# ECCO Adjoint & State Estimation

Let’s examine some key concepts related to the most fundamental application of the adjoint for ECCO: state estimation or optimization.

**Forward model**uses computer code (i.e., algorithms) to simulate changing conditions in a forward direction, for example ocean currents moving heat over time (click to see an example).

**Model – data differences**are used to determine the "misfit" between the forward model's calculation and observation data (e.g., ocean temperature measured by an in-water sensor).

**Cost function**measures how well a forward model's output matches observations (e.g., weighted quadratic function of the model-data differences). It's important to minimize the cost, which can be thought of as computation time.

**Control variables**have values that are subject to change. Examples include initial conditions or boundary conditions. Control variables can be adjusted to change the output of a forward model.

**Adjoint models**can be used to adjust control variables and reduce model-data differences. Not only that, adjoint solutions can provide a measure of

*sensitivity*of the cost function to the physical variables of the system. The adjoint is used to calculate the

*gradient of the cost function (see next)*with respect to the control variables.

**Gradient of the cost function**is important information because the ECCO model is nonlinear and has billions of variables! Thus, iterative searches for the optimal solution (i.e., minimized cost function) need

*directional information*– e.g., which way is downhill?

Now that we understand the various components of the optimization problem, let's see how they **fit together** in a high-level conceptual diagram.

Control variables set the stage for the forward model run. Model "data" (i.e., output) is compared to its real-world counterpart, observation data. Differences between the model and observation data are calculated as a key part of the cost function. Minimizing the cost function is achieved by analyzing its gradients, which are provided by the adjoint model.