POTENTIAL METHODS 2016-2017 Part 2c Data interpolation Carla Braitenberg Trieste University, DGM Home page: http://www2.units.it/~braitenberg/ e-mail: berg@units.it (tutto il ppt eseguito 25.10.2016)

Motivation Acquisition of observed data: irregular spacing, in some cases near to regular Data processing methods as Fourier transform, filters, display methods require regular grid Gridding : operation of interpolating uneven data onto regular grid. Distance between nodes: cell size or grid interval. Assumption: spatial variations are continuous

Illustration interpolation

Interpolation approaches Data are affected by noise: allow the interpolation algorithm to fit data within a predefined limit to reduce noise effect Interpolation: use data inside window centered on node. Shape of window to be defined, e.g. Circular. Larger windows: short wavelength variation is lost Two ways to establish node value a) statistical b) using simple function

Statistical interpolation method Average of points in window: median ("middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order) weighted mean: Weight: inverse of distance between node and point: d-k, with k=1,2 Nearest neighbor: assign value on node equal to nearest data point

Function for interpolation Spline function: a few points are fitted with a function, giving minimum curvature Cubic spline: polynomial of 3° order. Case of acquisition in subparallel lines. Use bi-directional gridding (see fig.)

Interpolation parameters and artefacts Cell-size: for regular or random data: Choose 1/2 distance between data points. If possible do not interpolate in areas of no-data. Rather signalize with a no-data value. Some programs can handle it. Features on single datapoints are unreliable (bull’s eye) Inspect interpolation with position of datapoints. Changes of data density can produce artefacts in frequency content

Airborne surveys Line spacing about 50 times the along-line data sampling, for reconnaissance, and 20 times for detailed survey. Cell size must be compromize of 1/5 to 1/3 of line spacing

Merging data sets Regrid to coherent grid Check for biases between data sets There should be overlap between data sets. Overlap controls bias and results at long wavelength Long wavelength, and consequently bias, can be also controlled by satellite observations of the same data type.

Data enhancement Averaging repeat measurements increaes overall precision Variations between repeat measurements should be due to noise. Adding noise should average to zero. Signal to noise ratio increases. Improved by a factor sqrt(N), with N=number of repeat measurements. Stacking is possible for a static field.

Illustration Mean, Mode, median for quasi symmetry and skew distribution of data

Illustration cubic splines

Gridding example with splines Bi-directional interpolation with splines

Filtering Operate on data set to enhance signal of interest. A) Convolution filter in 1D or 2D B) spectral filter C) statistical properties filter D) gradient filter: enhance changes along certain direction. Show strike of structures

Filter 1

Convolution illustration

Filter operator illustration

Illustration of a digital filter kernel in 2D

2D spatial domain filter- directional gradient Equal to convolution with filter hkn: