The Elephant in the Room:

Unsubstantiated Complacency Regarding the ‘Assay Exchange’ Paradigm – Sampling Uncertainties with Hidden Economic Consequences

Kim H. Esbensen

By Kim H. Esbensen
Research Professor and Consultant

Duncan Aldwin Vogel

By Duncan Aldwin Vogel
Global Expert in Weighing, Sampling and Testing of Traded Commodities

A full version of this article has been published in Spectroscopy Europe


The assay exchange paradigm is an integral element in many contractual agreements stipulating how business transactions rely on comparison of two independent assay results for commercial trading purposes. It turns out that the current Assay Exchange vs. splitting gap paradigm incurs no less than two unrecognised sampling uncertainties, which leads to hidden adverse economic consequences at least for one – and sometimes for both – contractual parties. The magnitude of this unnecessary uncertainty is never estimated, which leaves management without information about potential economic losses – a breach of due diligence. However, all that is needed to resolve this critical issue is stringent adherence to the Theory of Sampling (TOS) by mandatory contractual stipulations of only accepting representative sampling and sub-sampling principles, leading to a community call to action.

Reasons Behind the ‘Assay Exchange’ Paradigm

The focus below is on a commercial process, for example, trading involving metal concentrates or a depositor delivering a consignment to a refining facility with the aim of refining various precious metals. The common issue is a need for fast accounting in trading, including in the second example, since the physical-chemical refining process operates on a much longer time scale than the desired business closure. In both cases, the speed and reliability of the accounting is of the essence.

The seller’s, or the depositor’s, assay results recorded on the suppliers’ waybills constitute the documentation relied upon for characterising the input physical movement of materials. However, these assays are generally not relied upon by contractual parties for the purposes of securing reliable output documentation used for business settlement. Instead, the sampling procedures used for providing the material for analysis are contractually the responsibility of both transaction parties individually. This means that contractual documentation, and comparison and/or reconciliation objectives are typically based on the resulting analytical results from two parties, each interested in optimising its own prospect in the commercial transaction. The assay exchange paradigm is designed to resolve the closing business settlement issues and interests on a ‘fair and equal’ basis.

The ‘Assay Exchange’ Paradigm

In commercial practice, an agreed assay is determined between the two parties via a negotiation, which is based on a complex technical background. The agreed assay is referred to as the ‘settlement assay’ and the process of negotiation is referred to as an ‘assay exchange’. The negotiation basis typically used is age-old and has remained effective and unchallenged for long:

  • A primary sample of the lot in question is split into three sub-samples intended for i) buyer; ii) seller; and iii) umpire.
  • Each party (seller and buyer) simultaneously advises the other of its assay result.
  • Either party is obliged to use the assay determinations produced by its laboratory (the parties can in fact submit any assay result they may desire, but there is an in-built near certainty for a punishment for a(ny) party wishing to tip the scales unilaterally hopefully in its own favour).[1]
  • If the two assays exchanged fall within a contractually specified range, ‘the splitting gap’, the mean of the two assays becomes the settlement assay – the end of business settlement – and the accounting department makes the necessary multiplications of tonnages, concentrations x unit prices, etc.
  • But if the difference between the two assays is greater than the ‘splitting gap’, an independent third-party umpire shall arbitrate, helping to determine a settlement assay in accordance with the contractually agreed procedure. Several variants of the details of this latter part of the paradigm exist, but the basic principle of a splitting gap determinant remains. This process is referred to as ‘going to umpire’.

[1] In some leading countries (e.g. Switzerland), providing the exact assay result is mandatory under the regulatory framework of the country, while in most countries, there is no such requirement. Also, bearing the umpire cost is barely an incentive not to manipulate the result. A typical analysis will cost US$1,000- US$2,000 per element, significantly less than the profit one can make by ‘tuning the result’ even slightly.

Figure 1. Conventional assay exchange paradigm – focus is exclusively on relative analytical reliability.

  • Details: The difference between the two reported assays is compared to the contractually agreed maximum ‘splitting gap’. If the difference between the reported assays is lower than this threshold, the average of the two assays becomes the settlement assay. If the difference is greater, the business paradigm dictates that the third sub-sample of the primary sample is assayed by the umpire laboratory (this sample will have been kept in secure storage until it is decided whether to include it in the assay exchange scheme or not). In this case, the middle of the three assays becomes the settlement assay. This is the most usual option, but there are occasionally minor variants in play. All these options have perceived, or real, advantages or disadvantages, according to the specific business philosophy of the parties involved.

Depending on the specific settlement option, one party ends up paying the umpire’s analysis fee, which may, or may not, be of economic importance relative to the sum total involved.1 Following the conventional scheme, the gains and losses to the buyer and seller will be the monetary value of the difference between the settlement assay and their respective own assays, adjusted for umpire fees for the ‘loosing party’.

