This article details the advanced signal acquisition mechanics and mathematical reconstruction algorithms behind Full Matrix Capture (FMC) and the Total Focusing Method (TFM), which represent the state-of-the-art in phased array ultrasonic imaging.
1. Principles of Full Matrix Capture (FMC)
In standard phased array configurations, group elements are fired simultaneously with specific delay profiles to form a single synthesized beam along a defined angle or focus depth. While fast, this physical beamforming discards substantial spatial-acoustic information.
Full Matrix Capture (FMC) is an alternative data acquisition strategy that systematically records the complete ultrasonic propagation matrix across every individual transmit-receive element combination.
The FMC Acquisition Process
For a phased array probe containing N elements:
- Element 1 is excited as a single transmitter. All N elements in the array (Elements $1 through N) receive the returning echoes simultaneously. The instrument records N independent A-scan waveforms.
- Element 2 is then excited as a single transmitter, and all N elements act as receivers, capturing another set of N A-scans.
- This sequence repeats sequentially up to Element N.
The resulting dataset is a raw matrix containing N \times N individual A-scans (the FMC Matrix).
FMC Raw Data Matrix [A_i,j(t)] (i: Tx Element, j: Rx Element)
Where A_i,j(t) represents the A-scan waveform where element i transmitted the pulse and element j received the echo. This matrix captures the total raw acoustic field data of the inspection zone, independent of any focal laws.
2. Total Focusing Method (TFM) Reconstruction Algorithm
The FMC matrix is a raw database that is not directly human-readable. The Total Focusing Method (TFM) is the post-processing algorithm that synthesizes these raw waveforms into a highly focused 2D cross-sectional image.
Mathematical Formulation
The TFM algorithm discretizes the target examination area into a grid of discrete pixels. For each pixel P(x, z) within the grid, the algorithm calculates the exact acoustic travel time from every transmitter i to the pixel coordinate, and back to every receiver j.
Let:
- d_Tx,i(P) be the distance from transmitter element i to pixel P(x, z).
- d_Rx,j(P) be the distance from receiver element j to pixel P(x, z).
- v be the acoustic velocity in the material.
The total round-trip travel time t_{i,j}(P) for a wave traveling from element i to pixel P and back to element j is defined as:
t_{i,j}(P) = \frac{d_{Tx, i}(P) + d_{Rx, j}(P)}{v}
The TFM engine extracts the amplitude value of the corresponding A-scan waveform A_{i,j} at the exact time offset t_{i,j}(P). The final intensity or amplitude I(P) of the pixel P is the summation of these extracted amplitudes across all transmit-receive combinations:
I(P) = \left| \sum_{i=1}^{N} \sum_{j=1}^{N} A_{i,j}(t_{i,j}(P)) \right|
By computing this summation for every pixel in the grid, the TFM reconstructs a unified cross-sectional image of the test object.
3. Advantages of FMC/TFM over Standard PAUT
The mathematical synthesis of TFM yields significant physical improvements in NDT inspections:
- Synthetic Focusing Throughout the Zone: Unlike standard PAUT which focuses at a single depth, TFM behaves as if the acoustic beam is focused at every individual pixel coordinate across the entire image area.
- Enhanced Spatial Resolution: The summation of N^2 signals significantly increases the signal-to-noise ratio (SNR) and allows the separation of closely spaced flaws.
- Characterization of Complex Geometry: Because raw data is preserved, TFM can calculate complex multi-mode wave paths (such as shear waves reflecting off the backwall before hitting the flaw) to construct images in highly complex geometric areas.