Pipe Roughness Identification of Water Distribution Networks: The Full Turbulent Case

Martin Horn; Martin Steinberger; Stefan Kaltenbacher

REVIEW 3 major objections 4 minor 28 references

Pipe Roughness Identification of Water Distribution Networks: The Full Turbulent Case

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under steady fully turbulent flow, individual pipe roughnesses in a water distribution network can be recovered by inverting the hydraulic equations using pressure measurements at a subset of nodes, provided enough independent measurement…

desk verdict Useful new formulation for per-pipe roughness identification from pressure measurements, but the convergence proof has a genuine flaw and the reliability claim outruns the evidence. read the letter →

arxiv 1908.07794 v2 pith:THBMFBZD submitted 2019-08-21 eess.SY cs.SY

Stefan Kaltenbacher , Martin Steinberger , Martin Horn This is my paper · ORCID

classification eess.SYcs.SY MSC 65H1093B30

keywords RoughnesscalibrationWaterdistributionnetworksParameteridentificationColebrook-WhiteDarcy-WeisbachHydraulicfrictionparametersNewton-RaphsonSteady-statehydraulics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Working in the steady state and assuming the flow in every pipe is fully turbulent, the paper claims that individual roughness values for all pipes of a water distribution network can be solved for directly from the hydraulic network equations, using pressure measurements at a subset of nodes and known nodal consumption. The key step is to rewrite the turbulent head-loss relation as the explicit Colebrook–White flow function $f_t(\epsilon,\Delta h)$, stack the nodal mass-balance equations over several loading conditions, and count unknowns: at least $n_m=\lceil n_\ell/n_p\rceil$ independent measurement sets are needed. The authors propose a modified Newton–Raphson scheme with an $\ell^1$-norm backtracking line search and multi-start initialization, and show on an eight-pipe example that all eight roughnesses and the unmeasured pressure heads are recovered, with the largest roughness error about 6% in the noise-free case. If correct, this makes per-pipe roughness calibration from a sparse pressure-sensor layout feasible in the fully turbulent regime, which would improve model-based leakage detection and localization.

What carries the argument

The load-bearing object is the stacked equation set (16). For each measurement set $i$, it combines nodal mass conservation $A x_Q(\boldsymbol\epsilon,h_N^{(i)})=\bar q^{(i)}$ with the head-loss relation $\Delta h^{(i)}=\tilde C_s h_s^{(i)}-A^T C_h^T y_h^{(i)}-A^T\bar C_h^T h_N^{(i)}-A^T z$ and defines each pipe flow by the explicit turbulent Colebrook–White function $f_t(\epsilon_j,\Delta h_j)$ from equation (12). The stacked residual $f(x)$ has a tall rectangular Jacobian $J(x)$ whose block structure consists of repeated incidence matrices $A$ multiplying blocks that are diagonal in the roughness derivatives and dense in the unmeasured-head derivatives. The paper's counting rule $n_{m,\min}=\lceil n_\ell/n_p\rceil$ is what balances equations against unknowns, and the modified Newton algorithm uses the left inverse $(J^T J)^{-1}J^T$ with a backtracking line search based on the $\ell^1$ norm, plus physical box constraints on roughness and unmeasured heads to steer iterates back to the real root.

What would settle it

Compute the numerical rank of $J(x^*)$ from equation (20) at the true roughnesses and the measurement sets in Table 2; if the rank is below $n_\ell+n_m(n_j-n_p)$, then some nonzero direction $d$ satisfies $Jd=0$ and two different roughness/pressure configurations produce identical measurements, disproving unique identifiability for that configuration. A simpler experimental check is to run Algorithm 2 with measurement sets that differ only by a uniform scaling of all consumptions: if the returned roughnesses vary with random starts, the sets are not sufficiently independent.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the inversion problem is solvable as stated: with sufficiently independent steady-state measurement sets, solving the stacked equation system (16) recovers the true roughness vector $\boldsymbol\epsilon\in\mathbb{R}^{n_\ell}_{\ge 0}$ together with all unmeasured pressure heads $h_N^{(i)}$ in every measurement set. The discovery is that the implicit Colebrook–White friction law can be converted into an explicit flow function $f_t$, which makes the Jacobian of the stacked residual computable and enables a Newton-type solver. The authors show that the minimum number of measurement sets is $n_{m,\min}=\lceil n_\ell/n_p\rceil$, independent of the number of nodes, and they provide two algorithms: a modified Newton method with step-length control, and an outer multi-start procedure that reinitializes only the roughnesses that leave their physical range. In the noise-free three-cycle simulation, the algorithms converge to the real root; the authors state in the conclusion that the algorithms find the real root reliably whenever independent measurement-sets are available.

