# Stochastic Models

ODTK makes available three types of stochastic sequences to model the measurement state parameters associated with facility measurement biases, satellite measurement biases, transponder biases, and retroreflector delays.

Choices for the stochastic model include:

• GaussMarkov = State will be modeled as a scalar exponential Gauss Markov sequence
• RandomWalk = State will be modeled using a Wiener (Brownian motion) sequence
• Vasicek = State will be modeled using a Vasicek stochastic sequence. This is a two-parameter model that solves for both a short-term and long-term bias.

The stochastic modeling is computed based on algorithms given in the ODTK Orbit Determination: Theorems & Equations document.

## GaussMarkov

The state estimated is an error to an input nominal (constant) value. The state will be modeled using a scalar exponential process. In the absence of measurements, the state estimate will decay towards zero over time while the covariance on the estimate will tend to an upper bound defined by an input sigma.

GaussMarkov Properties
Property Description
Constant The constant (or nominal) value associated with the state parameter.
InitialError An initial estimate to the actual error in the "Constant". Nominally this will be 0.
Sigma Root- variance of the initial error in the "Constant". Also can be thought of as the root-variance of the error in the nominal value in the absence of measurements.
HalfLife The "half life" of the Gauss Markov state error estimate. This controls how fast the state error estimate will decay in the absence of measurements; the shorter the halflife the faster the decay.

## RandomWalk

The state estimated is an error to an input nominal (constant) value. The state will be modeled using a Wiener (Brownian Motion) process. In the absence of measurements, the state estimate will not change but the covariance on the estimate will continue to grow.

RandomWalk Properties
Property Description
Constant The constant (or nominal) value associated with the state parameter.
InitialError An initial estimate to the actual error in the "Constant". Nominally this will be 0.
InitialSigma Root- variance of the initial error in the "Constant".
DiffusionCoefficient Determines the amount of process noise to be added to the state covariance in going from time t1 to t2. The amount of process noise added will be a2Δt, where a = the diffusion coefficient and Δt = |t2 - t1|. Properly, the units on a are (bias units/time1/2). For example for a range bias the units might be meters/sec1/2. In the ODTK implementation only the bias units are displayed; the time units are implicitly taken to be seconds. So, again using a range bias as an example, a DiffusionCoefficient input of 0.1 meter will result in a process noise addition of 36 meter2 across 1 hour (3600 sec).

## Vasicek

The state will be modeled using a Vasicek stochastic sequence. This is a two-parameter model that solves for both a short-term error and a long-term mean error to an input nominal (constant) value.

Vasicek Properties
Property Description
LongTerm.Constant The constant (or long-term mean) value associated with the state parameter.
LongTerm.InitialEstimate An initial estimate to the error in the long term value. Nominally this will be 0.
LongTerm.Sigma Root- variance of the initial error in the "Constant".
LongTerm.ErrorThreshold The modeling of long-term mean error does not include the addition of any process noise to this parameter. Therefore over time the covariance of this parameter will approach zero. To avoid numerical issues associated with a zero covariance this input property is included to prevent the long-term mean error root-variance from going below some threshold. Process noise will be added to keep the error root-variance at or above this threshold. Note that for measurement biases this value is always input and displayed as a one-way value.
LongTerm.PNStep Used in conjunction with "LongTerm.ErrorThreshold". When the long-term mean error root-variance falls below the threshold, process noise will be added to set the error root-variance to LongTerm.ErrorThreshold + LongTerm.PNStep. Note that for measurement biases this value is always input and displayed as a one-way value.
ShortTerm.InitialEstimate An initial estimate to the actual error in the "Constant". Nominally this will be 0.
ShortTerm.Sigma Root- variance of the short-term error where the variance is about the long term mean.
ShortTerm.HalfLife The "half life" of the short-term state error estimate. This controls how fast the state error estimate will decay towards the long term mean in the absence of measurements; the shorter the halflife the faster the decay.

## Scripting Examples

The following are VBScript examples of setting the stochastic model inputs for measurement biases and transponder biases.

'''''''''''''''''''''''''''''''''''''''''''''''''''''
' Set Facility Range Bias to use GaussMarkov Model
'''''''''''''''''''''''''''''''''''''''''''''''''''''
set measStats = facility.MeasurementStatistics

set msIter = measStats.FindByName("Range")
If msIter.IsSafeToDeReference() then
set ms     = msIter.Dereference()
set model  = ms.Type.BiasModel
model.Type = "GaussMarkov"
model.Constant.Set 20,"m"
model.InitialEstimate.Set 0,"m"
model.Sigma.Set 10,"m"
model.HalfLife.Set 60,"min"
end if

'''''''''''''''''''''''''''''''''''''''''''''''''''''
' Set Facility Doppler Bias to use Vasicek Model
'''''''''''''''''''''''''''''''''''''''''''''''''''''
set measStats = facility.MeasurementStatistics

set msIter = measStats.FindByName("Doppler")
If msIter.IsSafeToDeReference() then
set ms     = msIter.Dereference()
set model  = ms.Type.BiasModel
model.Type = "Vasicek"
model.LongTerm.Constant.Set 10,"cm/sec"
model.LongTerm.Sigma.Set 4,"cm/sec"
model.LongTerm.ErrorThreshold.Set 1.0e-6,"cm/sec"
model.LongTerm.PNStep.Set 2.0e-6,"cm/sec"
model.ShortTerm.InitialEstimate.Set 0,"cm/sec"
model.ShortTerm.Sigma.Set 1,"cm/sec"
model.ShortTerm.HalfLife.Set 5,"min"
end if

'''''''''''''''''''''''''''''''''''''''''''''''''''''
' Set Transponder Bias to use Random Walk Model
'''''''''''''''''''''''''''''''''''''''''''''''''''''
transponder.BiasData.Type = "Time Units"
set model  = transponder.BiasModel
model.Type = "RandomWalk"
model.Constant.Set 1000,"nsec"
model.InitialEstimate.Set 0,"nsec"
model.InitialSigma.Set 1,"nsec"
model.DiffusionCoefficient.Set 0.02,"nsec"

ODTK 6.5