hector-ts 3.0

A collection of programs to analyse geodetic time series

Hector

Table of Contents

1. Introduction
2. Code Description
   1. Installation
   2. Directory Structure
3. Bugs/Future Work

1. Introduction

Hector is a software package that can be used to estimate a trend in time series with temporal correlated noise. Trend estimation is a common task in geophysical research where one is interested in phenomena such as the increase in temperature, sea level or GNSS derived station position over time. The trend can be linear or a higher degree polynomial and in addition one can estimate periodic signals, offsets and post-seismic deformation. Together they represent the model that is fitted to the observations.

It is well known that in most geophysical time series the noise is correlated in time ([Agnew, 1992](https://agupubs.onlinelibrary.wiley.com/doi/10.1029/91GL02832); [Beran, 1992](https://www.amazon.com/Statistics-Long-Memory-Processes-Monographs-Probability/dp/0412049015)) and this has a significant influence on the accuracy by which the model parameters can be estimated. Therefore, the use of a computer program such as Hector is advisable. Hector assumes that the user knows what type of temporal correlated noise exists in the observations and estimates both the model parameters and the parameters of the chosen noise model using the Restricted Maximum Likelihood Estimation (RMLE) method. Since for most observations the choice of noise model can be found from literature or by looking at the power spectral density, this is sufficient in most cases.

Instead of using Hector, one can also use the [CATS](https://www.ngs.noaa.gov/gps-toolbox/cats.htm) software of Williams (2008). Another alternative is the program [est_noise](https://github.com/langbein-usgs/est_noise) of Langbein (2010). Recent versions include some tricks from Bos et al. (2013) to deal with missing data but with a different way to construct the covariance matrix (Langbein, 2017). Hector is a complete rewrite of [Hector](https://teromovigo.com/hector/) which is written in C++. The reason for changing the programming language was the need to make maintenance of the code easier. The Hector (P for Python) has around 8 times less lines of code than Hector (C++). In addition, Hector could not run on Windows and installation on a Mac computer was difficult. Hector is a truly cross-platform application. I have tried to keep the way Hector works similar to that of the C++ version to smooth the transition

The reason Hector is fast is because it makes use of the symmetry in the covariance matrix (i.e., a Toeplitz matrix) to apply fast methods to compute its inverse. Non-stationary noise is approximated by a noise model that becomes stationary at the very, very low frequencies and in this way also a Toeplitz covariance matrix is generated. This is a nutshell the core reason why Hector is fast. Another part is due to clever usage of the Fast Fourier Transform (FFT).

In the book by [Montillet and Bos (2020)](https://link.springer.com/book/10.1007/978-3-030-21718-1#about) more examples on the analysis of geodetic time series with temporal correlated noise can be found.

The next secion explains how to install Hector on your computer, the best way to organise your files and the recommended work flow to analyse the time series. For more detailed information, see the Wiki pages on this site.

2. Code Description

List of programs provided by the Hector software package. Details can be found in the Wiki-pages.

Name	Description
estimatetrend	Main program to estimate a linear trend.
estimatespectrum	Program to estimate the power spectral density from the data or residuals using the Welch periodogram method.
modelspectrum	Given a noise model and values of the noise parameters, this program computed the associated power spectral density for given frequency range.
removeoutliers	Program to remove outliers from the data.
findoffsets	Program to find the epoch of a possible offset in the time series.
simulatenoise	Program to files with synthetic coloured noise.
date2mjd	Small program to convert calendar date into Modified Julian Date.
mjd2date	The inverse of date2mjd.
msfgen	MSF: MJD - SOD - Format. Small program to combine a json file with metadata and a text file with data in columns into a single binary file.
msfdump	Inverse of msfgen. Small program that reads a msf-file and creates a json file with metadata and a text file with data in columns.

2.i Installation

Following Python customs, it is best to create a virtual environment by typing on the command line:

python3 -m venv env
source env/bin/activate

This creates the virtual environment (called and stored in env) which is then activated. Next, on the command line type:

(env) pip install hectorp

That should be it. You can now go to the directory where you have your project and run the hectorp executables. Once you are done, you can exit your virtual environment with:

deactivate

2.ii Directory Structure

The following directory structure is recommended to automate the analysis of your time series with Hector:

ori_files
obs_files
pre_files
fin_files

The ori_files directory is needed if your time series does follow the Hector formats. Hector comes with scripts to convert formats (e.g., enu, pos, sol) into the mom-format which are then stored in the obs_files directory.

The obs_files directory contains the time series files in mom/msf-format. Normally one wants to remove outliers and the cleaned time series are stored in pre_files. These are then analysed with 'estimatetrend' to create a copy of the observations with the fitted model in the fin_files directory. If you look at example/ex1 then this will become clearer.

The advantage of adapting this directory structure is that you can just type estimate_all_trends which will look for all files in the obs_files directory, remove the outliers, estimate the trends and finally estimates the power spectral density of the residuals.

3. Bugs/Future Work

Of course, one selling point of Hector is its speed. Having a source code that is easy to maintain is all very well for me, but the user does not care about that. Fortunately, the numpy and scipy libraries are optimised which results in good comparison with the C++ version, see table below.

N	C++ (s)	Python (s)
1000	5	2.4
3000	6	6.7
5000	7	13.6
8000	12	56.5
5000 10%	16	63.8
8000 10%	33	81.0
5000 20%	26	70.0
8000 20%	60	140.0