Load-bearing premise

Everything rests on Assumption 5, that the available measurement sets are "sufficiently linearly independent" so the stacked Jacobian has full column rank; this is asserted rather than proved, and the counting rule alone does not guarantee it for a given topology, sensor placement, or load pattern.

Editorial extensions

If this is right

The counting rule $n_{m,\min}=\lceil n_\ell/n_p\rceil$ means the required number of loading scenarios depends only on the ratio of pipes to pressure sensors, not on the number of nodes.
With that many sufficiently independent measurement sets, all $n_\ell$ roughnesses and all $n_m(n_j-n_p)$ unmeasured pressure heads are identifiable from pressure measurements alone.
Using the explicit turbulent flow function $f_t$ avoids iterating on the implicit friction factor inside each function evaluation, which keeps the Jacobian construction tractable.
Adding more measurement sets beyond the minimum, with clearly separated measured heads, improves robustness to noise and helps distinguish the true root from local minima, at the cost of a larger equation set.
Extending the method to real networks requires a sufficiently smooth explicit model for laminar and transitional flows, as the paper states in its outlook.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not prove that Assumption 5 holds for a given network; before relying on the calibration, an operator should check the numerical rank of the stacked Jacobian for the chosen sensor placement and load patterns.
Because nodal consumption is assumed known, field application will likely be limited by consumption estimation error; fireflow tests, in which hydrant outflows are deliberately measured, are the natural setting where this assumption is approximately met.
The block structure of the Jacobian suggests each measurement set contributes an independent block coupled only through the shared roughness vector, so the solver could be parallelized or solved with a Schur-complement reduction on large networks.
A deterministic alternative to the random multi-start outer loop, such as continuing from one load scenario to the next by gradually changing consumption, might recover the global root with fewer function evaluations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know before reading.

The useful part is the explicit Colebrook-White turbulent flow expression (12) and the calibration problem (16) built on it. That formulation, together with the counting rule nm,min = ceil(n_l/n_p) in (17), is new and cleanly stated. The paper also does well in making its assumptions explicit, including the unrealistic but clearly acknowledged assumption that nodal consumption is known, and in restricting the scope to full turbulent flow. The one detailed example is reproducible in principle and shows the method can work when the rank condition holds.

The weak part is the solver analysis. The paper claims the algorithms reliably find the real root, but the supporting calculation in Eq. (26) is wrong for the overdetermined case. The Jacobian is rectangular, the Newton step uses the left inverse, and the directional derivative along that step is -sign(f)^T J (J^T J)^-1 J^T f, not -sign(f)^T f. The line search therefore rests on a false premise. The algorithm may still work, and the example suggests it does, but the convergence proof as written does not establish the claimed reliability. The stress-test note lands.

Other soft spots are proportional. Assumption 5, the linear independence of measurement sets, is asserted rather than proved, and it is genuinely load-bearing: the counting rule is only necessary, not sufficient. The paper does not verify the rank condition for its example beyond the successful run. The evaluation is a single noise-free simulation generated with the authors' own dynamic model, whose equilibrium is equivalent to the steady-state equations being inverted; that is not circular numerically, but it does not test generalizability. There is no noise study, no comparison against existing implicit or explicit calibration methods, and no shipped code. Those are standard gaps, not fatal flaws in the problem formulation itself.

Bottom line: the mathematical setup and the counting argument are worth publishing, and the explicit turbulent flow formulation is a genuine contribution. But the reliability claim needs a corrected descent analysis, rank verification, and noise experiments before it can be taken seriously. The paper deserves a serious referee, not a desk reject. I would send it to review with the expectation of heavy revision.

