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

ODTK computes the stochastic modeling 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. ODTK will model the state 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 zero (0)
Sigma	Root variance of the initial error in the "Constant"; 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; controls how fast the state error estimate will decay in the absence of measurements; the shorter the half-life, the faster the decay

RandomWalk

The state estimated is an error to an input nominal (constant) value. ODTK will model the state 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 zero (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/time^1/2). For example, for a range bias, the units might be meters/sec^1/2. 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 meters will result in a process noise addition of 36 meter² across one hour (3600 sec).

Vasicek

ODTK will model the state 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	This is the constant (or long-term mean) value associated with the state parameter.
LongTerm.InitialEstimate	This is an initial estimate to the error in the long term value. Nominally this will be 0.
LongTerm.Sigma	This is the 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, use this input property to prevent the long-term mean error root variance from going below some threshold. ODTK will add process noise to keep the error root variance at or above this threshold. For measurement biases, this value is always input and displayed as a one-way value.
LongTerm.PNStep	Use this in conjunction with "LongTerm.ErrorThreshold". When the long-term mean error root variance falls below the threshold, ODTK adds process noise to set the error root variance to LongTerm.ErrorThreshold + LongTerm.PNStep. For measurement biases, this value is always input and displayed as a one-way value.
ShortTerm.InitialEstimate	This is an initial estimate to the actual error in the "Constant". Nominally this will be 0.
ShortTerm.Sigma	This is the root variance of the short-term error, where the variance is about the long-term mean.
ShortTerm.HalfLife	This is te "half-life" of the short-term state error estimate. It controls how fast the state error estimate will decay toward the long-term mean in the absence of measurements; the shorter the half-life, 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"