stateVector Concepts

The stateVector class is used to design and build state vector ensembles. The class implements stateVector objects, which act as design templates for state vector ensembles. Users can edit a template to describe an ensemble that matches their requirements. When finished, they can then use the template to automatically build a state vector ensemble that conforms to its design.

The stateVector design templates allow you to design a state vector in a flexible manner, while limiting the manipulation of large data arrays. To facilitate the design process, we will need some vocabulary to help us describe state vector ensembles in a systematic way. The remainder of this page introduces some key concepts that will allow us to flexibly design diverse state vector ensembles.

State Vectors

State vectors are a fundamental concept for ensemble-based assimilation methods. In brief, a state vector is a collection of climate variables and parameters that describe the state of the climate system over some period of time. These variables are reshaped into a vector of data values, hence a “state vector”. Typically, state vectors are constructed from climate model output, so each state vector provides one potential description of the climate system (or at least, the part of the system that we’re interested in). There is no strict definition for the contents of a state vector, but they often contain one or more climate variables at a set of spatial points. In some cases, a state vector may include a trajectory of several points in time - for example, successive months of the year, or successive years after a volcanic event.

In the context of assimilation, state vectors will generally include two types of variables:

1. Reconstruction targets, and

2. Climate variables used to run proxy forward models

In some cases, a variable can fall into both categories.

Whatever the application, a state vector can include data from a variety of climate variables at different spatial and temporal resolutions. Figure 1 illustrates an example state vector.

Figure 1: A state vector with temperature, mean temperature, and precipitation variables.

This example state vector includes data derived from two different climate variables (temperature and precipitation), data with different spatial resolutions (global-mean and spatial gridded), and different temporal resolutions (annual and monthly). Most likely, this state vector describes the state of the climate system for some modeled year.

Ensembles

The DASH toolbox focuses on ensemble DA methods, which rely on state vector ensembles. A state vector ensemble is a collection of multiple state vectors organized into a matrix. Since each individual state vector represents a possible description of the climate system, an ensemble provides an empirical distribution of possible climate states. For paleoclimate applications, the state vectors in an ensemble are typically selected from:

1. Different time steps,

2. Different climate models, and/or

3. Different model runs

and we often refer to the different state vectors in an ensemble as ensemble members.

Continuing the previous example, a small ensemble for our state vector might look like:

Figure 2: An example ensemble (N=5) selected from different time points and/or model runs.

Here, each column is a different state vector (ensemble member). Each ensemble member has data for the temperature, mean temperature, and precipitation variables, but from different time steps. In the case of ensemble member 5, the ensemble member is from the same time step as ensemble member 1, but from a different climate model run.

State Vector Variables

The stateVector class uses data from gridfile data catalogues to build state vector ensembles. In this tutorial, we will often refer to state vector variables. In the context of DASH, we define a state vector variable as some subset of data from a gridfile. Multiple variables can be derived from the same gridfile, and each individual variable forms a contiguous block of the state vector.

Returning to our earlier figure:

Figure 1: A state vector with temperature, mean temperature, and precipitation variables.

we can see that the example state vector includes three variables: temperature (T), global mean temperature (Tmean), and precipitation (P). The T and Tmean variables are probably derived from the same gridfile (and possibly the P variable as well).

Note that the definition of a state vector variable says nothing about specific climate variables. In fact, a given state vector variable can even include multiple climate variables, so long as those climate variables are derived from the same gridfile. (This can occur when a gridfile includes a var dimension with multiple variables). We’ll discuss state vector variables again in the next coding session. For now, the important point is that each variable is built from data in an associated gridfile.

Ensemble and State Dimensions

In the DASH toolbox, we will often refer to “state dimensions” and “ensemble dimensions”. These dimensions refer to the gridfile dimensions associated with each state vector variable. In brief, an ensemble dimension is a gridfile dimension used to select different ensemble members. For example, the time and run dimensions are often ensemble dimensions, because we often select different ensemble members from different time steps and/or model runs. Ensemble dimensions have constant metadata values along each column of an ensemble. Equivalently, ensemble dimensions have constant metadata values for each ensemble member. Ultimately, the metadata values over all the ensemble dimensions forms a unique metadata coordinate for each ensemble member.

Zooming in on the temperature (T) variable in our example ensemble:

Figure 3: Ensemble dimensions for a state vector ensemble.

we can see that time and run are the ensemble dimensions, because individual ensemble members are selected from different (time, run) coordinates. Each ensemble member (column) is associated with fixed time metadata and fixed run metadata, although these metadata values change between individual ensemble members (columns). Note that the time = 1 metadata value is repeated in ensemble members 1 and 5, but that the total metadata coordinate (time=1, run=1) and (time=1, run=2) is unique to each ensemble member.

By contrast, state dimensions are the remaining gridfile dimensions - that is, they are the dimensions that are not used to select ensemble members. State dimensions have constant metadata values along each row of a state vector ensemble (this is somewhat the inverse of ensemble dimensions). Continuing the example:

Figure 4: State dimensions for a state vector ensemble.

we can see that lat and lon are the state dimensions, because each individual state vector elements are selected from different (lat, lon) coordinates. Each state vector element (rows) is associated with fixed lat metadata and fixed lon metadata, although these metadata values can change between individual rows.

As you design variables in a state vector ensemble, you will eventually specify each gridfile dimension as either a state dimension or an ensemble dimension. This specification will help the stateVector class to automatically build a state vector ensemble from a particular design template. A common breakdown of dimensions is:

• State Dimensions: lon, lat, lev, site, and/or var

• Ensemble Dimensions: time and/or run

However, this is just a rule of thumb and not a strict requirement. Depending on your application, any dimension could be a state dimension or ensemble dimension.

Sequences

In some cases, you may want an ensemble dimension to also have some structure down the state vector. This most often occurs when you want a state vector to include several sequential points in time, and so we refer to this case as a sequence.

For example, you may want a state vector to include data from each month of the year. Although the time metadata associated with each ensemble member will refer to a particular year, individual state vector elements will also be associated with some time metadata - in this case, the month of the ensemble member’s year.

Analogously, you may want a state vector to include data from several years following a major event (such as a volcanic eruption). Although the time metadata associated with each ensemble member will refer to the eruption, individual state vector elements will also be associated with time metadata - in this case, the number of elapsed years since the event.

Returning to the previous figures, let’s say I want the T temperature variable to include the spatial temperature grid from each of June, July, and August. In this case, the state vector ensemble for the T variable would have the following structure:

Figure 5: An example of a sequence for the time dimension. Each ensemble member (column) has time metadata listing the associated year. Each state vector element (row) has time metadata indicating a particular month.

We can see that time is an ensemble dimension and each ensemble member (column) has fixed time metadata listing the associated year. However, each state vector element (row) also has fixed time metadata listing the associated month. Because time is an ensemble dimension, but also has structure along the state vector, this state vector exhibits a sequence for the time dimensions.