For a reader: someone working on model calibration or leakage detection in water networks, or on identifiability of inverse problems with redundant measurements, will get value from the formulation and the counting rule. I would not cite the solver, but I would cite the explicit flow expression if I needed it.

Referee Report

3 major / 4 minor

Summary. The paper proposes an explicit inversion scheme for identifying per-pipe roughness coefficients in water distribution networks under fully turbulent flow. The authors formulate the steady-state hydraulic equations with Darcy-Weisbach friction and the implicit Colebrook-White relation rewritten in explicit turbulent flow form, obtaining a system (16) whose unknowns are the per-pipe roughnesses and the unmeasured nodal pressure heads for each measurement set. They derive the minimum number of independent measurement sets as nm = ceil(n_l / n_p), propose a modified Newton-Raphson algorithm with an L1 line search and a multi-start outer loop (Algorithms 1 and 2), and test it on a synthetic three-cycle network with eight pipes, three pressure sensors, and three measurement sets, reporting recovery of all roughnesses with a maximal deviation of about 6%.

Significance. The problem formulation is a useful addition to explicit calibration methods: it avoids the separate simulation-optimization loop typical of implicit calibration, makes the role of the number of measurement sets explicit, and the Jacobian structure in Eq. (20) is clearly laid out. The counting rule for independent measurement sets is simple and algorithmically relevant. However, the central reliability claim in Section 7 is not supported to the stated level. The convergence argument for the algorithm contains a mathematical error, identifiability is assumed rather than proven, and the numerical evidence is a single noise-free synthetic example. The contribution is therefore promising but requires substantial strengthening before the claimed reliability can be accepted.

major comments (3)

[Section 5.2, Eq. (26)] The descent-direction calculation in Eq. (26) is invalid for the rectangular system used here. With the left-inverse Newton direction delta_x = -J^+ f, where J^+ = (J^T J)^{-1} J^T and J has size n_m n_j by (n_l + n_m(n_j - n_p)), the directional derivative of v = ||f||_{L1} is -sign(f)^T J J^+ f, not -sign(f)^T f. Since J J^+ is the orthogonal projection onto the range of J, the equality used in Eq. (26), and consequently g'(0) in Eq. (29) and the acceptance criterion (28), hold only when f lies in the column space of J, which is not generally true at a non-root iterate of an overdetermined system. The line-search control in Algorithm 1 therefore rests on an unverified premise, and the claim in Section 7 that the algorithms 'enable to find the real root reliably' is not supported.
[Assumption 5 and Eq. (20)] The full-column-rank condition on the stacked Jacobian in Eq. (20) is asserted, not proved. The counting inequality in Eq. (17) is necessary but not sufficient: it does not guarantee that a given network topology, sensor placement, and set of load variations produce a linearly independent stacked Jacobian. The paper does not verify the rank condition for the three-cycle example and does not characterize network or measurement conditions under which Assumption 5 holds. Because this assumption is the only mechanism separating pipe roughnesses from unmeasured nodal pressure heads, the identifiability of the problem is not established; if the rank condition fails, Eq. (16) is underdetermined and the algorithm cannot distinguish roughness from pressure unknowns.
[Section 6, Tables 3-4] The numerical evaluation is limited to a single noise-free synthetic scenario, yet Assumption 7 concerns measurement noise and the conclusion claims reliability under independent measurement sets. No noisy case is presented, and the authors themselves note in Section 6.3 that with measurement noise 'one certainly loses the capability to identify the real root by only looking at v(x)'. Furthermore, the simulated measurements are generated using the authors' own dynamic model from [15,16], whose equilibrium is equivalent to the steady-state equations (2) used for inversion; this is a self-consistency check rather than an independent validation. A Monte Carlo study over noise realizations, initial values, and network configurations, together with rank verification for each configuration, would be needed to substantiate the reliability claim.

minor comments (4)