This traditional assay exchange scheme is designed to determine a settlement with ease, clarity and speed under the tacit assumption that the settlement assay will always lie close to a target lot’s ‘true’ metal content and that all analytical differences are exclusively a result of relative analytical ability. This assumption is incorporated in Figures 1 and 2 – while Figures 3 to 6 portray the more realistic assay exchange setup pointed to in this article, explicitly acknowledging a sampling-before-analysis variance that will always also be present. This is explained in full detail below.

While in the real world the two-party setup has many manifestations, e.g. buyer vs. seller, loading port vs. discharge port sampling, analytical Lab A vs. analytical Lab B (Figure 2), the principal issue is identical, a determining assay exchange. In the following, the example of sampling and certification at two ports (seller’s loading port vs. buyer’s discharging port) is used (issues are identical for seller vs. buyer and two labs or more).

In the real world, which is always fraught with sampling uncertainties, the critical issue is that potential non-representative sampling impacts independently at the two ports. Potential biased sampling, as well as other sampling deficiencies, if/when present, will unavoidably result in significant, increased uncertainties (blue: avr. +/- 2 std in Figure 2). The point here is that the magnitudes of these inflated sampling-plus-analysis variances are never known within the conventional paradigm.

Figure 2. Two-party dilemma – double jeopardy. Non-representative sampling compromises assay results independently at the both loading and discharge ports. Compromised sampling performance will unavoidably result in larger-than-necessary uncertainties (blue: avr. +/- 2 std)

The Elephant in the Room

In the overwhelming number of cases, there are no sampling stipulations associated with the mandate: “the primary sample is divided into three sub-samples intended for the seller, the buyer, and the umpire”. Thus, the adverse impact from inferior sampling is not included in the conventional assay exchange paradigm. Rather, since assay exchanges are financial negotiations, generally they accept whatever the prevailing sampling conditions are. Below is explained how this means that the standard scheme can in fact produce a settlement assay significantly different from the actual (‘true’) metal content in the lot. This is termed the mismatch vs gap error (MvG).

MvG (Mismatch vs Gap) risk: Difference between assay values from two opposing parties, compared to the magnitude of a contractually agreed upon maximum ‘splitting gap’. The absolute assay difference is compared to a mutually agreed splitting gap range, regardless of the general level of the average gap concentration level, which leaves analytical accuracy stranded as a victim of economic expediency.

While the MvG is an acknowledged risk for/by both parties, in the interest of a quick business resolution, most parties are usually eager to get to the settlement assay without further ado, thus accepting the MvG (knowingly or unrecognised) to get the payment for goods delivered effectuated as fast as possible. This status quo is presumably a reflection that both parties consider this a symmetrical risk, not worth elaborating much upon for every single transaction in view of the magnitude of accumulated day-to-day business totals. Status quo for the, often hidden, MvG error is that assessment hereof is almost never included in the commercial contract stipulations.

In contrast to this complacency, the real state of affairs is described below and shown in Figures 3 to 6.

Figure 3. Functionality of assay exchange paradigm, the case of ‘go to umpire’, after which the middle assay will be the settlement value. Note the tacit assumption that the splitting gap is centred on the ‘true lot concentration’. (Illustration of the ‘no umpire case’ can be found in the original publication

The elephant in the room is the tacit, unwarranted assumption that the contractual splitting gap is always centred on the true average lot concentration. Note, for example, in Figure 3 that an acceptable settlement assay is easily reached via the assay exchange scheme regardless of whichever general analytical level is bracketed by the interval spanning the three samples involved.

But this assumption is severely challenged by the fact that the crucial primary sample (which is immediately divided into three sub-samples) is in fact sampling FROM a heterogeneous lot/materials.

Theory of Sampling (TOS) interlude: All materials in the realm of technology, industry, processing, trading, commerce … for which Testing, Inspection and Certification (TIC) is on the agenda, are heterogeneous – it is only a matter of degree. The Theory of Sampling (TOS) has for over 70 years proved the severe danger involved in assuming that there is no sampling error involved when extracting the primary sample. But this is not the place to detail TOS. There are ample sources of excellent background literature available, e.g. references [1-6] and further references herein.

However, the reality is even more complex. The full scenario behind the assay exchange paradigm is shown in Figure 4, emphasising no less than two sampling operations, each with its own sampling/sub-sampling errors and uncertainties involved, all before analysis.

Figure 4. Full assay exchange reality, acknowledging two sampling stage uncertainty impacts: PSE, SSE (red arrows). PSE: = Primary Sampling Error; SSE: = Secondary Sub-sampling Error.

