Attenuation of noise bursts and glitches

Let $\bold h$ be an abstract vector containing as components the water depth over a 2-D spatial mesh. Let $\bold d$ be an abstract vector whose successive components are depths along the vessel tracks. One way to grid irregular data is to minimize the length of the residual vector $\bold r_d(\bold h)$: 

$$\bold 0 \quad\approx\quad \bold r_d \quad=\quad \bold B \bold h \ -\ \bold d$$ (20)

where $\bold B$ is a 2-D linear interpolation (or binning) operator and $\bold r_d$ is the data residual. Where tracks cross or where multiple data values end up in the same bin, the fitting goal (20) takes an average. Figure  is a display of simple binning of the raw data. (Some data points are outside the lake. These must represent navigation errors.)

Some model-space bins will be empty. For them we need an additional ``model styling'' goal, i.e. regularization. For simplicity we might minimize the gradient.  

$$\begin{array} {lllll} \bold 0 &\approx& \bold r_d &=& \bold... ...0 &\approx& \bold r_h &=& \epsilon \nabla \bold h \end{array}$$ (21)

where $\nabla=\left ( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right)$ and $\bold r_h$ is the model space residual. Choosing a large scaling factor $\epsilon$ will tend to smooth our entire image, not just the areas of empty bins. We would like $\epsilon$ to be any number small enough that its main effect is to smooth areas of empty bins. When we get into this further, though, we'll see that because of noise some smoothing across the nonempty bins is desireable too.