J. Kissel, O. Patane, B. Lantz

After seeing my post of the current (2025-03-19) performance of the H1ISIBS in LHO:83470, Brian -- in his LHO:83473 comment -- rightly cautioned Oli to beware the difference between 
    (1) a "statistical" or "incoherent" model of the CART2EUL projection to the suspension point, where 
        . one takes the ASDs of the CART DOFs (which are inherently only containing amplitude information, no phase relation between channels), 
        . multiplies them by the CART2EUL coefficients, and 
        . takes the quadrature sum 
      to form an ASD model of the euler basis motion,
   vs.
    (2) a "linear combination" or "coherent" model of the CART2EUL project to the suspension point, where 
        . the time-series of each CART DOF are multiplied by the CART2EUL coefficients, 
        . the time-series are then coherently summed (where "coherently" summed just means the amplitude AND phase relationship between the channels has been preserved), and 
        . then an ASD is taken of that 
      to form an ASD model of the euler basis motion. 

He states 
    - "if the DOFs are independent (which maybe they are, and maybe they are not), then using the quadruture sum of the ASDs, (1), is a reasonable thing to do." and 
    - "I think this difference [between (1) and (2)] not going to impact any of your calculations"

I'd not seen a comparison of these two models either at all or in a long time, every chamber + SUS combination is different, and I had the data, so I made the comparison.

I'll discuss the 6 Euler Basis plots in reverse-traditional order, because they're easiest to understand progressively that way.

YAW
This plot is uninteresting, because the BS projection matrix from CART to EUL has only one unity element, mapping RZ directly to Yaw. 
However, it lets me introduce what I'll be plotting.
In the upper panel, this shows the both models of ASDs and the underlying Cartesian components multiplied by the CART2EUL matrix element. 
As expected here, the thick black dashed ASD -- the coherent sum (2) model -- is identical to that think magenta dashed ASD -- the incoherent sum model (2).

The lower panel is the ASD ratio of the linear sum (2) divided by the incoherent sum (1).
Of course, for this DOF, the two models are identical, so this ASD ratio is identically 1.0 across the whole frequency band.
With me so far? Good. :-P

PITCH
Here, because the Beam Splitter suspension is mounted in the center of the ISI BS optical table, yaw'd 45 degrees, RX and RY map to PITCH via sqrt(2) with the same sign.
But the RX and RY performance of the ISI BS is slightly different, so the ratio between (2) and (1) is interesting.
Most notably around the HEPI cross-beam foot resonance (traditionally called the "HEPI Pier resonance" prior to 2014; see LHO:13505) -- the broad feature at ~7 Hz -- where the ASD ratio shows that the incoherent sum model (1) under predicts Sus. Point displacement by a factor of ~1.35x w.r.t. the coherent sum model (2).
And then at some other feature at ~17 Hz, the incoherent sum model (1) is over predicting the Sus. Point displacement by ~(1/0.8) = 1.25x.

ROLL
OK, now flip the sign of the contribution of RY, and watch the coherent sum drop -- fascinating! The contribution of that same ~7 Hz feature is now dramatically over-predicted by the incoherent sum, by a factor of ~(1/0.4) = 2.5x. Are these two the inverse of each other? No! I don't show it explicitly, but comparing (2)/(1) for roll (the inverse of what's plotted) and (1)/(2) for pitch, the 7 Hz number is 0.74x and 0.52x respectively, so markedly different!  

VERTICAL
Now we're getting really interesting -- for vertical, Z is mapped one-to-one, but RX and RY are contributing in opposite sign, and with only *roughly* the same magnitude [m/rad] CART2EUL coefficient.
The incoherent sum (1) is overestimating the vertical displacement by as much as a factor of ~(1/0.2) = 5x where the vertical motion is limited by RX and RY between 0.5 and 3 Hz.
Wow!
I won't look type thru the rest of the plot, because the plot describes it best, but boy is it more interesting than I thought it would be.

TRANSVERSE
With transverse, even though this degree of freedom "doesn't matter" for the beam splitter, now we're cooking with 5 contributing Cartesian degrees of freedom and except for RZ they're all contributing at interesting levels.
Again, you reading the plot is more useful than me describing it here, but it's quite interesting that the linear sum (2) predicts more motion between 0.6 Hz and 3.5 Hz and the incoherent sum (1) predicts more motion overestimates the motion between 3.5 to 15 Hz.

LONGITUDINAL
Finally, the DOF we work the hardest on, shows contribution from all 5 Cartesian degrees of freedom. A lucky-coincidence perhaps, but it looks like the models are about the same for most of the frequency region, and the incoherent sum (1) is over-predicting the displacement between 3.5 to 15 Hz, which is re-enforcing Brian's comment.

WHAT DOES IT ALL MEAN?
Brian is, again, definitely right to call out that the linear sum (1) model is a better model of the displacement of the Sus. Point than the incoherent sum.
But, both I (and perhaps even he) definitely wasn't expecting factors of 2x discrepancy, let alone factors of 5x.
So, I think I might make Brian's conclusion from LHO:83473 a little stronger -- the difference between models will impact the calculations of the Bigger Beam Splitter Suspension (BBSS) performance, so for the update to the seismic input motion, I'll *not* just update the performance from the ~2005 seisBSC.m estimate to the current 2025 real *cartesian* performance incoherently projected to the Sus Point, but instead update it to the current 2025 real *euler* Sus. Point performance computed in the front-end.