Maximilian Weigand 17. April 2019
Andreas Hense Homework
Petra Friederichs InvMod SS 2021
Solve the following exercises using the JupyterHub. Submit a running Python 3 Jupyther Notebook (.ipynb),
and (optionally) a PDF of a fully executed Jupyter notebook to the corresponding eCampus exercise. Please refer
to the eCampus exercise for the deadline.
Total number of points: 20
1 The ONE measurement (4 Points)
The weight of two objects is to be estimated. The only measurement available is a total measurement of both
objects combined, m1 +m2 = 2kg.
1.1 How many solutions has the equation m1 +m2 = 2kg? (Points: 1)
1.2 Is the problem unique, under-, mixed-, or overdetermined? (Points: 1)
1.3 Compute the minimum-length estimate mest = [m1,m2] (Points: 2)
2 Ambiguity in the measurements (5 Points)
You want to estimate the weight of two objects. Measurement of the first object yields m1 = 1 kg, the second one
yields m2 = 2kg. Both objects combined weigh to 2kg.
2.1 Compute the least-squares estimate mest of both objects. (Points: 5)
3 Zero offset profiling (11 Points)
A zero offset seismic depth sounding is conducted with the aim of detecting the sub-surface topography of a
bedrock, overlain with sediments of known seismic velocity. The surface is located at z = 0, and 9 measurements
are conducted at positions of 2, 4, 6, 12, 14, 16, 22, 24, 26 m.
A simplified seismic model view is assumed: A seismic signal is generated at a source location (z=0) and the signal
is then assumed to travel downwards and be reflected at the top of the bedrock. A receiver at the source location
then picks up the reflected signal. Measurements are thus travel times (in s) of the seismic wave from source to
bedrock and back. The p-wave velocity of the sediment layer is assumed to be v = 2500m/s.
Based on other deliberations a three segmented bedrock is assumed, with constant depths in the ranges between
0 and 10 m, 10 and 20 m, 20 and 30 m (right figure).
3.1 Formulate the forward model operator G (Points: 5)
Assume a straight wave propagation (ray path).
3.2 Determine the bed rock depths using the least-squares approach (Points: 2)
A measurement yields the following 9 measurement: 0.076, 0.07, 0.083, 0.124, 0.118, 0.122, 0.101, 0.097, 0.096 s.
3.3 Compute the covariance matrix (Points: 4)
Assume independent, normally-distributed data noise with a standard deviation of 5µs.
• Plot the covariance matrix using the imshow command, add a colorbar and a colorbar label “variance-
• Compute the standard deviations and the confidence intervals for the model parameter estimation
• Create two more plots that show the standard deviations and confidence intervals using imshow (i.e., convert
the covariance matrix to these quantities while knowing that only the diagonal entries have meaning here).
• For each model parameter, print the estimated bedrock depth and its confidence interval in the notation:
depth +- confidence-interval