Power Profile on the Water

The Power Profile on the Water: A Canoe-Specific Method for Critical Power and W' from Short Maximal Efforts

Why this post?

In our post Metabolic Thresholds on the Water↗, we determined the full set of metabolic thresholds for the same two canoe sprinters, namely the first and second lactate thresholds (LT₁ and LT₂), the two ventilatory thresholds (VT₁ and VT₂) and the modeled maximal lactate steady state (MLSS), the highest power at which blood lactate maintains a stable steady state. All of these values require a blood sample, a gas exchange measurement or a metabolic model. None of this is available in everyday training, but the power measured in the boat is.

That post also ended with an open problem. The previously assumed critical power (CP) substantially exceeded the measured LT₂ consensus value in both athletes and came close to the power at V̇O_2max, the maximal oxygen uptake (Table 1). A sustainable endurance boundary so far above the second lactate threshold is physiologically implausible, provided the measured LT₂ really marks the point at which lactate production exceeds its removal and no steady state can be maintained.

Table 1 shows how far both previous estimates, the simple fit and the 3-min all-out test examined below, lie above the measured thresholds.

Table 1 The previous critical power estimates in their physiological context

The old fit therefore provides no sound basis for further analysis. This post first clarifies where the error comes from and how large it is. It then presents a procedure tailored to canoe sprint that delivers physiologically plausible values for CP and W' even without long maximal efforts.

What the models do and where their validity ends

In the heavy and severe domains, the power-duration relationship can be described as a hyperbola, a curve that first drops steeply and then levels off toward a limit. That limit is the critical power, and the curvature describes the anaerobic work capacity W' (Monod & Scherrer, 1965; Morton, 1996; Poole et al., 2016). The basics are covered in the second post of our power-based training series↗, and the measurement of propulsive power in the first↗. The classic models are intended for durations between 2 and 15 min, relying on pure mathematical extrapolation beyond that range (Leo et al., 2022), as Figure 3 below shows.

The data

The profile is based on three maximal test batteries on the water, all in a K1 and all with power measured by the PaddlePulse.

• Test 01 with two sprints of 15 s and one maximal effort over 6 min. It covers both the short and the long end in a single session.
• Test 02 with one maximal effort over 3 min and one over 1 min. It fills the gap in between.
• Test 03, a 3-min all-out test. Its best values feed into the best-power curve like those of the other tests, while its own estimate of CP and W' serves only as a cross-check.

Why the longest effort lasted only six minutes

The literature is clear. A robust estimate of CP and W' calls for three to five maximal efforts between 2 and 15 min, with the shortest between 2 and 5 min and the longest between 12 and 15 min (Leo et al., 2022). Efforts under 2 min do not reliably reach V̇O_2max and therefore fall outside the severe domain. Efforts longer than ~15 min are not recommended, because glycogen depletion and motivation then start to influence the result (Leo et al., 2022). In addition, exhaustion at critical power typically occurs within 20 to 30 minutes regardless (Jones et al., 2019), so even longer efforts leave the range the model describes. At six minutes, our longest effort sits at the lower end of this window, leaving the upper duration boundary completely unanchored.

Nevertheless, our longest effort lasted only six minutes. There were three reasons for this.

• Race relevance. The Olympic distances of 200, 500 and 1000 m correspond to race times of roughly 35 s to about four minutes. For 1000 m, 262 s has been reported in elite kayakers (Pilotto et al., 2019). A maximal effort over 15 min has no counterpart in competition outside canoe marathon.
• Specific training background. Sprint paddlers almost never perform maximal efforts of this length in training. A 15-minute all-out effort requires a pacing strategy they have never practiced, and an unfamiliar test format of this kind systematically underestimates true performance.
• Course conditions. The tests took place on a river, which simply lacks a stretch long enough for an evenly paced maximal effort of 12 to 15 min, a problem compounded by river flow, bends, bridge passages and boat traffic.

This limitation is exactly why the simple fit failed in both cases, and the procedure presented here has to compensate for it. It therefore imposes physiologically derived limits on the fit, so that a plausible critical power can be obtained even from mostly short maximal efforts.

