Abstract
The SOC change index, defined as the normalized difference between the actual Soil Organic Carbon and the value assumed at an initial reference year, is here tailored to the RothC carbon model dynamics. It assumes as a baseline the value of the SOC equilibrium under constant environmental conditions. A sensitivity analysis is performed to evaluate the response of the model to changes in temperature, Net Primary Production (NPP), and land use soil class (forest, grassland, arable). A non-standard monthly time-stepping procedure has been proposed to approximate the SOC change index in the Alta Murgia National Park, a protected area in the Italian Apulia region, selected as a test site. The SOC change index exhibits negative trends for all the land use considered without fertilizers. The negative trend in the arable class can be inverted by a suitable organic fertilization program here proposed.
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Notes
\(\frac {\partial \rho ^{(1)}(r)} {\partial Temp^{(1)}} =\frac {106.06}{47.91} (k_{a}(Temp^{(1)}))^{2} k_{b}(Acc^{(1)})k_{c}(r)\) \(\frac {e^{\frac {106.06}{Temp^{(1)}+\frac {106.06}{log(46.91)} -Temp^{(0)} } }}{(Temp^{(1)}+\frac {106.06}{log(46.91)} -Temp^{(0)})^{2}}>0\).
\(\displaystyle \frac {\partial \rho }{\partial r}(Temp^{(1)},r)= k_{a}(Temp^{(1)})k_{b}({Acc^{(1)}}) N_{b} \displaystyle \frac {e^{x(r)}}{r^{2}\left (1+e^{x(r)}\right )^{2}},\) \( x(r)=\displaystyle \frac {30(r-1)}{r}\)
In a leap year \(t_{m}^{(n)}:= t_{0}+nT+ \displaystyle \frac {T}{366} \displaystyle \sum \limits _{i=1}^{m} N_{i}\) and N2 = 29.
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Acknowledgements
The authors would like to thank the referee for valuable comments which helped to improve the presentation of the results. This study was carried out within the framework of the H2020 project “Earth Observation Training in Science and Technology at the Space Research Centre of the Polish Academy of Sciences’—’EOTiST”’. The project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 952111. The work of Angela Martiradonna has been carried out within the project “Modelli differenziali per la salvaguardia della biodiversità minacciata dalle specie invasive nelle aree protette. Utilizzo di dati satellitari per l’analisi di scenario e il controllo ottimo nel Parco Nazionale dell’Alta Murgia” funded by “Research for Innovation” (REFIN) progamme n. 0C46E06B- Regione Puglia, Italy.
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Appendix
Appendix
1.1 A.1 Proof of Theorem 2
By plugging the expression of P(t0 + nT) into the Eq. 12, for all t ∈ [t0 + nT,t0 + (n + 1)T], and \(n=1,2,\dots \), we have
Thus,
Recalling the relation between P(t0) and c(t0) in Eq. 9 that yields
we have
1.2 A.2 Proof of Theorem 3
From
it results that, for t = t0 + T + 𝜖,
consequently,
Notice that the eigenvalues of A, are negative and distinct from each other (λ1 < λ2 < λ3 < λ4 < 0), therefore there is a nonsingular matrix \(V\in \mathbb {R}^{4\times 4}\) such that A = V− 1EV, where E is the diagonal matrix whose diagonal components are the eigenvalues of A. Consequently, φ(B(ε)) is diagonalizable and its eigenvalues are positive and less than one. In fact,
where \(\varphi \left (\varepsilon \rho ^{(1)}(r) E\right )\) is diagonal and its components are 0 < φ(ερ(1)(r)λi) < 1, for \(i=1,\dots ,4\) because ερ(1)(r)λi < 0.
