I never really intended the FFT stuff to go on for as long as it did, since that sort of thing wasn’t really what I was planning as the focus for this Data Analysis in Haskell series. The FFT was intended primarily as a “warm-up” exercise. After fourteen blog articles and about 10,000 words, everyone ought to be sufficiently warmed up now…

Instead of trying to lay out any kind of fundamental principles for data analysis before we get going, I’m just going to dive into a real example. I’ll talk about generalities as we go along when we have some context in which to place them.

All of the analysis described in this next series of articles closely follows that in the paper: D. T. Crommelin (2004). Observed nondiffusive dynamics in large-scale atmospheric flow. J. Atmos. Sci. 61(19), 2384–2396. We’re going to replicate most of the data analysis and visualisation from this paper, maybe adding a few interesting extras towards the end.

It’s going to take a couple of articles to lay out some of the background to this problem, but I want to start here with something very practical and not specific to this particular problem. We’re going to look at how to gain access to meteorological and climate data stored in the NetCDF file format from Haskell. This will be useful not only for the low-frequency atmospheric variability problem we’re going to look at, but for other things in the future too.

### The NetCDF file format

The NetCDF file format is a “self-describing” binary format that’s used a lot for storing atmospheric and oceanographic data. It’s “self-describing” in the sense that the file format contains metadata describing the spatial and temporal dimensions of variables, as well as optional information about units and a bunch of other stuff. It’s a slightly intimidating format to deal with at first, but we’ll only need to know how a subset of it works. (And it’s much easier to deal with than HDF5, which we’ll probably get around to when we look at some remote sensing data at some point.)

So, here’s the 30-second introduction to NetCDF. A NetCDF file contains dimensions, variables and attributes. A NetCDF dimension just has a name and a size. One dimension can be specified as an “unlimited” or record dimension, which is usually used for time series, and just means that you can tack more records on the end of the file. A NetCDF variable has a name, a type, a list of dimensions, some attributes and some data. As well as attributes attached to variables, a NetCDF file can also have some file-level global attributes. A NetCDF attribute has a name, a type and a value. And that’s basically it (for NetCDF-3, at least; NetCDF-4 is a different beast, but I’ve never seen a NetCDF-4 file in the wild, so I don’t worry about it too much).

### An example NetCDF file

That’s very abstract, so let’s look at a real example. The listing below shows the output from the ncdump tool for one of the data files we’re going to be using, which stores a variable called geopotential height (I’ll explain exactly what this is in a later article – for the moment, it’s enough to know that it’s related to atmospheric pressure). The ncdump tool is useful for getting a quick look at what’s in a NetCDF file – it shows all the dimension and variable definitions, all attributes and also dumps the entire data contents of the file as ASCII (which you usually want to chop off…).

netcdf z500-1 {
dimensions:
longitude = 144 ;
latitude = 73 ;
time = 7670 ;
variables:
float longitude(longitude) ;
longitude:units = "degrees_east" ;
longitude:long_name = "longitude" ;
float latitude(latitude) ;
latitude:units = "degrees_north" ;
latitude:long_name = "latitude" ;
int time(time) ;
time:units = "hours since 1900-01-01 00:00:0.0" ;
time:long_name = "time" ;
short z500(time, latitude, longitude) ;
z500:scale_factor = 0.251043963537454 ;
z500:_FillValue = -32767s ;
z500:missing_value = -32767s ;
z500:units = "m**2 s**-2" ;
z500:long_name = "Geopotential" ;
z500:standard_name = "geopotential" ;