Following TOS, when sampling heterogeneous materials (aggregate or solid materials and mixtures, materials with significant grain-size contrasts …), there is every reason to take notice – and specially to take appropriate operational precautions – regarding the impact of the dominant primary lot sampling error (PSE) [1-6]. And there are equally serious reasons to take appropriate precautions regarding the subsequent ‘sample division’ producing the three tacitly ‘assumed equal’ samples for the seller, the buyer and the umpire, for which there will be a Secondary Sub-sampling Error (SSE).

Assay Exchange Paradigm – the Grim Reality

The degree to which it has been possible to reduce the Primary Sampling Error (PSE) will determine the general analytical concentration level in the primary lot sample, which, therefore may be different from the ‘true lot concentration’ to some, generally unknown, degree (see Figures 5 and 6). Appropriate precautions first and foremost include TOS’s ability and success in eliminating the Incorrect Sampling Errors (ISE), which is the necessary condition for unbiased sampling [1-6]. This similarly applies to the subsequent sample preparation and division, ibid. Of these, the primary sampling errors (variance) will usually contribute with a dominating uncertainty contribution, but accidental residual heterogeneity within the primary sample may also contribute significantly with appreciable uncertainty contributions regarding ‘sample division’, then assuredly making the three replicate samples unequal to an unknown degree. These are issues very rarely understood or acknowledged.

There is a logical order to these complementary influences, as follows. The degree of incomplete primary sampling bias elimination will lead to a location of the splitting gap, which is manifested by a deviation from the assumed centring on the true lot concentration, as shown in Figures 5 and 6. This will be the situation regardless of whether the paradigm leads to ‘go to umpire’ or not.

Figure 5. Realistic splitting gap location, with a deviating analytical level as a function of the degree of incomplete sampling bias elimination and other ISE deficiencies. (Illustration of the ‘no umpire’ case can be found in the original publication)

Mismatch vs. Gap Error (MvG)

A deviation between the settlement assay and the ‘true lot’ concentration is termed the ‘Mismatch vs. Gap’ error (MvG). Adding in the sample triplication division error (sub-sampling bias and/or variance), the relative disposition of the three analytical results cannot be ignored. The tripartite assay results from Lab A, Lab B and Lab Umpire will depend on to which degree the primary within-sample heterogeneity has been successfully reduced/eliminated by appropriate TOS action before (thorough mixing) and during the practical sample division (proven, documentable sub-sampling).

The full MvG error can be illustrated with graphic clarity (Figure 6). Note that within the conventional assay exchange paradigm, the MvG uncertainty is tacitly always assumed to be zero. The MvG uncertainty constitutes an economic risk, which needs to be managed, but this is manifestly not a concept envisaged in the conventional assay exchange paradigm.

Figure 6. Illustrating the MvG risk, showing three alternative analytical levels for the splitting gap – a case of going to umpire. The MvG error is only (close to) zero in the case of vanishing primary sampling uncertainty. (Illustration of the ‘no umpire’ case can be found in the original publication.)

Figures 5 and 6 show that the location of the contractual splitting gap can be significantly displaced from the assumed closeness to the true lot concentration (location in the Y-axis direction), which is caused by the degree of a primary sampling bias that has not been successfully mitigated. This applies both to the ‘no umpire’ as well as to the ‘go to umpire’ cases (only the latter illustrated here).

Note that the assay exchange scheme is followed regardless of this uncertainty, giving rise to potentially significant MvG errors with potentially significant economic losses.

The Point

The point to be made is that most current assay exchange practices do not include mandatory means to deal with sampling influences in the splitting gap accounting scheme.

Thus, there will always be a real, non-vanishing risk of settling a business transaction based on the assay exchange paradigm at a level which may actually lie significantly distanced from the true metal lot content, aLot, which is tantamount to a significant settlement bias. The deviation, the MvG error/risk, may be small (low heterogeneity materials and/or acceptable sampling performance) or it may be large (significantly to excessively heterogeneous materials; non-representative sampling competence/equipment), but the point is that this is studiously unknown to the contractual parties.[2]

[2] On record are a few discriminate sellers requesting the buying party to share the details of the sampling procedure, and possibly to have that procedure accredited following ISO 17025. And even when this is not the case, sellers may ask a supervisor, or an external TIC company, to monitor the sampling process and guarantee its robustness. Such measures will help to limit the risk of bias.

Awareness is good, see for example [7], but advice and practical ‘what-to-do-about-it’ tools are often missing. A comprehensive analysis of TOS as a determining element in risk assessment and risk management was published recently, in which all necessary and sufficient actions to remedy the adverse issues delineated above were presented: “Framing the Theory of Sampling (TOS) in Risk Assessment” [8].

It matters not that the economic value of this unmanaged risk may be small, because it may just as well be substantial (even large). This is entirely a function of the managed, or unmanaged, sampling errors, uncertainties and risks involved. Small effects may perhaps be wished for in status quo, but the real magnitude will forever be unknown, when the involved parties are studiously unaware.