Now, let us consider the norm on \(\mathbb {R}^{4}\) such that for all vector \(\mathbf {x}\in \mathbb {R}^{4}, \|\mathbf {x}\|:=\|V\mathbf {x}\|_{2}\) and the corresponding matrix norm that for all \(M\in \mathbb {R}^{4\times 4}, \|M\|= \|VMV^{-1}\|_{2}\). We have that
Moreover, since all norms on finite-dimensional vector spaces are equivalent, there exists a constant c > 0 such that for all vector \(\mathbf {x}\in \mathbb {R}^{4}\) it results that ∥v∥≤ c∥x∥2, in particular ∥a(g)∥ < c∥a(g)∥2 = c. So,
1.3 A.3 Proof of Theorem 4
Since the sensitivity of Δsoc to Temp(1) is defined as \(s_{\Delta soc,Temp^{(1)}}=\mathbbm {1}^{\intercal } \mathbf {s}_{\Delta \mathbf {\overline c},Temp^{(1)}}\), let us begin by obtaining the initial value problem for \(\mathbf {s}_{\Delta \mathbf {\overline c},Temp^{(1)}}\). According to Eq. 15, applied to Eq. 17, we have that for all t ∈]t0 + T,t0 + 2T]
where
Thus, for all t ∈]t0 + T,t0 + 2T]
By multiplying both sides of the previous equation by \(\mathbbm {1}^{\intercal }\), and by recalling that \(\mathbbm {1}^{\intercal } A = -\delta \mathbf {k}^{\intercal }\), and \(\mathbbm {1}^{\intercal }\mathbf {a}^{(g)}=1\), Eq. 21 is proved.
For proving the second part of the statement, let us consider the expression of \({\Delta } \mathbf {\overline c} (t) \) in Eq. 19.
By setting \(\psi (t):=\mathbf {k}^{\intercal }\varphi \left (\rho ^{(1)}(r) A(t-t_{0}-T)\right )\mathbf {a}^{(g)}\), we have that
and, by replacing 𝜗(1) with its definition in Eq. 18, Eq. 21 becomes
Consider that \(\psi (t_{0}+T)= \mathbf {k}^{\intercal } \mathbf {a}^{(g)} >0\), then, by continuity, there is a number \(\bar \epsilon >0\) such that ψ(t) > 0 for all \(t\in ]t_{0}+T, t_{0}+T+\bar \epsilon ]\). By defining \(k_{min}:=\min \limits _{i}{\mathbf {k}_{i}}\), then \(\mathbf {k}^{\intercal } \mathbf {s}_{\Delta \overline {\mathbf {c}},Temp^{(1)}}\geq k_{min}\mathbbm {1}^{\intercal } \mathbf {s}_{\Delta \mathbf {\overline c},Temp^{(1)}}\), and \(\delta (t-t_{0}-T) \psi (t)N_{P}^{(1)}>0\) for all \(t\in ]t_{0}+T,t_{0}+T+\bar \epsilon ]\). It follows that
for all \(t\in ]t_{0}+T,t_{0}+T+\bar \epsilon ]\).
Notice that the function (t − t0 − T)ψ(t) is positive for all \(t\in ]t_{0}+T,t_{0}+T+ \bar \epsilon ]\) and it is equal to zero at t = t0 + T. Since \(\displaystyle \frac {1}{\delta \rho ^{(1)}(r)}>0\), there exists an 𝜖 > 0 such that \((t-t_{0}-T) \psi (t) \leq \displaystyle \frac {1}{\delta \rho ^{(1)}(r)}\) for all t ∈]t0 + T,t0 + T + 𝜖]. Thus, exploiting the positivityFootnote 1 of \(\displaystyle \frac {\partial \rho ^{(1)}(r)}{\partial Temp^{(1)}}\), we have that
The solution of the Cauchy problem \(\displaystyle \frac {dx}{dt}=-\rho ^{(1)}(r)\delta k_{min} x\), with x(t0 + T) = 0, is the function x(t) ≡ 0, for all t ∈ [t0 + T,t0 + T + 𝜖]. Since \(s_{\Delta soc,Temp^{(1)}}(t_{0}+T)= x(t_{0}+T) =0 \), we have that \(s_{\Delta soc,Temp^{(1)}}\leq x(t)=0\), for all t ∈ [t0 + T,t0 + T + 𝜖].
1.4 A.4 Proof of Theorem 5
At first, let us consider the sensitivity of \({\Delta } \mathbf {\overline c} \) to \(N_{P}^{(1)}\), which satisfies the following initial value problem
according to Eq. 15 applied to Eq. 17.
By recalling that \(\mathbbm {1}^{\intercal } A =-\delta \mathbf {k}^{\intercal }\) and \(\mathbbm {1}^{\intercal } \mathbf {a}^{(g)}=1\) it is easy to see that \(s_{\Delta soc,N_{P}^{(1)}}\) satisfies the initial value problem (22).