// global attributes:
:Conventions = "CF-1.0" ;
:history = "Sun Feb  9 18:46:25 2014: ncrename -v z,z500 z500-1.nc\n",
"2014-01-29 21:04:31 GMT by grib_to_netcdf-1.12.0: grib_to_netcdf /data/soa/scra
tch/netcdf-web237-20140129210048-3022-3037.target -o /data/soa/scratch/netcdf-web237-20140129210411-3022
-3038.nc" ;
data:

longitude = 0, 2.5, 5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, 25, 27.5, 30,
32.5, 35, 37.5, 40, 42.5, 45, 47.5, 50, 52.5, 55, 57.5, 60, 62.5, 65,
67.5, 70, 72.5, 75, 77.5, 80, 82.5, 85, 87.5, 90, 92.5, 95, 97.5, 100,
102.5, 105, 107.5, 110, 112.5, 115, 117.5, 120, 122.5, 125, 127.5, 130,
132.5, 135, 137.5, 140, 142.5, 145, 147.5, 150, 152.5, 155, 157.5, 160,
162.5, 165, 167.5, 170, 172.5, 175, 177.5, 180, 182.5, 185, 187.5, 190,
192.5, 195, 197.5, 200, 202.5, 205, 207.5, 210, 212.5, 215, 217.5, 220,
222.5, 225, 227.5, 230, 232.5, 235, 237.5, 240, 242.5, 245, 247.5, 250,
252.5, 255, 257.5, 260, 262.5, 265, 267.5, 270, 272.5, 275, 277.5, 280,
282.5, 285, 287.5, 290, 292.5, 295, 297.5, 300, 302.5, 305, 307.5, 310,
312.5, 315, 317.5, 320, 322.5, 325, 327.5, 330, 332.5, 335, 337.5, 340,
342.5, 345, 347.5, 350, 352.5, 355, 357.5 ;

latitude = 90, 87.5, 85, 82.5, 80, 77.5, 75, 72.5, 70, 67.5, 65, 62.5, 60,
57.5, 55, 52.5, 50, 47.5, 45, 42.5, 40, 37.5, 35, 32.5, 30, 27.5, 25,
22.5, 20, 17.5, 15, 12.5, 10, 7.5, 5, 2.5, 0, -2.5, -5, -7.5, -10, -12.5,
-15, -17.5, -20, -22.5, -25, -27.5, -30, -32.5, -35, -37.5, -40, -42.5,
-45, -47.5, -50, -52.5, -55, -57.5, -60, -62.5, -65, -67.5, -70, -72.5,
-75, -77.5, -80, -82.5, -85, -87.5, -90 ;

As shown in the first line of the listing, this file is called z500-1.nc (it’s contains daily 500 millibar geopotential height data). It has dimensions called longitude, latitude and time. There are variables called longitude, latitude, time and z500. The variables with names that are the same as dimensions are called coordinate variables and are part of a metadata convention that provides information about the file dimensions. The NetCDF file format itself doesn’t require that dimensions have any more information provided for them than their name and size, but for most applications, it makes sense to give units and values for points along the dimensions.

If we look at the longitude variable, we see that it’s of type float and has one dimension, which is the longitude dimension – this is how you tell a coordinate variable from a data variable: it will have the same name as the dimension it goes with and will be indexed just by that dimension. Immediately after the line defining the longitude variable are the attributes for the variable. Here they give units and a display name (they can also give information about the range of values and the orientation of the coordinate axis). All of these attributes are again defined by a metadata convention, but they’re mostly pretty easy to figure out. Here, the longitude is given in degrees east of the prime meridian, and if we look further down the listing, we can see the data values for the longitude variable, running from zero degrees to 357.5°E. From all this, we can infer that the 144 longitude values in the file start at the prime meridian and increase eastwards.

Similarly, the latitude variable is a coodinate variable for the latitude dimension, and specifies the latitude of points on the globe. The latitude is measured in degrees north of the equator and ranges from 90° (the North pole) to -90° (the South pole). Taking a look at the data values for the latitude variable, we can see that 90 degrees north is in index 0, and the 73 latitude values decrease with increasing index until we reach the South pole.

The time coordinate variable is a little more interesting, mostly because of its units – this “hours since YYYY-MM-DD HH:MM:SS” approach to time units is very common in NetCDF files and it’s usually pretty easy to work with.