From raw recordings to the data points used in the fit

Taking the best mean power for every duration across all three tests gives the best-power curve (mean maximal power curve). This alone raises the maximal three-minute power from 258 to 298 W (Athlete 01) and from 167 to 184 W (Athlete 02). However, not every point on this curve represents a discrete effort. The best 20 s window of a six-minute effort, for example, is merely a subsegment of it. Only genuine, discrete maximal efforts are therefore included in the fit, identified in three steps without fixed time windows and without a pre-fitted model.

1. Discrete bout versus subsegment. On the work-duration axis, every subsegment of an evenly paced effort lies on a straight line through the origin, while a genuine maximal effort lies above it on the upper convex hull, the upper envelope of all points (Drake et al., 2024).
2. Selection by quality. The longest effort is always retained, as it alone anchors the asymptote. After that, selection depends on the height above the athlete's own trend and on a clear subsequent drop in power. Durations immediately adjacent to an already selected point (e.g., 57 s vs. 58 s) are excluded to prevent redundant clustering.
3. Check. The decay limit and the W' pair test remove durations that cannot have been maximal (appendix).

Figure 5 shows the selected points as large circles, just as the app does. In the app, any effort can be deselected or a new one added, such as a race result. In its standard mode, the sensor smooths the first seconds, so this range is mathematically de-smoothed to remove the filter effect and recover the true peak power. It nevertheless remains the least certain part of the profile. The PaddlePulse also offers an unsmoothed mode for this purpose.

Table 2 Maximal efforts included in the model fit

The longest effort carries the most weight in the fit, while efforts under 30 s carry less (weighting). How the de-smoothed peak power is used is explained in the appendix.

The 3-min all-out test as a cross-check

The 3-min all-out test is considered a time-efficient alternative to the classic multi-test approach. Instead of fitting a hyperbola, it reads two values directly from the power trace. The mean power of the last 30 s is taken as critical power and the work done above that level as W' (Burnley et al., 2006; Vanhatalo et al., 2007, Figure 1). This only holds under two assumptions. W' must be essentially used up within the first 150 s or so, and the athlete has to go flat out from the very first stroke. Leo et al. (2022) generally advise against single tests, because they only represent one point on the power-duration curve, and they explicitly do not recommend the 3-min all-out test either, as it can overestimate the power-duration relationship in the severe domain.

Figure 1 Schematic of the 3-min all-out test. The mean power of the last 30 s is taken as critical power, the hatched area above that level as W'. Adapted from Sreedhara et al. (2019, Fig. 4).

First check: was it really all out?

Starting too fast is not an execution error in this format, because the protocol demands exactly that. The critical execution error is the opposite, submaximal pacing, which artificially inflates critical power. We evaluate this with a ratio curve R(t). From the de-smoothed signal, two best-power curves are built for 1 to 180 s, one from the all-out test alone and one from Tests 01 and 02, and R(t) is their ratio. A curve that is flat and close to 1 from the start indicates a genuine all-out effort, whereas one that climbs from below indicates pacing. In the following figures and analyses, a duration is classified as paced if the all-out test stays below 95% of the other efforts there (Figure 2).

The analysis reveals a clear contrast between the two athletes. For Athlete 01, R is close to 1 after only a few seconds and even above 1 on average, so in the all-out test he outperformed his other efforts over seconds and minutes. His test was therefore essentially all out. For Athlete 02, by contrast, R only meets the criterion after almost a full minute. Her test is a textbook example of a paced all-out test. She started well below her potential, leaving disproportionate reserve capacity for the final 30 s. That is exactly what makes the test unusable as the sole basis in this case, because if W' is not used up, the end power lies above the asymptote. Critical power is then systematically overestimated (162 instead of 132 W) and W' underestimated (4.20 instead of 6.89 kJ).

Figure 2 Pacing in the 3-min all-out test, Athlete 01 at the top and Athlete 02 at the bottom, both on the same axes. Blue is the best-power curve from the all-out test alone, dark gray the one from Tests 01 and 02, both from the de-smoothed signal. Dashed gray on the right axis is their ratio R(t), and the dotted line marks the 0.95 criterion. Shaded red are the durations at which the all-out test stayed below 95% of the other efforts, in other words where the athlete paced the effort. For Athlete 01 this applies only to the first six seconds, for Athlete 02 to almost the entire first minute.

Second check: is the result plausible at all?