[Section 4.2, Eq. (17)] The derivation of nm,min = ceil(n_l / n_p) is compressed; the underlying break-even condition m n_p >= n_l is not stated explicitly, which would make the counting argument easier to verify.
[Section 5.1, Eq. (22b) and Eq. (23)] The statement that the derivative of Delta_h with respect to h_N is constant is correct given Eq. (16b), but it would be clearer to state that this constancy relies on the measured heads y_h and source heads h_s being fixed inputs in the inversion.
[Algorithm 1, line 17 and Remark 4] The treatment of roughness values as absolute values is described only informally in Remark 4; the manuscript should specify exactly which components are reflected and how this reflection affects the Jacobian computation in later iterations.
[General presentation] The manuscript contains numerous typographical and formatting errors, for example 'ndeﬁne' in the Notation section and inconsistent units in the presentation of v(x) in Table 4; a careful proofread would improve readability.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant derivation circularity; the inversion is built on external physical relations, with only a minor in-sample validation via the authors' own dynamic model.

other [Section 6, Simulation Example (measurement generation and model equivalence)]
"In order to produce an independent set of steady-state configurations (“measurements”), a dynamic model is utilized which has been derived in [15, 16] while varying the desired consumption qd. ... In this context, the equivalence of the solution of (2) to the equilibrium of the dynamic model has been proven in [15, Theorem 2]."

The simulated measurements used to validate the inversion are generated from the authors' own dynamic model [15,16], and a self-citation, [15, Theorem 2], is used to assert that the equilibrium of that model coincides with the steady-state equations (2) being inverted. Thus the measurement data are in-sample for the same model family: the true parameter vector x* is a root of the residual (16) by construction, so the algorithm's successful recovery is a consistency check rather than independent confirmation of the reliability claim. This does not affect the derivation of the inversion, which is based on the external Darcy-Weisbach and Colebrook-White relations plus an explicit rank assumption; it is a minor validation-circularity rather than a circular derivation.

full rationale

The central derivation is self-contained and not circular. The paper sets up the inversion by combining standard external physical relations: Darcy-Weisbach (6), Colebrook-White (8), and the derived turbulent flow expression (12). It then builds the residual (19) and its Jacobian (20) directly from these equations, and the unknowns (pipe roughnesses and unmeasured heads) are exactly the variables appearing in those equations. There is no fitted parameter renamed as a prediction, and no result is defined in terms of the quantity it purports to derive. The rank/identifiability condition is explicitly stated as Assumption 5 and the counting rule (17) is only a necessary condition; the paper does not pretend to prove uniqueness from the counting alone. The only self-referential element is the simulation example, where the 'measurements' are produced by the authors' own dynamic model [15,16] whose steady-state solution is equivalent to the inverted equations (2) by [15, Theorem 2]. This makes the numerical validation in-sample for the model family and weakens the demonstrated reliability, but the paper is transparent that this is a simulation example and the derivation itself does not reduce to its inputs. The separate concern about Eq. (26) concerns the correctness of the descent-direction argument for a rectangular Jacobian; it is a mathematical validity issue, not a circularity of derivation.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central claim rests on eight numbered assumptions in Sections 2 and 3. The most fragile is Assumption 5 (full rank or independence of measurement sets), which is the identifiability condition and is asserted rather than proven. Several practical assumptions (known consumption, negligible minor losses, negligible noise, full turbulence) are acknowledged to be unrealistic in real networks. No new physical entities are introduced.

free parameters (4)

Algorithm initial roughness fraction = 0.01d (1% of pipe diameter)
The initial guess for roughness is chosen as 1% of each pipe's diameter (Section 6.2), not derived from data; convergence of the Newton scheme depends on this and on the random restarts.
Roughness search upper bound = 0.05d (5% of diameter)
The physically useful range for roughness is set to 0-5% of pipe diameter, referencing the Moody chart (Section 5.3); this bound is a hand-chosen constraint and not a measured quantity.
Initial h_N bounds = min/max of neighboring measured heads
Lower and upper bounds for unmeasured pressure heads are chosen from surrounding sensor values (Sections 5.3 and 6.2), a heuristic that influences the solution region.
Newton line-search and stopping constants = 10^-4 Armijo constant, 0.1 and 0.5 step bounds, eps_f=1e-7, eps_x=5e-7, 1000 iterations
These numerical constants from Algorithm 1 and Numerical Recipes [28] control convergence; they are hand-chosen and no sensitivity analysis is given.