Finally, we get on to the data variable, z500. This is defined on a time/latitude/longitude grid (so, in the data, the longitude is the fastest changing coordinate). The variable has one slightly odd feature: its type. The types for the coordinate variables were all float or double, as you’d expect, but z500 is declared to be a short integer value. Why? Well, NetCDF files are quite often big so it can make sense to use some sort of encoding to reduce file sizes. (I worked on a paleoclimate modelling project where each model simulation resulted in about 200 Gb of data, for a dozen models for half a dozen different scenarios. In “Big Data” terms, it’s not so large, but it’s still quite a bit of data for people to download from a public server.) Here, the real-valued geopotential height is packed into a short integer. The true value of the field can be recovered from the short integer values in the file using the add_offset and scale_factor attributes – here the scale factor is unity, so we just need to add the add_offset to each value in the file to get the geopotential height.

Last of all we have the global attributes in the file. The most interesting of these is the Conventions attribute, which specifies that the file uses the CF metadata convention. This is the convention that defines how coordinate variables are represented, how data values can be compressed by scaling and offsetting, how units and axes are represented, and so on. Given a NetCDF file using the CF convention (or another related convention called the COARDS metadata convention), it’s pretty straightforward to figure out what’s going on.

So, how do we read NetCDF files into a Haskell program to work on them? I’ve seen a few Haskell FFI bindings to parts of the main NetCDF C library, but none of those really seemed satisfactory for day-to-day use, so I’ve written a simple library called hnetcdf that includes both a low-level wrapping of the C library and a more idiomatic Haskell interface (which is what we’ll be using).

In particular, because NetCDF data is usually grid-based, hnetcdf supports reading data values into a number of different kinds of Haskell arrays (storable Vectors, Repa arrays and hmatrix arrays). For this analysis, we’re going to use hmatrix vectors and matrices, since they provide a nice “Matlab in Haskell” interface for doing the sort of linear algebra we’ll need.

In this section, we’ll look at some simple code for accessing the NetCDF file whose contents we looked at above which will serve as a basis for the more complicated things we’ll do later. (The geopotential height data we’re using here is from the ERA-Interim reanalysis project – again, I’ll explain what “reanalysis” means in a later article. For the moment, think of it as a “best guess” view of the state of the atmosphere at different moments in time.) We’ll open the NetCDF file, show how to access the file metadata and how to read data values from coordinate and data variables.

We need a few imports first, along with a couple of useful type synonyms for return values from hnetcdf functions:

import Prelude hiding (length, sum)
import Control.Applicative ((<$>)) import qualified Data.Map as M import Foreign.C import Foreign.Storable import Numeric.Container import Data.NetCDF import Data.NetCDF.HMatrix type VRet a = IO (Either NcError (HVector a)) type MRet a = IO (Either NcError (HMatrix a)) As well as a few utility imports and the Numeric.Container module from the hmatrix library, we import Data.NetCDF and Data.NetCDF.HMatrix – the first of these is the general hnetcdf API and the second is the module that allows us to use hnetcdf with hmatrix. Most of the functions in hnetcdf handle errors by returning an Either of NcError and a “useful” return type. The VRet and MRet type synonyms represent return values for vectors and matrices respectively. When using hnetcdf, it’s often necessary to supply type annotations to control the conversion from NetCDF values to Haskell values, and these type synonyms come in handy for doing this. #### Reading NetCDF metadata Examining NetCDF metadata is simple: Right nc <- openFile "/big/data/reanalysis/ERA-Interim/z500-1.nc" putStrLn$ "Name: " ++ ncName nc
putStrLn $"Dims: " ++ show (M.keys$ ncDims nc)
putStr $unlines$ map (\(n, s) -> "  " ++ n ++ ": " ++ s) $M.toList$ flip M.map (ncDims nc) $\d -> show (ncDimLength d) ++ if ncDimUnlimited d then " (UNLIM)" else "" putStrLn$ "Vars: " ++ show (M.keys $ncVars nc) putStrLn$ "Global attributes: " ++ show (M.keys $ncAttrs nc) let Just ntime = ncDimLength <$> ncDim nc "time"
Just nlat = ncDimLength <$> ncDim nc "latitude" Just nlon = ncDimLength <$> ncDim nc "longitude"