As a stand-alone estimate, the test gives 229 W (W' 12.53 kJ) and 162 W (W' 4.20 kJ), even more than the already too high simple fit (Table 1). The literature expects critical power to lie only about 7% above the maximal lactate steady state, with study means between 1 and 16% (Dekerle et al., 2003; Jones et al., 2019), and our measured LT₂ consensus value is of the same order. Gaps of 39 and 53% ▲ are incompatible with that. The test is also debated in the literature. In the laboratory, end-test power sometimes agreed well with conventionally determined CP, but in other cases only for certain test variants (Vanhatalo et al., 2007; Wright et al., 2017). Other studies found the resulting CP to be too high, with participants exhausted after only 12 to 15 min on average (McClave et al., 2011; Bergstrom et al., 2013), and in the field the test overestimated the maximal metabolic steady state (Maunder et al., 2022).

The cause is not the same for both methods. The simple fit fails because of extrapolation, since it has to estimate an asymptote from data no longer than six minutes. The all-out test fails because of its own assumptions, since it fits no curve at all but assumes that W' is used up after about 150 s and that the remaining power equals the asymptote. If W' is not fully depleted by the end or the athlete paced the effort, the end power is too high. Athlete 02 exhibited this pacing artifact prominently, whereas Athlete 01 adhered closely to all-out criteria.

Four models and where they apply

So far we have used a single model, the hyperbola, and it produced an implausible result in both cases. Before adding extra assumptions, it is worth stepping back to ask which descriptions of the power-duration relationship are available and what each is meant for.

The starting point is a simple observation. If you plot maximal power against duration, the curve falls continuously, and the longer the effort, the flatter it gets. The models diverge fundamentally in how they extrapolate this decline at extended durations. Figure 3 shows the intensity domains together with the duration ranges each model is intended for. Within the measured window, all models lie close together and only diverge at the two ends. Exactly where they differ, we have no data, and that is precisely where the quantity we want to estimate lies.

Figure 3 The power-duration relationship with the intensity domains and, below it, the duration ranges for which the 2-parameter, the 3-parameter and the omni-domain model are intended. The shaded colors mark the intensity domains here and do not follow the color code of the text, and critical power is shown as a dashed gray line.

The hyperbola forces the decline toward a finite asymptotic limit, the critical power, and interprets the area above it as a finite store W'. The 2-parameter model (Monod & Scherrer, 1965) is its leanest form, the 3-parameter model (Morton, 1996) adds a finite maximal power P_max, and the omni-domain model (Puchowicz et al., 2020) lets power keep falling beyond a threshold duration T* instead of leveling off at a constant value.

Power-law modeling takes a fundamentally different approach. It goes back to Kennelly (1906), was popularized in running by Riegel (1981) and tested against the critical power models on large data sets by Drake et al. (2024). It describes the curve as a pure power function, P(t) = S · t^(E−1), and therefore has no asymptote at all. The hyperbola approaches critical power for long durations, whereas the power-law curve keeps falling toward zero. If you plot work instead of power against duration, the hyperbola becomes a straight line with slope CP and intercept W', while the power law stays curved (Drake et al., 2024). This is exactly the view our selection of key efforts relies on.

For our question, the key finding of Drake et al. (2024) is that the good fit of the hyperbola between 2 and 15 min is no proof that the power-duration relationship is hyperbolic, because within this window a hyperbola simply approximates a power law well. Outside it, the hyperbola behaves unrealistically, and attempts to fix this by attaching a second curve create a kink at the junction that has no physiological counterpart. Figure 4 shows both side by side, the two parameters of the power-law model on the left and the kink on the right.

Figure 4 Left, the two parameters of the power-law model. S shifts the curve up or down and corresponds to the power for one second, E determines how quickly it falls. A sprint specialist (solid) starts higher and loses more, an endurance specialist (dashed) starts lower and holds on longer. Right, measured best efforts of a cyclist with three composite omni-domain models (blue) and the power-law model (pink). The blue curves are hyperbolic below the threshold duration T* and decline above it, creating the marked kink at the junction. Adapted from Drake et al. (2024, Figs. 3 and 10), combined and recolored.

The power-law model describes the power-duration relationship with two parameters. The level parameter S sets the height of the whole curve and equals the power the model predicts for one second. The endurance exponent E indicates how strongly power falls with duration. The left part of Figure 4 illustrates what both mean. The reciprocal of the exponent, F = 1/E, is called the fatigue factor and goes back to Riegel (1981). The values for our two athletes, and why they cannot be compared directly with fatigue factors from running, are given in the appendix.

