Volumetric Refinement and Smoothing, Depth Enforcement

Overview

The volumetric refinement and smoothing procedure lightly adjusts mesh bathymetry to improve:

  • Volume fidelity — better alignment between mesh element volumes and DEM reference.

  • Smoothness — elimination of single-node pits and sharp valleys that degrade numerical stability.

Typical changes are small (0.1–0.2 m), but isolated pits may be smoothed more. A shoreline regularization step prevents accidental drying of tidally wetted areas. After refinement, depth enforcement ensures hydraulic features such as channels and levees are properly represented where DEMs are incomplete. Removing unresolved dips and pits also helps prevent switching in the numbers of levels in a local sigma coordinate mesh, which reduces undulation in the vertical grid. For the most part these settings are “set-it-and-forget-it”. For a Bay-Delta mesh we provide reasonable defaults in the preprocessor.

The method is available both as a standalone script and as an integrated step in the BayDeltaSCHISM preprocessor.

Mathematical Formulation

Volume fidelity:

\[J_\text{vol} = \sum_e \left( V_e(h) - V^\text{DEM}_e \right)^2\]

Smoothness (total variation):

\[J_\text{TV} = \sum_{(i,j)} \lvert h_i - h_j \rvert\]

Combined objective:

\[J = \lambda_{L2} J_\text{vol} + \mu J_\text{TV}\]

where:

  • \(\lambda_{L2}\) controls the weight on volume fidelity.

  • \(\mu\) (often referred to as tv_weight) controls the weight on total variation (smoothness).

Note

The weights are chosen so that the two terms are balanced on comparable scales. In practice, this means adjustments to \(\lambda_{L2}\), tv_weight, and l2_weight are the most meaningful levers, but these are rarely changed once defaults are tuned.

The TV term is non-differentiable at zero. In practice a subgradient is used: rather than a true slope, it provides a proxy direction (raise or lower a node) that reduces roughness. This simple but powerful idea allows TV regularization to work in large-scale mesh optimization.

Depth Enforcement

Following smoothing, targeted enforcement rules may be applied:

  • Maximum depth (carve channel):

    \[h_i = \min(h_i, h_\text{max,poly})\]

  • Minimum depth (protect levee crest):

    \[h_i = \max(h_i, h_\text{min,poly})\]

These adjustments are applied within polygons or along polylines defined in the YAML configuration.

Shoreline Discovery Seeds

As part of the shoreline regularization process, seed points are used to identify wetted areas that should remain connected to open water.

  • For the Bay-Delta domain, default seeds are already embedded in the code (specified in UTM coordinates).

  • For other domains, users must explicitly supply seeds in deep, reliably wetted areas and update the epsg field to match their coordinate system.

This ensures that shoreline constraints are applied consistently during the refinement process.

Preprocessor Integration

In the BayDeltaSCHISM preprocessor, this step is configured under the grid: section. Parameters are usually factored into a separate file and included using the extended YAML !include syntax.

Example:

grid:
  href:
    name: href.gr3
    default: 0.0
    smooth: true

  depth_optimization:
    method: volume_tvd
    include: depth_opt_params_volume_tvd.yaml

Inline Parameters (Annotated)

The included file depth_opt_params_volume_tvd.yaml contains tuned defaults. For clarity, the main options are shown below with short annotations.

# Target CFL-like condition for stability (0.5–1.0 typical)
cfl_target: 0.9

# Initial timestep size for iterations
dt_init: 1.0

# Number of optimization iterations
niter: 25

# Weight for prioritizing progress on volume fidelity
lambda_l2: 1.0

# Weight of total variation (smoothness) term
tv_weight: 0.5

# Optional additional L2 regularization weight
l2_weight: 0.1

# Clipping tolerance (null disables clipping)
clip_eps: null

# Logging level (INFO or DEBUG)
log_level: INFO

Both this file and the top-level templates (such as main_bay_delta.yaml) are included in the standard BayDeltaSCHISM distribution.

Examples

Figures below illustrate typical applications.

Sacramento River and Threemile Slough

Sacramento River and Threemile Slough

Fig. 1 Optimization removes isolated pits and improves smoothness. Pre-optimization contours are light, post-optimization heavy.

Middle River and northern Mildred Island

Middle River and Mildred Island

Fig. 2 Improved volume fidelity and channel continuity. Pre-optimization contours are light, post-optimization heavy.