You can apply several filtering methods with user-specified settings to remove outliers and to smooth the position estimates. The smoothed position estimates (filtered values) are numerically displayed in the Filtered Results views. You can graphically compare them with the unfiltered results in the Chart views.

Outlier filtering using the median

The median is the middle value in a list ordered from smallest to largest. Using a median filtering technique you can smooth the data (offsets to coordinates) and remove outliers from the data, while preserving significant changes within curves ("edges"). The technique applied uses a sliding time window over either 3 or 5 epochs (selectable) and a minimum definition for outliers. If the value of the central epoch exceeds the user-defined offset threshold, the filter replaces this value by the median of the samples in the window. Thus, one outlier (with a window size of three epochs) or two consecutive outliers (with a window size of five epochs) will be removed, while a jump in the values will be kept.

When combining several filter methods, the median filter will be applied before other data smoothing methods, such as the weighted mean or the Kalman filter.

Weighted Mean

The weighted mean filter takes a specified period of processed data, computes a weighted mean from the set of data and forwards only this average to the depending modules. The contents of the set of data is time dependent: You can define a "moving" period which is used for averaging the data. For each epoch, all data older than the period defined in the Filter period setting is skipped and only the period up to the last available, synchronized epoch is taken into account. All data is weighted with their variances.

Kalman Filter

The processing engines provide position estimates in North, East, and height, and the corresponding 3x3 variance-covariance matrix. A Kalman filter can be used to filter the data set considering the covariance information.

The estimates of the station's position are used to estimate the position's time derivatives under the assumption of a linear, time discrete equation of motion. The state vector in the implemented Kalman filter has the general form

,

where the vector y has the length 3 and holds North, East, and height components. Besides the position vector y, the vector x holds the derivatives of y, i.e., velocity and acceleration.

The vector x has the length 3 in case of a Kalman filter with order 0, the length 6 for order 1 and the length 9 for order 2. The Kalman filter is setup in a way that a driving noise input is driving the highest derivative only. The general form of the driving noise is

.

In case of the order 0 only the w₁ is used, the vector elements for North, East, and height are identical. In case of the order 1 only the w₂ is used, the w₁ part of the vector is zero. In case of the order 2, the w₁ and w₂ parts are zero, only the w₃ part is used. For detailed information on the values and units of the driving noise see the description of the settings of the Filter Settings category.

Use median filter: If set to Yes, a median filter will be applied before other data smoothing methods, such as the weighted mean or the Kalman filter. Default: Yes.

Sliding time window size: Specifies the size in epochs of the sliding time window used by the median filtering method. The sliding time window always includes the last available epoch and the epochs before. A larger window filters more outliers, but also increases the epoch delay. With 3 epochs selected, a single outlier will be detected and removed; two consecutive outliers will stay in the data. With 5 epochs selected, two consecutive outliers will be removed and thus, a jump in the data might be detected one epoch later.

Outlier size: If a position differs from the median of data in the current time window more than the value set here, it is defined as an outlier and is replaced by the median. Default value: 0.05 meters.

Filtering method: Specifies the filter type for data smoothing. Available selections are Weighted mean and Kalman filter.

Tip - If you do not want to use any filtering, keep the default Weighted mean filtering method and select Unfiltered as the size of the sliding time window.

Sliding time window size: This setting is only available for the weighted mean filter. This filter takes a specified period (sliding time window) into account and uses the values available for this period (the set of data) to compute the weighted mean. This setting specifies the period the set of data covers. The sliding time window always includes the last available epoch and the epochs before. Possible selections are unfiltered (no data smoothing) or periods between 5 seconds and 4 hours. The larger the set of data, the more the data is smoothed. The default value is 1 minute.

Kalman filter characteristic: This setting is only available for the Kalman filter. It specifies a characteristic movement of the position you can expect within the network of selected stations. For more details on the available selections see the following table.

Selection	Order of model; driving noise	Characteristics
Static	Order = 0, driving noise = 1* 10-14	No motion expected. Highest accuracy, comparatively large delay of reaction to position changes. The resulting graph converges to a straight line.
Moderate movement (default selection)	Order = 2, driving noise = 1* 10-21	Slow or moderate motion expected. Fair accuracy and reaction time. The resulting graph smoothly follows the input data.
Sudden movement	Order = 0, driving noise = 1* 10-5	Sudden movements expected. Fast reaction time to position changes, comparatively high noise. The resulting graph closely follows the input data.
User defined parameters	Depending on additional user defined settings	Depending on user defined setting

Note - Trimble recommends user-defined parameters only to extremely experienced users who fully understand the impact of the values onto the results.

Order of model: Specifies the order of model and thus, whether the results are estimates of the position, the velocity, or the acceleration. Allowed orders are 0 (only position), 1 (position and velocity), and 2 (position, velocity and acceleration).

Driving noise: Specifies, depending on the order of the model, the variance of the noise in the position, velocity, or acceleration. The lower the value is, the higher the influence of past measurements is weighted. The unit of the driving noise depends on the order of the model: (m/sⁿ)² / s, where n is the order of the model. See the following table.

Selected Order	Unit of driving noise
0	m2/s
1	m2/s3
2	m2/s5