Table 3 summarizes which of these models enter the analysis and which do not.

Table 3 The models of the power-duration relationship used here

Why the fit needs constraints

The core difficulty is easy to state. The longest maximal effort lasts six minutes, whereas the asymptote of the hyperbola describes a state of unlimited duration. As a completely free parameter, it is poorly constrained by such short data, and the optimization then tends to place it too high, as happened in the previous fits. Three additional pieces of information are therefore added, derived in the appendix, along with a set of checks that mainly catch unfavorable data sets.

1. Physiologically bounded parameter ranges

Limits are set for CP, W' and P_max within which the optimization works. These limits are not cosmetic. They are the alternative to an optimum that is mathematically better but physiologically impossible. An asymptote above a power the athlete demonstrably could not hold for seven minutes cannot be the power that, according to the model, can be held indefinitely. Of course, indefinitely applies only within the model. In reality, critical power can usually be sustained for 20 to 40 min (Jones et al., 2019; Chorley & Lamb, 2020). The parameter boundaries for CP and P_max are deliberately broad, and in the primary model neither athlete's fitted parameters hit a boundary constraint. Only the unsupported fit for Athlete 02 ends with W' at the lower edge of the corridor described in the next section.

2. The corridor for the anaerobic work capacity

W' is primarily a model parameter, not a measured energy store. It is related to muscular energy stores without being identical to them (Skiba et al., 2015; Chorley & Lamb, 2020). The following calculation should therefore be read as a corridor and explicitly not as a measurement of W'.

Within this limitation, the bound can be estimated from quantities that do not come from the power profile, namely the active muscle mass, the measured efficiency and store sizes from the literature. The corridor consists of an alactic component from the high-energy phosphates and a lactic component from glycolysis, and their sum gives the combined corridor.

3. Support points where the data have gaps

The third building block uses the power-law model. It is fitted to the same efforts as the hyperbola but needs no asymptote and therefore remains well-behaved beyond the longest effort (Figure 4, right). Beyond the longest measured effort, support points are therefore placed on the power-law model, additional points that guide the hyperbola where measurements are missing. The remaining gaps on the duration axis, at the short end and between widely spaced efforts, are filled with minimum powers the model must not fall below. Where the best-power curve has a measured value, the minimum power is exactly that value, because the athlete demonstrably held it. In Figure 5, support points are shown as open circles and minimum powers as open triangles, and the details are in the appendix.

4. Checks that apply to unfavorable data sets

The building blocks described so far assume that the selected efforts really were maximal and that the test battery is long enough. For data sets where this does not hold, such as training recordings instead of tests, the procedure includes further safeguards, such as a decay limit, a W' pair test and a lower bound for CP, which are described in the appendix. None of them is triggered for the two athletes in this study, and their results are identical with and without them.

The prior mode. If the efforts cover too narrow a part of the duration axis or there are too few of them, the power-law model and W' are gradually pulled toward typical values, which was not necessary for either athlete here. In the app↗, this prior depends on the athlete profile, that is the chosen discipline from 200 m sprint to marathon and the performance level, leaning toward a large W' and a faster decline for a sprinter and the opposite for a marathon paddler. Profile and performance level only act in this mode, that is only when the efforts alone cannot decide.

Results of the model comparison

The two tables below are the main results of this post. For each athlete, they show all three usable models side by side, with the previous value as a reference. The primary model is highlighted in blue.

Two rows share the same model type. Without support points means the 3-parameter model is fitted to the measured efforts alone, within the parameter limits and the W' corridor. With support points adds the points from the previous section. The difference between the two rows therefore shows exactly what the additional information contributes.

Table 4 Model comparison, Athlete 01

Table 5 Model comparison, Athlete 02

*The power-law model has no asymptote. The value shown is the power at the 20-min reference duration. All R² values and deviations are calculated on the measured efforts only, so that no model is rewarded for hitting its own support points.

Compared with the previous value, critical power falls by 32 W (15%) ▼ for Athlete 01 and by 12 W (9%) ▼ for Athlete 02, while W' more than doubles ▲ for him and increases by roughly 28% ▲ for her. This is inevitable, because a hyperbola with a lower asymptote requires a larger curvature parameter (W'), and any statement about one parameter is always a statement about the other as well. The primary model lies 5.1% below and 7.7% above the 20-minute power of the power-law model, respectively. What the support points achieve is shown in the P(30 min) column. There, the unsupported hyperbola lies 15 and 32% above the power-law model, the supported one only 11 and 22%, and Figure 5 shows how the support points hold the primary curve between the two in the gray region without measurements.