To complete the proof, let us define \(k_{max}:=\max \limits _{i}{\mathbf {k}_{i}}\). Thus,
for all t ∈]t0 + T,t0 + 2T]. Since \(s_{\Delta soc,N_{P}^{(1)}}(t_{0}+T)= 0\), we have that \(s_{\Delta soc,N_{P}^{(1)}}\geq 0\) for all t ∈ [t0 + T,t0 + 2T].
1.5 A.5 Proof of Theorem 6
Let us begin by obtaining the initial value problem for \(\mathbf {s}_{\Delta \mathbf {\overline c},r}\). According to Eqs. 17 applied to Eq. 17, we have that
where
and \(\mathbf {v}:= [1, -1, 0, 0]^{\intercal }\). Thus, we have that
By multiplying both sides of the above equation by \(\mathbbm {1}^{\intercal }\), and recalling that \(\mathbbm {1}^{\intercal } A = -\delta \mathbf {k}^{\intercal }\), and \(\mathbbm {1}^{\intercal } \mathbf {v}=0\), Eq. 23 is proved.
For the second part of the proof, as in the proof of Theorem 4, there exists an 𝜖 > 0 such that for all t ∈]t0 + T,t0 + T + 𝜖] the sign of the function
is the same as the sign of 𝜗(1). For this reason, we distinguish the two cases: 𝜗(1) ≥ 0 and 𝜗(1) < 0. Let us observe that \(\displaystyle \frac {\partial \rho ^{(1)}(r)}{\partial r}>0\)Footnote 2 so that, when 𝜗(1) ≥ 0, it results
Since sΔsoc,r(t0 + T) = 0, we have that sΔsoc,r(t) ≤ 0 for all t ∈ [t0 + T,t0 + T + 𝜖]. If 𝜗(1) < 0, then \(\begin {array}{ll} \displaystyle \frac {d s_{\Delta soc,r} }{dt} &\geq -\rho ^{(1)}(r)\delta k_{max}s_{\Delta soc,r} \end {array}\) so that, as sΔsoc,r(t0 + T) = 0, then sΔsoc,r(t0 + T) ≥ 0, for all t ∈ [t0 + T,t0 + T + 𝜖] and this completes the proof.
1.6 A.6 Thornthwaite’s formula for estimating the potential evapotranspiration
We need to estimate the potential evapotranspiration pet(t), [mmmonth− 1], estimated by means of the Thornthwaite’s formula which is expressed, for the n th year, on a monthly basis at the instants \(t_{m}^{(n)}:= t_{0}+nT+ \displaystyle \frac {T}{365} \displaystyle \sum \limits _{i=1}^{m} N_{i}\) with \(m = 1,\dots , 12\) and Ni denoting the number of days of the i th month of the n th year,Footnote 3 as follows:
In the above formula, \(L_{d,m}^{(n)}\) and \(Temp_{d,m}^{(n)}\) represent the average day length (hours) and the average daily temperature of the m th month of the n th year, respectively. Finally, In is the heat index for the n th year given by
where \(Temp_{k}^{(n)}:= \displaystyle \frac {\displaystyle {\int \limits }_{t_{k-1}^{(n)}}^{t_{k}^{(n)}} Temp(s) ds}{t_{k}^{(n)}- t_{k-1}^{(n)}}\) is the k th monthly mean temperature, for \(k=1,\dots , 12\). Finally,
1.7 A.7 Estimation of the accumulate soil moisture deficit
The accumulate soil moisture deficit in the n th year, is also estimated on a monthly basis at the instants \(t_{m}^{(n)}:= t_{0}+nT+ \displaystyle \frac {T}{365} \displaystyle \sum \limits _{i=1}^{m} N_{i}\) with \(m = 1,\dots , 12\). Then \(Acc(t_{m}^{(n)},M)=0\) for all \(m=1,\dots ,\bar m\) such that \(pet(t_{m}^{(n)})\leq rain(t_{m}^{(n)})\), while
for \(m=\bar m +1,\dots ,12\).
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Diele, F., Luiso, I., Marangi, C. et al. Evaluating the impact of increasing temperatures on changes in Soil Organic Carbon stocks: sensitivity analysis and non-standard discrete approximation. Comput Geosci 26, 1345–1366 (2022). https://doi.org/10.1007/s10596-022-10165-3
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DOI: https://doi.org/10.1007/s10596-022-10165-3