We open a file using hnetcdf’s openFile function (here assuming that there are no errors), getting a value of type NcInfo (defined in Data.NetCDF.Metadata in hnetcdf). This is a value representing all of the metadata in the NetCDF file: dimension, variable and attribute definitions all bundled up together into a single value from which we can access different metadata elements. We can access maps from names to dimension, variable and global attribute definitions and can then extract individual dimensions and variables to find information about them. The code in the listing above produces this output for the ERA-Interim $Z_{500}$ NetCDF file used here:

Name: /big/data/reanalysis/ERA-Interim/z500-1.nc
Dims: ["latitude","longitude","time"]
latitude: 73
longitude: 144
time: 7670
Vars: ["latitude","longitude","time","z500"]
Global attributes: ["Conventions","history"]

#### Accessing coordinate values

Reading values from a NetCDF file requires a little bit of care to ensure that NetCDF types are mapped correctly to Haskell types:

let (Just lonvar) = ncVar nc "longitude"
Right (HVector lon) <- get nc lonvar :: VRet CFloat
let mlon = mean lon
putStrLn $"longitude: " ++ show lon ++ " -> " ++ show mlon Right (HVector lon2) <- getS nc lonvar [0] [72] [2] :: VRet CFloat let mlon2 = mean lon2 putStrLn$ "longitude (every 2): " ++ show lon2 ++ " -> " ++ show mlon2

This shows how to read values from one-dimensional coordinate variables, both reading the whole variable, using hnetcdf’s get function, and reading a strided slice of the data using the getS function. In both cases, it’s necessary to specify the return type of get or getS explicitly – here this is done using the convenience type synonym VRet defined earlier. This code fragment produces this output:

longitude: fromList [0.0,2.5,5.0,7.5,10.0,12.5,15.0,17.5,20.0,22.5,25.0,
27.5,30.0,32.5,35.0,37.5,40.0,42.5,45.0,47.5,50.0,52.5,55.0,57.5,60.0,
62.5,65.0,67.5,70.0,72.5,75.0,77.5,80.0,82.5,85.0,87.5,90.0,92.5,95.0,
97.5,100.0,102.5,105.0,107.5,110.0,112.5,115.0,117.5,120.0,122.5,125.0,
127.5,130.0,132.5,135.0,137.5,140.0,142.5,145.0,147.5,150.0,152.5,155.0,
157.5,160.0,162.5,165.0,167.5,170.0,172.5,175.0,177.5,180.0,182.5,185.0,
187.5,190.0,192.5,195.0,197.5,200.0,202.5,205.0,207.5,210.0,212.5,215.0,
217.5,220.0,222.5,225.0,227.5,230.0,232.5,235.0,237.5,240.0,242.5,245.0,
247.5,250.0,252.5,255.0,257.5,260.0,262.5,265.0,267.5,270.0,272.5,275.0,
277.5,280.0,282.5,285.0,287.5,290.0,292.5,295.0,297.5,300.0,302.5,305.0,
307.5,310.0,312.5,315.0,317.5,320.0,322.5,325.0,327.5,330.0,332.5,335.0,
337.5,340.0,342.5,345.0,347.5,350.0,352.5,355.0,357.5] -> 178.75

longitude (every 2): fromList [0.0,5.0,10.0,15.0,20.0,25.0,30.0,35.0,40.0,
45.0,50.0,55.0,60.0,65.0,70.0,75.0,80.0,85.0,90.0,95.0,100.0,105.0,110.0,
115.0,120.0,125.0,130.0,135.0,140.0,145.0,150.0,155.0,160.0,165.0,170.0,
175.0,180.0,185.0,190.0,195.0,200.0,205.0,210.0,215.0,220.0,225.0,230.0,
235.0,240.0,245.0,250.0,255.0,260.0,265.0,270.0,275.0,280.0,285.0,290.0,
295.0,300.0,305.0,310.0,315.0,320.0,325.0,330.0,335.0,340.0,345.0,350.0,
355.0] -> 177.5

