Stochastic Models
The Ansys Systems Tool Kit® (STK®) application 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.
STK processing computes the stochastic modeling based on algorithms given in the ODTK Orbit Determination: Theorems & Equations document in the Ansys Orbit Determination Tool Kit (ODTK®) Help.
GaussMarkov
The state estimated is an error to an input nominal (constant) value. The STK software will model the state using a scalar exponential process. In the absence of measurements, the state estimate will decay toward zero over time while the covariance on the estimate will tend to an upper bound defined by an input sigma.
| GaussMarkov Properties | |
|---|---|
| Property | Description |
| Sigma | Root variance of the initial error in the drag or SRP constant |
| Half-Life | 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 software 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 |
| Sigma | Root variance of the initial error in the drag or SRP constant |
| Diffusion Coefficient | 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. 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
STK software 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 |
| Sigma | Specify the root variance of the initial error in the drag or SRP constant. |
| HalfLife | Specify the half-life of the state error estimate, which controls how fast the state error estimate will decay in the absence of measurements. The shorter the half-life, the faster the decay. |
| Long Term Sigma | This is the root variance of the initial error in the Constant. |
| Long Term Error Threshold | 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. STK software 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. |
| Long Term PN Step | Use this in conjunction with Long Term Error Threshold. When the long-term mean error root variance falls below the threshold, STK software adds process noise to set the error root variance to Long Term.Error Threshold + Long Term PN Step. For measurement biases, this value is always entered and displayed as a one-way value. |
Stochastic density correction parameters
The following table describes the parameters for stochastic density modeling:
| Parameter | Description |
|---|---|
| Density Correction Half Life | Enter the time that it takes for dρ/ρ to decay to one-half its value in the absence of measurements. The quantity dρ/ρ is the estimated relative correction to the atmospheric density, where the latter is calculated via the selected nominal density model. |
| Density Correction Sigma Scale | Enter a multiplicative scale factor for the atmospheric density uncertainty computed by STK software. The computed density
uncertainty is based on the historical performance of the CIRA 1972 (Jacchia 1971) atmospheric density model considering the
current satellite state. This is available for Earth-orbiting satellites only.
AGI recommends a nominal value of 1.0 for most applications; use of other values should be considered experimental. |
| Density Ratio Root |
Enter the reciprocal of the exponent to be applied to the ratio of the local atmospheric density divided by the baseline density. STK software computes the density ratio as (current density / baseline density)^(1/DRR), where DRR is the DensityRatioRoot. STK software computes the local density using the current solar flux and geomagnetic activity, while computing the baseline density using average conditions. This is available for Earth-orbiting satellites only. AGI recommends a nominal value of 1.0 for most applications; use of other values should be considered experimental. |
| Density Ratio Increase Threshold |
Specifies a threshold for the stepwise change of a ratio of the density evaluated at perigee at the current solar and geomagnetic conditions to an evaluation of the density at mean solar and geomagnetic conditions. Recommended values are:
This is available for Earth-orbiting satellites only. |
| Add Process Noise | This check box is not available in the STK application. It controls the addition of white process noise in two components (specified via the OutOfPlaneFraction and the InPlaneFraction attributes) normal to the Earth-fixed velocity direction in the ODTK application. |
| Out Of Plane Fraction |
This is a read-only parameter that the ODTK application uses but the STK application does not. When you choose to add drag process noise, ODTK processing adds the acceleration noise in the direction normal to the Earth fixed velocity and computes the normal to the satellite position vector as the magnitude of the nominal drag acceleration multiplied by the Out Of Plane Fraction. |
| In Plane Fraction |
This is a read-only parameter that the ODTK application uses but the STK application does not. When you choose to add drag process noise, ODTK processing adds the acceleration noise in the direction normal to the Earth fixed velocity and in the plane defined by the satellite position and computes the Earth fixed velocity as the magnitude of the nominal drag acceleration multiplied by the In Plane Fraction. |
N-Plate Stochastic parameters
The following table describes the parameters applied to each plate group for N-Plate stochastic modeling:
| Parameter | Description |
|---|---|
| Name | Enter the name assigned to the plate group you want to specify. |
| Nominal Value | Enter the value of the nominal scale factor correction for the plate group. |
| EstimateParameter | Select the check box if you want the software to include this plate group in processing. |
| HalfLife | Specify the half-life of the stochastic modeling for this plate group. |
| Sigma | Enter the short-term uncertainty in the estimated scale factor correction. |
| LongTermSigma | Enter the long-term uncertainty in the estimated scale factor correction. |