I read a number of papers yesterday and looked at some of TiN's data. I've had a long think about the system this morning.
Error terms I have identified so far:
temperature
pressure
humidity
age
thermal gradient
current level
surface leakage
resonant modes in the enclosure
The errors introduced by changes in temperature,pressure and humidity have demonstrated hysteresis, particularly affecting the resistors. As a consequence, to mathematically correct for errors they introduce would require a complete record of the environmental changes. It would also require developing models which would require an extraordinary amount of data. The effects are near linear with small hysteresis for small excursions, but quickly become problematic. The most appropriate means of addressing these is to control the environment as tightly as possible.
While not appropriate for the requirements posited, in general a thick wall enclosure machined from a block of 1008 low carbon steel, filled with dry nitrogen and sealed is the most economical and easiest option to fabricate. The thick walls have the additional benefit of providing good magnetic and electric shielding. Such an enclosure should be annealed after fabrication which requires a very specific temperature cycle over 1-6 hours. Unless one happens to have the requisite machine tools and furnace, such effort is only relevant to an ultra precision reference because of the cost
If one is willing to forgo very tight pressure control, a lightweight steel enclosure can serve for controlling temperature and humidity at an economical cost while still providing some damping of pressure changes. An American pattern .50 BMG ammo should provide a satisfactory enclosure, allowing plenty of room for insulation and a large battery for hot shipped travel standard use.
Thermal gradient and current levels are probably first order effects. Irrespective of the enclosure temperature, device heating leads inevitably to thermal gradients. Thus it is useful to minimize currents as much as is consistent with electrical noise. In the case of a buried zener reference such as the LM399, minimizing the heater current will serve as a better proxy than using a temperature sensor. However, such an approach may necessitate logging temperature so as to correct for hysteresis and aging effects upon the resistors.
Current levels are a significant factor both in resistor aging and hysteresis. The latter can be avoided by avoiding changes in current level. The effect on aging must be dealt with mathematically.
Surface leakage is not something I have any real sense of and undoubtedly varies widely depending upon board cleaning, materials etc. That will require some investigation. It may well be a 3rd order error term that can be neglected. For a high precision device I should expect that a ceramic substrate would be desirable.
Zener diodes are intrinsically broadband noise sources. Placed in a metal enclosure will it will develop resonances controlled by the enclosure dimensions. Without adequate filtering this can interact with the sampling to produce errors. While estimates are readily made of the frequencies, power levels and radiation efficiency have a dominant effect. This argues for a PCB layout which minimizes radiation.
The level of noise which I have observed in the data from TiN suggests to me that it might be better to forgo the PLC sampling and collect measurements at the maximum rate the DMM can make and then use a Fourier transform to determine the DC value. If one discards all but a random subset of the data, then the requirements for compressive sensing are met and aliasing does not take place. The use of the Fourier transform has a rather significant advantage in that it should permit making readings to 8.5 digits with a 5.5 digit DMM by taking a sufficiently long sample. There are a number of mathematical constraints which must be satisfied, but this is often the case.
I think I should note that one can acquire data with arbitrary amplitude resolution using a single bit ADC. Sam Allen published a good bit of work on the subject in the '70s. In his case he was acquiring very high fold vibroseis data (1024 channels when others were only collecting 48). The rapid evolution of seismic recording systems overtook him and the technique is largely forgotten. It's actually a trivial application of the work of Norbert Wiener.
As regards the choice of resistors I have nothing more to say at this time other than they need to be "good enough". There is wide variation among manufacturers and processes in the behavior of resistors and their sensitivity to environmental factors. I shall eventually have three LM399s running in a controlled chamber which use low spec resistors. One is running the best matched pair which had an initial resistance before soldering within 0.02%. Temperature coefficients were not measured.
As regards the mathematics. In pore pressure prediction a common approach is to use a reference function which is the sum of a constant and an exponential with a constant exponent. As I had a vast amount of data I could see that such a form was not a particularly good fit. It looks pretty good with a small data sample from a few wells, but not when you have thousands. Ultimately I found that I could get a much better fit by using a function of the same form, but with a polynomial exponent. Were I to do the same project today, I should construct a vast array of curves calculated from the physics of fluid flow in porous media and then use a sparse L1 pursuit to find the best solution. None of the papers I've read to date suggest that any of the authors knew how to do the sort of approximation I was doing over 10 years ago. There may well be a group which has. If so I look forward to reading their work.