Figure 5 Power-duration profile of Athlete 01 (top) and Athlete 02 (bottom), laid out as in the app. The best-power curve and key efforts (large circles) are colored by their deviation from the primary model, dark red below and violet above, with ±3% spanning the scale. Blue is the primary model, pink dashed the power-law model and dotted gray the model without support points. Open circles are support points, open triangles minimum powers and the diamonds the peak power. The gray area marks the range without a maximal effort.

The colors in Figure 5 also show where the hyperbola reaches its limits. For Athlete 01, the curve lies slightly above the measured values between 20 and 60 s, for Athlete 02 below at her two longest efforts. One reason may be that she did not paddle her one- and three-minute efforts fully maximally, as the first minute of her 6-min test was only 6% below her best one-minute power. In the plausibility check, the primary model nevertheless passes all criteria for Athlete 02 and narrowly misses one for Athlete 01. The key practical point is that three efforts are not enough for a stable profile. A profile only becomes reliable with a minimum of five distinct efforts.

How long does the longest effort need to be?

To estimate this, we progressively truncated the best-power curves of both athletes in the app↗. Below 60 s, the app does not return a result, because no asymptote can be estimated from sprints alone. With a longest effort of 1 to 4 min, critical power was still off by up to a little over 10%, from about 5 min only by a few percent. To anchor the asymptote in a measurement rather than in model support, add a maximal effort of 10 to 15 min. With two athletes and truncated curves only, this is a guide, not a validation, since each truncated curve is compared with the fit to the full data.

The profile and the metabolic thresholds

Table 6 Power profile and metabolic thresholds side by side

For the ratios and the time to CP, green marks the range expected from the literature, yellow its upper part and red values outside it.

The new critical power lies above the measured LT₂ consensus value for both athletes, by 8% for Athlete 01 and 24% for Athlete 02. Compared with 27 and 36% before, this is a clear improvement, but for Athlete 02 the gap remains relatively large and sits in the upper part of the range reported in the literature. Compared directly with the modeled maximal lactate steady state, the quantity the literature refers to, the gap is 4 and 5%. Studies report an average of about 7%, with study means between 1 and 16% (Dekerle et al., 2003; Jones et al., 2019). Both values therefore fall within the range the literature leads us to expect, whereas the previous ones were well above it at 22 and 15% ▲ above MLSS and 27 and 36% ▲ above LT₂. Equating critical power, MLSS and the second lactate threshold works as an approximation but is wrong as a definition, which is why this post keeps the three quantities separate throughout.

None of the models in this article uses a measured threshold as an input. Efforts, power-law model, support points and hyperbola come from the power data alone, and the measured thresholds are used only for the comparison in Table 6 and for the context criteria of the plausibility check. A power meter can therefore deliver a complete power profile, but it does not replace a threshold test.

Limitations

• The edges of the data window. No maximal effort lasted longer than six minutes, whereas 12 to 15 min is recommended (Leo et al., 2022), so the asymptote relies on model support rather than on a measurement. At the short end, the first seconds are smoothed, which makes P_max an imprecise measure.
• The long-duration end relies on the model. The support points come from the power-law model fitted to the same data, and the longest efforts require more W' than the hyperbola provides. Beyond the measured range, the curve should therefore be interpreted with caution, especially where the power-law model describes the measured efforts less well. The comparison with and without support points in Tables 4 and 5 gives an impression of their influence.
• W' as a quantity. The corridor spans a factor of about three and depends on efficiency and the active muscle fraction, neither of which has been validated specifically for kayaking. Even between different models, W' varies widely for the same data (Sreedhara et al., 2019). The reported W' is therefore of limited use for W' balance calculations, and for Athlete 02 it clearly underestimates the work she performed above CP in her longest effort.
• Sample size. Two athletes without repeated measurements, although CP and W' already vary by about 3 and 8% between repeated laboratory time trials (Triska et al., 2017), and for Athlete 02 a paced all-out test that still provided one of her four key efforts. For Athlete 01, six of the seven key efforts come from the same test and are therefore not independent of one another.