assumptions (8)

domain assumption Assumption 1: The network graph is connected, has no self-loops, and has at least one source node.
Needed for rank(A)=nj and for the cycle-equation formulation; standard for distribution networks.
domain assumption Assumption 2: The friction function is strictly monotonic, continuous, at least once differentiable, and vanishes at zero flow.
Suffices for the steady-state network equations to have a unique flow solution; a standard functional property.
domain assumption Assumption 3: Minor losses in all pipes are negligible.
Reduces the unknowns to one roughness per pipe; common in the field but unverified for real networks with fittings and valves.
domain assumption Assumption 4: Pipe dimensions, topology, nodal elevation, source heads, and nodal consumption are known.
The paper explicitly admits that known consumption is unrealistic today; the whole inverse formulation depends on it.
ad hoc to paper Assumption 5: The nm measurement sets are sufficiently linearly independent, i.e., the stacked Jacobian in (20) has full column rank.
This is the central identifiability assumption. It is asserted, not proved; no topology, sensor placement, or loading conditions are derived that guarantee the rank condition.
domain assumption Assumption 6: Each measurement set is taken in steady state.
Required for the steady-state equations to apply; fireflow tests are intended to approximate this.
domain assumption Assumption 7: Pressure and fireflow measurement noise has zero mean and is much smaller than the pressure drops.
All simulations are noise-free; the assumption is acknowledged unrealistic for real sensors.
domain assumption Assumption 8: Every pipe flow in every measurement set is fully turbulent, Re >= 4000.
Needed for the Colebrook-White flow expression (12); laminar and transitional flow regimes are excluded.

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Pith. "Pith review of Pipe Roughness Identification of Water Distribution Networks: The Full Turbulent Case." pith.science (2026). https://pith.science/paper/THBMFBZD

@misc{pith2026190807794,
  author       = {Pith},
  title        = {Pith review of: Pipe Roughness Identification of Water Distribution Networks: The Full Turbulent Case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/THBMFBZD}},
  note         = {Machine review of arXiv:1908.07794}
}

read the original abstract

This paper proposes a technique to identify individual pipe roughness parameters in a water distribution network by means of the inversion of the steady-state hydraulic network equations. By enabling the reconstruction of these hydraulic friction parameters to be reliable, this technique improves the conventional model's accuracy and thereby promises to enhance model-based leakage detection and localization. As it is the case in so-called fireflow tests, this methodology is founded on the premise to measure the pressure distributed at a subset of nodes in the network's graph while assuming the nodal consumption to be known. Beside of the proposed problem formulation, which is restricted to only allow turbulent flow in each of the network's pipes initially, developed algorithms are presented and evaluated using simulation examples.

Figures

Figures reproduced from arXiv: 1908.07794 by the authors.

**Figure 1.** Two-Cycle/Loop Network. a network with nℓ = 5 pipes numbered by P = {1, 2, . . . , 5}, nj = 3 inner nodes numbered by k ∈ I = {1, 2, 3} and one source, i.e. ns = 1, providing constant pressure head hs from reservoir R. Suppose that consumers are sitting at nodes k = 2 and k = 3 and thereby consume q2 and q3 (m3/s). Then,   −1 −1 0 0 1 1 0 −1 1 0 0 1 1 −1 0   | {z } A     Q1 . . . Q5     | {z } xQ =   0… view at source ↗

**Figure 2.** 3-cycle network with pressure sensors at red color [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Norm of f(x) plotted along the parameter space of ǫ7/d7 and h (3) N,5 regarding the network in figure 2. There is a particularly weak slope towards x ∗ along the ǫ7/d7 axes in the L2 norm of figure 3a, whereas the L1 norm in figure 3b shows the overall highest slope towards the real root x ∗ . This result is consistent, even when varying different roughnesses (one of the nℓ pipe roughnesses) and different h (i) N,j … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Figures showing nm = nm,min = 3 “measurement” -sets for the roughness identification of the 3-cycle network in figure 2. Figure 4a shows that the real consumption q follows the reference qd , whereas the steady-state consumption values applied for (16) were displayed …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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work page doi:10.1061/(asce)0733-9372(1988)114:1(120 1988

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T. Walski, Technique for Calibrating Network Models, Jour- nal of Water Resources Planning and Management - ASCE 109. doi:10.1061/(ASCE)0733-9496(1983)109:4(360)

work page doi:10.1061/(asce)0733-9496(1983)109:4(360 1983

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