying it to this model.

Consider a problem with

Numerous algorithms have been developed to improve this brute force approach.
These optimisation algorithms begin with an initial guess of the input
parameters and iterate, then generate improved estimates until they terminate,
hopefully at the optimised solution. The authors refer the reader to

Most of these optimisation algorithms require the gradient of the objective
functional with respect to the input parameters,

Obtaining the adjoint model begins by applying the chain rule to

Equation

However, suppose that

This expression can be substituted for

A simple property of inner products,

Gathering the left-hand side of the inner product into a new variable,

Equation (

As mentioned above,

The Marnoso-arenacea Formation spans 17 to 7 Ma (Late Burdigalian to
Tortonian) and is over 3500 m thick

The sandstone depths measured for Bed 1.1 along the Pietralunga and
Ridracoli structural elements orientated approximately parallel to the
palaeoflow. This has been reconstructed from Fig. 5 in

In this section, an optimisation algorithm is used to select model parameters
that produce an output deposit that best matches part of Bed 1.1 in the
Marnoso-arenacea Formation, as recorded by

The deposit consists of sandstone and mudstone components. The focus here
will be on attempting to recreate only the sandstone portion of the deposit.
It is likely that ponding effects have influenced the shape of the mud
deposit in this bed

The initial conditions are based upon the analytical solution for a
non-depositional flow at a non-dimensional time,

The non-dimensional particle-settling velocity,

The beginning of the basin is defined as being at the front of the current at

The aim here is to reduce the difference between the deposit profile generated
by this model and the target deposit profile from field measurements. To do
this,
we need to map the non-dimensional, transformed results from the model back to
the observation space. We also only measure the variation over the length of the
measured deposit. Therefore, the functional that we will aim to minimise,

To calculate this functional,

It is important to note that at the end of the simulation,

The calculation of

This can be approximately transformed into the model coordinate system as

The gradient computation was verified using the Taylor remainder convergence
test. Let

Even small errors in the derivative destroy the second-order convergence in
Eq. (

Taylor remainders

The above convergence was succesfully carried out for the implemented adjoint
model with a number of different controls and functionals. As an example,
Table

With confidence that the forward and backward models are working, optimisation
of the input parameters,

These bounding constraints are chosen based upon very loose limits of expected values that each parameter may possibly take. The principal purpose of these bounds is to avoid invalid negative values being generated for any of the parameters.

The nonlinear optimisation library, IPOPT

The aim is to recreate the sand deposit by modelling only the sand in the
flow using a single average grain size,

Values of the parameters over the optimisation iterations against the
value of the objective functional,

The dimensional deposit output

The criteria for finishing the parameter optimisation is based upon the relative
change in

These optimised values are not completely acceptable. The value for

With the exception of the sediment diameter, the optimised values are fairly similar to those chosen as input values. This confirms that the input parameters chosen were sensible predictions of the starting conditions for the gravity current. To test this hypothesis, we ran the same situation starting from a number of alternative initial conditions. We found that there are indeed a number of local minima. An optimisation with initial conditions

Figure

The dimensional deposit output

The existence of alternative minima must always be considered when running optimisations of this type. It is important to have a good understanding of the problem to choose sensible initial starting conditions and also to assess the resultant optimised values. A regularisation approach would avoid this problem but assumes prior knowledge about the target profile.

A clear omission from the model is the presence of mud in the suspension. The presence of mud will significantly alter the energy budget of the flow. A mud sediment class can easily be included so that the model produces more realistic optimised values. This is detailed below.

Investigating the effect of including mud in the sediment mixture can be
achieved relatively simply by including an additional transport equation with
a form identical to Eq. (

Finally,

The initial condition needs to be altered to include the new sediment class.
The initial vertically averaged volume fraction of sand is changed to

The set of optimised input parameters is redefined as

The input parameters are normalised as detailed in Sect.

Values of the parameters from the model with both mud and sand sediment
classes over the optimisation iterations against the value of the objective
functional,

Optimised dimensional deposit output from the model with both mud
and sand sediment classes,

The optimisation of the model with two sediment classes is completed in 17
iterations with a final functional value of

The fit with the measured data is still poor towards the end of the deposit.

The final optimised values are

A comparison of these results to those obtained without a mud sediment class
shows that the
value of

It is also interesting to assess the sensitivity of the model to variations in
the input parameters by analysing the final gradient of the objective
functional,

It is indeed found that changing this value has very little effect on the
obtained deposit. The same simulation is run with the mud diameter decreased by
2 orders of magnitude such that the input parameter values are

The resulting functional value is

Optimised dimensional deposit output from the model with both mud
and sand sediment classes,

Although the sandstone deposits generated by the single and two sediment class
models are very similar, the properties of the turbidity currents that produced them
are very different (Fig.

The time evolution of the dimensionalised variables for three simulations: a
simulation with a single sand sediment class and optimised input parameters
to match the Bed 1.1 sand deposit, a simulation with sand and mud classes
and optimised input parameters to match the Bed 1.1 sand deposit, and a
simulation with sand and mud classes and the same optimised input parameters
but a mud diameter 2 orders of magnitude smaller. The results are shown
against the dimensional time,

The simulated turbidity currents that produced

The height of the current in the optimised simulation with both sand and mud
sediment classes is

The model also neglects variations in the bed profile. The gradient of the sea
floor in the basin where the Marnoso-arenacea Formation was created was
substantially less than 1 degree

This paper has presented a novel implementation of the shallow water equations for modelling density currents using a mixed finite-element formulation. The model has been differentiated to allow for parameter optimisation using gradient-based optimisation techniques and the use of gradient information in sensitivity analyses.

The proposed model is based upon simplifying shallow water sediment-laden density current assumptions and has been used here to recreate a low-volume deposit from the Marnoso-arenacea Formation in Italy with some success. However, the lack of many key flow processes within the current model, including bedload transport and re-entrainment, has arguably led to optimised parameters values which would be improved upon with a more complete underlying model.

This paper has demonstrated the power of gradient-based optimisation methods for determining the set of input parameters that best fits a particular turbidity current deposit. Since the input parameters are rarely known with any accuracy for these flows, optimisation represents a sensible way to better estimate these values.

Future development of the model could enable more complex boundary conditions and add parameterisations for ambient fluid entrainment, bed erosion, and bedload transport. This will increase the capacity for the model to recreate a range of deposits found in the field, while the parameter space will grow significantly. The optimisation techniques presented in this paper will allow for the efficient selection of optimised values for a large parameter space.

The model implementation and the test setups described in this paper are
freely available as a separate git repository on bitbucket:

Dolfin-adjoint and all FEniCS core components are licensed under the GNU LGPL as published by the Free Software Foundation, either version 3 of the licence or (optionally) any later version.

This work was supported by the Natural Environment Research Council grant NE/K000047/1 and the Research Council of Norway through a Centre of Excellence grant (project number 179578) and the FRIPRO Program (project number 251237) to the Center for Biomedical Computing at Simula Research Laboratory (project number 179578). Edited by: A. Sandu Reviewed by: two anonymous referees