What comes next

After this post, we have two independent descriptions of the same athlete, a set of metabolically determined thresholds and a purely mechanically determined power-duration profile. Both consist of points and curves, not ranges. The final post brings them together and derives a canoe-specific intensity scale from them, using several independent approaches whose results are then compared and combined. A training zone table with three, five and seven zones, calculated from your own recordings, is already available in the PaddlePulse app↗.

Appendix

This appendix contains the background to the individual building blocks that would have overloaded the main text, followed by two reference tables.

The corridor for the anaerobic work capacity

W' is defined as the finite amount of work that can be done above critical power. This definition says nothing about what it corresponds to physiologically. It is commonly attributed to two stores, an alactic one from high-energy phosphates and a lactic one from glycolysis (Green, 1995; Gastin, 2001). The available evidence supports this attribution but does not fully explain W'. Skiba et al. (2015) found that the available W' was closely linked to the breakdown of creatine phosphate but not to pH, and that creatine phosphate recovered much faster than W'. The phosphate store alone therefore does not explain W'. During repeated muscle contractions, fatigue developed only slowly below the critical limit, the counterpart of critical power, but rapidly above it until the task had to be stopped (Burnley et al., 2012). This also suggests that a distinct, limited reserve is used up above this limit. As in the main text, the following calculation should therefore be read as a corridor and not as a measurement of W'.

The reference quantity is the active muscle mass. It is derived from the skeletal muscle mass of the segmental bioimpedance measurement, of which only part is active in kayaking, and can be estimated from fat-free mass when no measurement is available (Janssen et al., 2000). The metabolically available energy is then translated into mechanical work using the individually measured mechanical efficiency.

Both stores are estimated per kilogram of active muscle from literature values.

• Alactic. The resting creatine phosphate content of the muscle, minus a small residue that remains even at exhaustion (Hultman et al., 1967; Bogdanis et al., 1996; Heck, 2022), converted into energy using the free energy of ATP hydrolysis, which drops below its resting value as fatigue progresses (Mader, 2003).
• Lactic. The amount of lactate the muscle tolerates until exhaustion (Heck, 2022), converted via the stoichiometry of glycolysis from glycogen, which yields 1.5 ATP per lactate formed (Brooks, 2012).

The lower and upper bounds of both stores give the floor and ceiling of the corridor, extended by a buffer and a reserve factor for oxygen stores and uncertainties. We deliberately keep the exact numbers a little secret here 😉. If you are particularly curious how they come about, feel free to get in touch. For Athlete 01 this gives 7.85 to 23.14 kJ, for Athlete 02 6.32 to 18.63 kJ.

The width of the corridors is striking, a factor of about three for both athletes, because even with additional physiological information the anaerobic work capacity remains a fuzzy quantity (Green, 1995; Chorley & Lamb, 2020; Leo et al., 2022). The corridor is therefore less an estimate of W' than an exclusion criterion. The values of the simple fit, 9.17 and 5.38 kJ, lie at its lower edge and below it, respectively, which is further evidence against that fit, and its upper bound is the yardstick of the W' pair test.

The support points

Peak power as a bound. Treated as an ordinary data point, the least certain value of the profile would distort CP and W' through the curvature at the short end. The de-smoothed peak therefore only caps P_max from above, and the fitted P_max of 579 and 509 W lies well below it (Figure 5, filled diamond).

Minimum powers in the gaps. Every stretch of the duration axis without a key effort receives one-sided minimum powers that the model must not fall below but can freely exceed. In the interior, this is the measured value of the best-power curve, and below the shortest effort an additional back-calculation with the flattest plausible endurance exponent according to Riegel applies, with the larger value taken. For Athlete 02, this also places points between 15 and 58 s and between 58 and 177 s, such as 262 W at 29.5 s (Figure 5, open triangles).

At the long end, points on the athlete's own power-law model. Beyond the longest effort, points from the power-law model are placed up to 30 min, and these are the only ones that enter as two-sided targets. There are two for each athlete, for Athlete 01 211 W at 706 s and 183 W at 1318 s. Their weight grows with the share of the axis the efforts leave uncovered but stays well below that of the measurements. They stabilize the asymptote without overriding or dominating empirical measurements.

The 30-min cap is not arbitrary. Jones et al. (2019) state that exhaustion at critical power occurs after at most about 20 to 30 min, and Drake et al. (2024) show that the hyperbola increasingly misbehaves beyond that.