The mean function used in above is defined as:

mean :: (Storable a, Fractional a) => Vector a -> a
mean xs = (foldVector (+) 0 xs) / fromIntegral (dim xs)

It requires a Storable type class constraint, and makes use of hmatrix’s foldVector function.

#### Accessing data values

Finally, we get round to reading the data that we’re interested in (of course, reading the metadata is a necessary prerequisite for this: this kind of geospatial data doesn’t mean much unless you can locate it in space and time, for which you need coordinate variables and their associated metadata).

The next listing shows how we read the $Z_{500}$ data into a row-major hmatrix matrix:

let (Just zvar) = ncVar nc "z500"
putStrLn $"z500 dims: " ++ show (map ncDimName$ ncVarDims zvar)
Right slice1tmp <- getA nc zvar [0, 0, 0] [1, nlat, nlon] :: MRet CShort
let (HMatrix slice1tmp2) =
coardsScale zvar slice1tmp :: HRowMajorMatrix CDouble
slice1 = cmap ((/ 9.8) . realToFrac) slice1tmp2 :: Matrix Double
putStrLn $"size slice1 = " ++ show (rows slice1) ++ " x " ++ show (cols slice1) putStrLn$ "lon(i=25) = " ++ show (lon @> (25 - 1))
putStrLn $"lat(j=40) = " ++ show (lat @> (nlat - 40)) let v @!! (i, j) = v @@> (nlat - i, j - 1) putStrLn$ "slice1(i=25,j=40) = " ++ show (slice1 @!! (25, 40))

There are a number of things to note here. First, we use the getA function, which allows us to specify starting indexes and counts for each dimension in the variable we’re reading. Here we read all latitude and longitude points for a single vertical level in the atmosphere (which is the only one there is in this file). Second, the values stored in this file are geopotential values, not geopotential height (so their units are m s-2 instead of metres, which we can convert to geopotential height by dividing by the acceleration due to gravity (about 9.8 m s-2). Third, the geopotential values are stored in a compressed form as short integers according to the COARDS metadata convention. This means that if we want to work with floating point values (which we almost always do), we need to convert using the hnetcdf coardsScale function, which reads the relevant scaling and offset attributes from the NetCDF variable and uses them to convert from the stored data values to some fractional numeric type (in this case CDouble – the destination type also needs to be an instance of hnetcdf’s NcStorable class).

Once we have the input data converted to a normal hmatrix Matrix value, we can manipulate it like any other data value. In particular, here we extract the geopotential height at given latitude and longitude coordinates (the @!! operator defined here is just a custom indexing operator to deal with the fact that the latitude values are stored in north-to-south order).

The most laborious part of all this is managing the correspondence between coordinate values and indexes, and managing the conversions between the C types used to represent values stored in NetCDF files (CDouble, CShort, etc.) and the native Haskell types that we’d like to use for our data manipulation activities. To be fair, the first of these problems is a problem for any user of NetCDF files, and Haskell’s data abstraction capabilities at least make dealing with metadata values less onerous than in C or C++. The second issue is a little more annoying, but it does ensure that we maintain a good cordon sanitaire between external representations of data values and the internal representations that we use.

### What’s next

We’re going to have to spend a couple of articles covering some background to the atmospheric variability problem we’re going to look at, just to place some of this stuff in context: we need to look a little at just what this study is trying to address, we need to understand some basic facts about atmospheric dynamics and the data we’re going to be using, and we need to take a look at the gross dynamics of the atmosphere as they appear in these data, just so that we have some sort of idea what we’re looking at later on. That will probably take two or three articles, but then we can start with some real data analysis.