Universal Resolution – Theory of Sampling (TOS)

One might perhaps worry that remedying the hidden MvG risk issues would demand a colossal effort – as checking for its magnitude at every commercial transaction would indeed be prohibitive.

However, all that is needed to resolve all issues delineated above is remarkably much simpler!

The entire array of debilitating issues regarding the conventional assay exchange scheme will conveniently go away if/when a mandatory statute is agreed upon by all parties only to use TOS-compliant representative sampling and sub-sampling/replicate sub-sample procedures throughout the full lot-to-aliquot pathway [1-6]. A two-sentence mandate that can be included in every relevant trade contract going forward will solve all problems: TOS to the fore!

"All sampling procedures invoked to secure primary samples, as well as all sub-sampling operations needed to produce the analytical aliquot, shall be 100% compliant with the principles of representative sampling as laid out by the Theory of Sampling, f.ex. codified in standard DS 3077 (2023). All sampling procuedures must be adequately and fully documented."

Figure 7. Credo


1. K.H. Esbensen, Introduction to Theory and Practice of Sampling. IM Publications Open, UK (2020). ISBN: 978-1-906715-29-8,

2. F.F. Pitard, The Theory of Sampling and Sampling Practice, 3rd Edn. CRC Press (2019). ISBN: 978-1- 138476486

3. G. J. Lyman (2019) Theory and Practice of Particulate Sampling – an Engineering Approach. Materials Sampling & Consulting. 540 p. ISBN 9781646333820

4. K.H. Esbensen and C. Wagner, “Theory of Sampling (TOS) versus measurement uncertainty (MU) – a call for integration”, Trends Anal. Chem. (TrAC) 57, 93–106 (2014). trac.2014.02.007

5. P. Gy, Sampling for Analytical Purposes. Elsevier, Netherlands (1998).

6. C.A. Ramsey, “Considerations for inference to decision units”, J. AOAC Int. 98(2), 288–294 (2015).

7. LBMA Assaying and Refining Conference 2021, 15 - 17 March 2021

8. (a) K.H. Esbensen & C. Paoletti (2022). “Framing TOS in risk assessment: an outreach perspective for the future”. Spectroscopy Europe/World, issue 34-8, p. 36-40. DOI: 10.1255/sew.2022.a26

(b) C. Paoletti and K.H. Esbensen, “Framing TOS in risk assessment: an outreach perspective for the future”, TOS Forum 11, 419–423 (2022).

(c) (Day 3/Session 9 (4 h:18 min:45 s)

[9] Clickable ref. to the original article in Spectroscopy Europe:[KE1]

Kim H. Esbensen

By Kim H. Esbensen
Research Professor and Consultant

Kim H. Esbensen, PhD, Dr (hon), has been research professor in Geoscience Data Analysis and Sampling at GEUS, the National Geological Surveys of Denmark and Greenland (2010–2015), chemometrics and sampling professor at Aalborg University, Denmark (2001–2015), professor (Process Analytical Technologies) at Telemark Institute of Technology, Norway (1990–2000 and 2010–2015) and professeur associé, Université du Québec à Chicoutimi (2013–2016). From 2015 he forged ahead as an independent researcher and consultant. A geologist/geochemist/metallurgist/data analyst by training, he has 20+ years experience in chemometrics, latterly focusing on representative sampling of heterogeneous materials, processes and systems: Theory of Sampling (TOS), PAT (Process Analytical Technology) and chemometrics. He has published more than 250 peer-reviewed papers and is the author of a widely used textbook in Multivariate Data Analysis. He was chairman of the taskforce behind the world’s first horizontal (matrix-independent) sampling standard DS 3077 (2013) and editor of the science magazine TOS forum and Sampling Column, while in 2020 he published the textbook: Introduction to the Theory and Practice of Sampling ( He can be contacted on

Duncan Aldwin Vogel

By Duncan Aldwin Vogel
Global Expert in Weighing, Sampling and Testing of Traded Commodities

Duncan Aldwin Vogel is a global expert in weighing, sampling and testing of traded commodities. During his study in business management at the International School of Economics, Rotterdam, Aldwin started building his pedigree in the renowned family inspection business Hoff & Co. Services BV that became part of Bureau Veritas in 2010. In 2022 Aldwin became Regional General Manager Europe for Alfred H. Knight and, in 2023, launched a TOS compliant Sampling Hub for circular commodity: Incinerator Bottom Ashes. His expertise covers all aspects of inspection, sampling and analysis starting from green field prospect requirements to fully implemented turn-key projects. Aldwin is highly experienced at all aspects of testing for Transportable Moisture Limit and was leader of the TML workgroup of the TIC Council. He is a delegate of the Netherlands on ISO Technical Committee 102 (Iron ore and direct reduced iron) and TC183 (Copper, lead, zinc and nickel ores and concentrates) where his focus is on sampling, sample preparation, moisture determination and TML. He can be contacted on