Weighting by duration range

For Athlete 01, five of the seven key efforts lie between 8 and 111 s, and with equal weighting per point, the long-duration domain governing the asymptote would be statistically overwhelmed five to one by short-duration efforts. Each effort is therefore weighted by the stretch of the logarithmic duration axis it represents. Several points in one range together count only slightly more than a single one, and efforts under 30 s are down-weighted. As a result, the longest effort carries the greatest weight for Athlete 01.

Checks for unfavorable data sets

These safeguards are intended for data sets in which not every effort was maximal or the test battery is too short, such as training recordings instead of tests.

• The decay limit. If the curve drops more steeply than human endurance allows, the end of the recording has been reached, and that part is cut off.
• The W' pair test. If two efforts together require more W' than the corridor allows, the longer one was not maximal. In the borderline range, it only counts with reduced weight.
• The short battery. If the longest effort is very short, there is no fit or only a cautiously bounded one with a warning (how long the longest effort should be).
• The weight cap. No single effort may determine the result on its own.
• The lower bound for CP. A critical power for which the longest effort would need an implausibly large anaerobic reserve is ruled out.

The fatigue exponent in detail

Table A1 Parameters of the power-law model and the resulting fatigue factors

Power-based and speed-based fatigue factors are not directly comparable. The reference values from running are based on speed. In a boat, however, the power required rises roughly with the cube of speed, so the same drop in speed looks about three times as steep on the power axis. Converting with the exponent derived from towing measurements on the K1 (Gomes et al., 2015) turns the seemingly extreme 1.303 and 1.298 into 1.082 and 1.081, very close to the value of 1.083 for female runners (Table A3). At first glance sprint paddlers appear to experience much steeper fatigue, but this is largely an artifact of dimensional units.

Converting between the power and the speed axis. The two exponents are linked through the power exponent n, and n = 3.062 gives the values in Table A1.

E_power − 1 = n · (E_speed − 1)

Gomes et al. (2015) described the passive drag of a towed K1 as a power function of speed, D = k · vⁿ. Because power is drag times speed, the exponent for power is n + 1, which averages 3.062 over the usable curves. During active paddling, drag is higher than in the towing test. How this affects the exponent remains open.

The fatigue factor is not a constant but a local slope. In the men's running world records, it falls from about 1.23 between 400 and 800 m to 1.06 between 5000 m and the marathon. Riegel's 1.077 is an average over race durations of about 3.5 to 130 min, while our fit covers 8 s to 6 min. That the two numbers coincide is encouraging, but not conclusive.

The lower half of Table A3 shows something unexpected. The K1 world best times decline much more steeply than our two athletes, with power-based factors of 1.40 to 1.58 instead of about 1.30. The conversion exponent hardly explains this, because a higher, active exponent would even widen the gap. Two explanations are more plausible. First, 200 to 1000 m only cover durations of about 35 s to three and a half minutes, where every power-duration curve is particularly steep, whereas our exponent is a slope averaged over 8 s to 6 min and the support points. Second, the long end of our profiles may be optimistic, because no measurement goes beyond six minutes and the support points merely extend the decline of the athlete's own power-law model. A single maximal effort of 12 to 15 min would resolve both questions.

The plausibility check

Whether a fit is usable is decided by a set of ratios whose bands come from a synthetic athlete. The six core criteria in Table A2 need no threshold measurement, and the strictest of them requires that the power held for the longest effort lies above the asymptote. Five further context criteria check the result against measured thresholds and correspond to the ratios in Table 6. The primary model passes all eleven for Athlete 02 and narrowly misses only the W'/CP band for Athlete 01. Reported but not scored is the duration at which the athlete's own power-law model reaches critical power, 25.1 min for Athlete 01, within the expected range of 20 to 40 min (Brickley et al., 2002; Jones et al., 2019), and only 14.5 min for Athlete 02.

Table A2 The six core criteria of the plausibility check and how the primary model performs

*Outside the band. **The band depends on the duration of the longest effort and is shown here for six minutes. The bands come from a synthetic athlete across a wide range of endurance exponents, sampled with the same durations as the real test batteries.

Table A3 Fatigue factors from world records, speed-based

Upper half after Riegel (1981, Table 1). Lower half from the K1 world best times, converted with n = 3.062.

References

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