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LAMBDA (least-squares ambiguity decorrelation adjustment)

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LAMBDA (Least-squares Ambiguity Decorrelation Adjustment) is a technique used in GPS (Global Positioning System) applications to resolve integer ambiguities, which is crucial in achieving high precision positioning. In this article, we will discuss what integer ambiguities are, the need for resolving them, how LAMBDA works, and its advantages and limitations.

Integer Ambiguities

GPS positioning relies on measuring the time taken for radio signals to travel from a GPS satellite to a GPS receiver on the ground. The receiver calculates the distance to the satellite based on the travel time, which is called a pseudorange. To obtain accurate positioning information, the receiver must also determine its own location in space, which is done by measuring the range to multiple satellites and using trilateration. The range measurements are affected by several sources of errors, including atmospheric delays, receiver noise, and multipath.

To correct for these errors, the receiver uses a technique called differential GPS, which involves comparing the measurements from a stationary reference receiver with those from a moving receiver. However, differential GPS only corrects for errors that affect both receivers equally. Other errors, such as satellite clock errors and orbital errors, affect all GPS users equally and cannot be corrected by differential GPS.

One of the most significant errors that affects GPS positioning is the integer ambiguity. Pseudorange measurements are affected by both the distance to the satellite and the integer number of carrier cycles between the satellite and receiver. The fractional part of the carrier phase is continuous and can be measured with high accuracy, but the integer part is ambiguous and cannot be determined directly from the measurements.

For example, if the carrier phase measurement is 123.4567 cycles, the integer part could be 123, 124, or any other whole number. This ambiguity affects the accuracy of the range measurement and can cause positioning errors of several meters.

Resolving Integer Ambiguities

Resolving integer ambiguities is essential for achieving high-precision GPS positioning. There are several techniques for resolving integer ambiguities, including:

  1. LAMBDA
  2. Integer aperture estimation
  3. Integer least squares
  4. Bootstrapping

In this article, we will focus on LAMBDA.

How LAMBDA Works

LAMBDA is a method for resolving integer ambiguities using least-squares estimation. It involves forming a set of linear equations based on the carrier phase measurements from multiple satellites and solving them for the integer ambiguities.

The basic idea behind LAMBDA is to use a linear combination of the carrier phase measurements to create a new observable that is independent of the integer ambiguities. This new observable is called a decorrelated phase measurement or DPM.

The DPM is calculated as follows:

DPM = b1 * Phase1 + b2 * Phase2 + ... + bn * Phasen

where Phase1, Phase2, ..., Phasen are the carrier phase measurements from n satellites, and b1, b2, ..., bn are weighting coefficients that depend on the geometry of the satellites and the receiver. The weighting coefficients are chosen to minimize the variance of the DPM.

The DPM can be written in matrix form as:

DPM = A * N

where N is a vector of integer ambiguities, and A is a matrix of coefficients that depend on the carrier phase measurements and the weighting coefficients.

The integer ambiguities can be solved by minimizing the residuals between the DPM and the linear combination of the pseudorange measurements, which are not affected by the integer ambiguities. This is done using a least-squares estimation technique, which involves minimizing the sum of the squared residuals.

The solution for the integer ambiguities can be written as:

N = (AT * W * A)-1 * AT * W * (P - b) where AT is the transpose of matrix A, W is a diagonal matrix of weights that depend on the measurement errors, P is a vector of pseudorange measurements, and b is a vector of biases that depend on the receiver and satellite hardware.

The LAMBDA technique involves iterating this least-squares solution several times, each time refining the weighting coefficients and biases to improve the accuracy of the integer ambiguity solution. The iterations continue until the residuals between the DPM and the pseudorange measurements are minimized.

Advantages of LAMBDA

LAMBDA has several advantages over other integer ambiguity resolution techniques:

  1. It is computationally efficient and can be implemented in real-time on a GPS receiver.
  2. It is robust to measurement errors and can handle large numbers of satellites.
  3. It is accurate and can resolve integer ambiguities to within a few millimeters, depending on the quality of the carrier phase measurements.
  4. It can be used in both single- and dual-frequency GPS receivers, making it widely applicable.

Limitations of LAMBDA

Despite its advantages, LAMBDA has some limitations that should be considered:

  1. LAMBDA assumes that the carrier phase measurements are uncorrelated, which may not be true in some situations, such as when the satellites are closely spaced or when the receiver is in a multipath-prone environment.
  2. LAMBDA requires a good estimate of the pseudorange errors, which may be difficult to obtain in some situations, such as when the receiver is in a highly dynamic environment.
  3. LAMBDA may fail to resolve integer ambiguities if the geometric configuration of the satellites and the receiver is poor, or if there are insufficient measurements to provide a good solution.

Conclusion

LAMBDA is a powerful technique for resolving integer ambiguities in GPS positioning. It works by forming a set of linear equations based on carrier phase measurements from multiple satellites, and solving them using a least-squares estimation technique. LAMBDA is computationally efficient, accurate, and robust to measurement errors. However, it has some limitations that should be considered, such as the assumption of uncorrelated carrier phase measurements and the requirement for a good estimate of the pseudorange errors. Overall, LAMBDA is an essential tool for achieving high-precision GPS positioning.