div_curl

The codes accompanying the paper "Modeling tangential vector fields on a sphere". Note that this description is out of date and needs to be updated.

The aim of this repository is to allow the reproduction of all numerical results reported in this paper.

Dependencies

Except the analyses by the multivariate Matern model (based on the R package RandomFields), all the codes are written in Matlab R2014a with Financial Toolbox installed. They also depend on the following packages written by other researchers.

• RBFSPHERE Matlab package http://math.boisestate.edu/~wright/montestigliano/index.html
• MEALPix 3.0 (The original link is broken)
• subtightplot http://www.mathworks.com/matlabcentral/fileexchange/39664-subtightplot
• Binned Scatter Plot http://www.mathworks.com/matlabcentral/fileexchange/19506-binned-scatter-plot

To ensure compatibility, the latter three packages are already included in the repository.

Usage

1. Download the codes by cloning the repository git clone https://github.com/minjay/div_curl.
2. Install BFSPHERE Matlab package by following the instructions on http://math.boisestate.edu/~wright/montestigliano/index.html.

Reproducing all numerical results in the paper

All main scripts are in the folders /MixedMatern/tests (Matlab) and /R_code (R), and all intermediate and final results are saved in the folder /results. Note that some codes take a long time to run even they are run parallelly on a server.

Simulation part

1. To generate Figure 1: run sim1.m in the Matlab Command Window.
2. To generate Figure 2: first run sim3_server.m and sim3_server2.m on a server to obtain the MLEs of 500 realizations. The former uses a regular grid with (n_lat, n_lon)=(10, 20), while the latter uses a regular grid with (n_lat, n_lon)=(15, 30). After that, run plot_sim.m to generate Figure 2.
3. To generate Table 1: first, run the command matlab -r "sim4_bootstrap(num)" on a server to obtain the MLEs of 200 bootstrap samples, where num is 1 to 10 since this procedure is done for 10 realizations. Then run compute_sim_ratio.m to compute the ratios between bootstrap standard errors and empirical standard errors and the median ratios for these 10 realizations.

Real data part

1. The real data are downloaded from the FTP: ftp://podaac-ftp.jpl.nasa.gov/allData/quikscat/L3/wind_1deg_1mo. The filenames are uas_QuikSCAT_L2B_v20110531_199908-200910.nc and vas_QuikSCAT_L2B_v20110531_199908-200910.nc. They represent u (zonal) and v (meridional) wind fields, respectively.
2. real_init.m preprocesses the real data, transforms the raw data to u and v residual fields, and generates Figure 3.
3. real_sim.m checks the multivariate normality of the u and v residual fields, and generates Figure 4.
4. real_fit.m fits the residual fields to the Mixed Matern model, and the results are summarized in the second column of Table 2, where the standard errors are computed by real_bootstrap.m. Both of them are run on a server. Whether the physical characteristics of the wind fields are captured by the Mixed Matern model is checked by real_sim.m. It generates Figure 5 showing the empirical and fitted cross-covariance functions.
5. The residual fields are also fitted to the parsimonious bivariate Matern model by Multi_Matern.R. It uses the R package RandomFields and runs on a server. The results are summarized in the first column of Table 2.
6. Figure 6 is plotted by real_kriging.m.
7. The cokriging cross-validation results shown in Table 3 are generated by real_kriging20.m (for the Mixed Matern model) and Multi_Matern_kriging.R, real_kriging20_BM.m (for the parsimonious bivariate Matern model). The first two scripts are run on a server.

Some results would vary slightly because different random numbers could be generated.