1. POTENTIAL METHODS 2016-2017 Part 2c Data interpolation
Carla Braitenberg
Trieste University, DGM
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(tutto il ppt eseguito )

2. 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

3. Illustration interpolation

4. 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

5. 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

6. 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.)

7. 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

8. 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

9. 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.

10. 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.

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

12. Illustration cubic splines

13. Gridding example with splines
Bi-directional interpolation with splines

14. 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

15. Filter 1

16. Convolution illustration

17. Filter operator illustration

18. Illustration of a digital filter kernel in 2D

19. 2D spatial domain filter- directional gradient
Equal to convolution with filter hkn: