docs

# DART Data Assimilation — Experiment Documentation
## Overview

This documentation covers data assimilation experiments conducted using
**DART (Data Assimilation Research Testbed)** — an open-source ensemble
DA framework developed by NCAR/UCAR.

Three models were studied in order of increasing complexity:

| Model | Variables | Dimensions | Purpose |
|-------|-----------|------------|---------|
| [Lorenz 63](lorenz63.md) | 3 (x, y, z) | 0D — 3 equations | Learn DA basics |
| [Lorenz 96](lorenz96.md) | 40 (x1–x40) | 1D — latitude circle | Test inflation and localization |
| [B-grid Solo](bgrid.md) | 28,200 (Ps, T, u, v) | 3D — global atmosphere | Real atmospheric DA |

---

## What is Data Assimilation?

Data assimilation (DA) mathematically combines two imperfect sources of
information to produce the best possible estimate of the true atmospheric state:

1. **A computer model** — simulates the atmosphere but drifts from reality
2. **Observations** — real measurements but sparse and noisy

DART uses the **Ensemble Kalman Filter (EnKF)** — running many slightly
different model forecasts (an ensemble) and using their spread to estimate
uncertainty, then correcting them toward observations.

---

## System Setup

All experiments were run on the **FSU Research Computing Center (RCC) HPC**
via the OnDemand web interface.

### Loading Required Modules

```bash
module load precompiled
module load intel/21
module load openmpi/4.1.6
module load netcdf/4.7.0
module load python/3
which mpif90   # verify — should return a path
```
Add to `~/.bashrc` to load automatically:

```bash
module load intel/21
module load openmpi/4.1.6
```

### DART Installation Path

```
/gpfs/home/ea25e/space/DART/
```

---

## Key Concepts

### Ensemble Size vs Model Size

| Term | Parameter | Meaning |
|------|-----------|---------|
| `model_size` | `&model_nml` | Number of variables in ONE model run |
| `ens_size` | `&filter_nml` | Number of parallel model runs |

### RMSE

Root Mean Square Error — measures how far the ensemble mean is from truth.
Lower = better. Computed separately for prior (before DA) and posterior (after DA).

### Spread/RMSE Ratio

- **Target: 1.0** — ensemble spread matches RMSE (well calibrated)
- **< 0.8** — underdispersive (ensemble too confident)
- **> 1.2** — overdispersive (ensemble too uncertain)

### Inflation

Artificially increases ensemble spread to prevent collapse over time.
Controlled by `inf_flavor` (method) and `inf_initial` (starting strength).

### Localization

Limits how far each observation's influence reaches spatially.
Controlled by `cutoff` in `&assim_tools_nml`.

---

## Navigation

- [Lorenz 63 Experiment](lorenz63.md)
- [Lorenz 96 Experiment](lorenz96.md)
- [B-grid Solo Experiment](bgrid.md)# Lorenz 96 Model

**Model directory:** `DART/models/lorenz_96/work/`  
**Date:** May 2026  

---
# Lorenz 63 Model

**Study:** Sensitivity of Observation Error and Observation Frequency in Chaotic Models  
**Model directory:** `DART/models/lorenz_63/work/`  
**Date:** May 2026  

---

## Objective

To investigate how **observation error variance (R)** and **observation frequency**
affect data assimilation performance in a chaotic system using the Lorenz 63 model and DART.

---

## What is Lorenz 63?

The Lorenz 63 model is a **3-variable nonlinear chaotic system** originally
designed to model atmospheric convection. It is highly sensitive to initial
conditions — small differences in starting state grow rapidly over time,
making it an ideal testbed for data assimilation methods.

### Governing Equations

```
dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz

Default parameters: σ=10, ρ=28, β=8/3
```

### Key difference from Lorenz 96

| Feature | Lorenz 63 | Lorenz 96 |
|---------|-----------|-----------|
| Variables | 3 (x, y, z) | 40 (x1–x40) |
| Dimensions | 0D — 3 equations | 1D — latitude circle |
| Obs file | Created interactively | Pre-built file |
| Purpose | Sensitivity studies | Inflation/localization |

---

## Data Assimilation Framework

DART was used to:

- Generate a synthetic nature run (truth)
- Create observations with Gaussian noise
- Run the Ensemble Kalman Filter (EnKF)
- Compute RMSE and spread diagnostics

---

## Workflow

### Step 1 — Compile

```bash
module load intel/21
module load openmpi/4.1.6
cd /gpfs/home/ea25e/space/DART/models/lorenz_63/work/
./quickbuild.sh
```

### Step 2 — Create Observation Sequence

```bash
./create_obs_sequence
```

Interactive prompts answered:

```
Input upper bound on number of observations : 1000
Input number of copies of data              : 0
Input number of quality control values      : 0
Input observation type                      : RAW_STATE_VARIABLE
Input error variance                        : (varied per experiment)
input a -1 if there are no more obs         : -1
Input filename for sequence                 : set_def.out
```

### Step 3 — Create Fixed Network Sequence

```bash
./create_fixed_network_seq
```

```
Input filename for network definition : (Enter — use set_def.out)
Regular repeating sequence            : 1
Number of observation times           : 1000
Initial time (days, seconds)          : 0 0
Period (days, seconds)                : 0 3600
Output filename                       : obs_seq.in
```

### Step 4 — Generate Truth and Observations

```bash
./perfect_model_obs
```

Runs the true Lorenz 63 trajectory forward and samples synthetic
observations with Gaussian noise. Produces `true_state.nc` and `obs_seq.out`.

### Step 5 — Run Data Assimilation

```bash
./filter
```

Runs the ensemble Kalman filter. Produces `preassim.nc` and `analysis.nc`.

---

## Experiment 1 — Observation Error Variance

### Purpose

Determine how changing observation error variance **R** affects
assimilation performance. Four values of R were tested.

### Key Concept

- **Large R** → DART trusts observations less → weaker corrections
- **Small R** → DART trusts observations more → stronger corrections

### Results

| R | Prior RMSE | Posterior RMSE |
|---|------------|----------------|
| 8.0 | 0.1780 | 0.1730 | 2.8% |
| 2.0 | 0.0981 | 0.0943 | 3.9% |
| 0.5 | 0.0535 | 0.0504 | 5.8% |
| 0.2 | 0.0339 | 0.0310 | **8.6% ✓ Best** |

### Conclusion

- Smaller observation error variance produced lower RMSE across all cases
- Posterior RMSE was always smaller than prior RMSE — assimilation always helped
- As R decreased from 8.0 to 0.2 the prior RMSE dropped by **81%**
- The filter gave more weight to observations when R was small — pulling
  the ensemble closer to truth

---

## Experiment 2 — Observation Frequency

### Purpose

Study how the number of observations affects forecast error growth
and assimilation performance.

### Setup

Two observation network sizes were tested:

| Setup | Observations | Description |
|-------|-------------|-------------|
| Sparse | 50 observations | Few measurements |
| Dense | 200 observations | More frequent updates |

### Key Findings

- **Fewer observations** caused larger forecast errors — less information
  to constrain the ensemble
- **More frequent observations** reduced error growth — ensemble tracked
  truth more closely
- Frequent updates prevented the chaotic model from drifting too far
  from the true state between corrections

---

## Evaluation Metrics

### RMSE (Root Mean Square Error)

Measures the difference between the ensemble mean and the true state.
**Lower RMSE = better assimilation performance.**

```
RMSE = sqrt( mean( (ensemble_mean - truth)^2 ) )
```

Computed separately for:
- **Prior RMSE** — before assimilating observations
- **Posterior RMSE** — after assimilating observations

### Ensemble Spread

Measures the uncertainty in the ensemble forecast — how spread out
the 20 members are around their mean. A well calibrated ensemble has:

```
Spread ≈ RMSE   (Spread/RMSE ratio close to 1.0)
```

---

## Overall Conclusions

| Finding | Detail |
|---------|--------|
| Observation error effect | Smaller R → better RMSE. R=0.2 gave 81% lower prior RMSE than R=8.0 |
| Observation frequency effect | More observations → better error control and truth tracking |
| Posterior always better | In all experiments posterior RMSE < prior RMSE — DA always helped |
| Chaotic sensitivity | Lorenz 63's chaos makes observation quality critical — small R matters a lot |

> **Key message:** In chaotic systems like Lorenz 63, the quality and
> frequency of observations strongly controls how well data assimilation
> can track the true state. This motivates careful observation network
> design in real atmospheric DA systems.

---

## Next Steps

- [ ] Test additional R values between 0.2 and 0.5
- [ ] Compare with Lorenz 96 sensitivity results
- [ ] Test impact of ensemble size on sensitivity to R
- [ ] Apply findings to guide observation setup in B-grid experiments

## What is Lorenz 96?

The Lorenz 96 model consists of **40 variables** arranged around a
latitude circle, each representing an atmospheric quantity at one site.
The variables influence each other through a single equation, producing
chaotic (weather-like) behavior when forcing F ≥ 8.

It is used as a **toy atmosphere** — simple enough to run in seconds
but realistic enough to test data assimilation methods.

---

## Governing Equation

```
dx(j)/dt = (x(j+1) - x(j-2)) * x(j-1) - x(j) + F

where j = 1...40  (cyclic: x(41) = x(1))

Terms:
  (x(j+1) - x(j-2)) * x(j-1)  →  advection
  - x(j)                        →  dissipation
  + F                            →  external forcing
```

---

## Key Parameters (input.nml)

| Parameter | Default | Namelist | Description |
|-----------|---------|----------|-------------|
| `model_size` | 40 | `&model_nml` | Number of variables |
| `forcing` | 8.0 | `&model_nml` | F ≥ 8 = chaotic behavior |
| `delta_t` | 0.05 | `&model_nml` | Internal RK time step |
| `time_step_seconds` | 3600 | `&model_nml` | 1 hour per DA cycle |
| `ens_size` | 20 | `&filter_nml` | Ensemble members |
| `cutoff` | 0.20 | `&assim_tools_nml` | Localization half-width |

---

## Observation Setup

Used pre-built observation file: `obs_seq_identity_1_20.in`

| Property | Value |
|----------|-------|
| Observed variables | x(1) to x(20) — first 20 of 40 |
| Unobserved variables | x(21) to x(40) — second 20 of 40 |
| Observation error variance | 1.0 for all observations |
| Observations per cycle | 20 |
| Total cycles | 1000 |
| Total observations | 20,000 |

---

## Workflow

### Step 1 — Compile

```bash
module load intel/21
module load openmpi/4.1.6
cd /gpfs/home/ea25e/space/DART/models/lorenz_96/work/
./quickbuild.sh
```

**Note:** Intel/21 compiler required. GNU compiler causes
`'-assume buffered_io'` error. OpenMPI/4.1.6 must be loaded after Intel.

### Step 2 — Set Up Observations
## Pre-built Observation Files

Three observation sequence files are available in the work directory.
Copy your preferred file to `obs_seq.in` before running `perfect_model_obs`:

| File | Observed | Error Variance | Description |
|------|----------|----------------|-------------|
| `obs_seq_identity_1_20.in` | x(1)–x(20) | 1.0 | 20 of 40 variables — used in these experiments |
| `obs_seq_identity_1_40_2.in` | x(1)–x(40) | 2.0 | All 40 variables — moderate noise |
| `obs_seq_identity_1_40_4.in` | x(1)–x(40) | 4.0 | All 40 variables — higher noise |

To switch observation files simply run:

```bash
cp obs_seq_identity_1_40_2.in obs_seq.in
```

Then rerun `perfect_model_obs` and `filter`.

Example;
```bash
cp obs_seq_identity_1_20.in obs_seq.in
```

Observes x(1)–x(20) with observation error variance = 1.0.

### Step 3 — Generate Truth and Observations

```bash
./perfect_model_obs
```

**Produces:**
- `true_state.nc` — true atmosphere (shape: 1000 × 1 × 40)
- `obs_seq.out` — synthetic observations with noise

### Step 4 — Run Data Assimilation

```bash
./filter
```

**Produces:**
- `preassim.nc` — prior ensemble (shape: 1000 × 20 × 40)
- `analysis.nc` — posterior ensemble (shape: 1000 × 20 × 40)
- `obs_seq.final` — observations with DA diagnostics

### Step 5 — Save Results

```bash
mkdir /gpfs/home/ea25e/Trials/L_96/exp1_no_inflation
cp preassim.nc analysis.nc obs_seq.final true_state.nc \
   /gpfs/home/ea25e/Trials/L_96/exp1_no_inflation/
```

---

## Experiments

### Experiment 1 — Baseline (No Inflation)

**Settings:** `inf_flavor=0` | `cutoff=0.20` | `ens_size=20`

| Variable Group | Prior RMSE | Post RMSE | Improvement |
|----------------|------------|-----------|-------------|
| All 40 variables | 1.819 | 1.795 | 1.3% |
| Observed x(1)–x(20) | 0.404 | 0.325 | **19.6%** |
| Unobserved x(21)–x(40) | 2.532 | 2.511 | 0.8% |

**Spread/RMSE ratio:** 0.653

---

### Experiment 2 — With Inflation

**Settings:** `inf_flavor=2` | `inf_initial=1.0` | `cutoff=0.20`

Change in `input.nml`:
```fortran
inf_flavor = 2,    0,    ! was 0, 0
inf_initial = 1.0, 1.0,
```

| Variable Group | Prior RMSE | Post RMSE | 
|----------------|------------|-----------|
| All 40 variables | 1.840 | 1.816 |
| Observed x(1)–x(20) | 0.397 | 0.317 |
| Unobserved x(21)–x(40) | 2.564 | 2.544 |

**Spread/RMSE ratio:** 0.750 — improved but still underdispersive ✗

**Key finding:** Inflation increased spread (1.187 → 1.379) as expected.
Slight increase in overall RMSE is normal — members pushed further from truth.

---

### Experiment 3 — Inflation + Wider Localization

**Settings:** `inf_flavor=2` | `inf_initial=1.0` | `cutoff=0.30`

Change in `input.nml`:
```fortran
cutoff = 0.30    ! was 0.20
```

| Variable Group | Prior RMSE | Post RMSE | 
|----------------|------------|-----------|
| All 40 variables | 1.799 | 1.777 |
| Observed x(1)–x(20) | 0.394 | 0.313 | 
| Unobserved x(21)–x(40) | 2.507 | 2.488 |

**Key finding:** Wider localization (cutoff=0.30 ≈ 12 variables either side)
allowed observations to reach further into the unobserved region.

---

## Key Findings

1. **Observed variables (x1–x20)** showed 19–21% RMSE improvement
   across all experiments — direct observation correction works well

2. **Unobserved variables (x21–x40)** showed only 0.8% improvement
   in all cases — the root cause is lack of observations, not settings

3. **Inflation** improved Spread/RMSE ratio from 0.653 → 0.750
   — moving toward the target of 1.0

4. **Wider localization** (cutoff=0.30) gave best observed RMSE (0.313)

5. **The dominant problem** is observation coverage — the recommended
   next step is observing all 40 variables using `obs_seq_identity_1_40_2.in`

---

## Output File Reference

| File | Shape | Contents |
|------|-------|----------|
| `true_state.nc` | (1000, 1, 40) | Truth trajectory |
| `preassim.nc` | (1000, 20, 40) | Prior ensemble + mean + sd |
| `analysis.nc` | (1000, 20, 40) | Posterior ensemble + mean + sd |
| `obs_seq.out` | — | Synthetic observations |
| `obs_seq.final` | — | Obs + model predictions |

---

## Visualization

Python scripts were developed in Jupyter on FSU RCC to visualize results.
Key plots produced:

- **Time series** — truth vs prior vs posterior for observed and unobserved variables
- **RMSE over time** — prior and posterior RMSE across 1000 cycles
- **Spread vs RMSE** — ensemble calibration diagnostic
- **Split RMSE** — separate panels for all 40, observed, and unobserved variables
- **Snapshot** — state space at final cycle with longitude axis (0°–360°)

---

## Next Steps

- [ ] Observe all 40 variables: `cp obs_seq_identity_1_40_2.in obs_seq.in`
- [ ] Test stronger inflation: `inf_initial = 1.05`
- [ ] Ensemble size sensitivity: `ens_size = 10, 20, 40, 80`
- [ ] Compare with Lorenz 63 results

# B-grid Solo Model

**Model directory:** `DART/models/bgrid_solo/work/`  
**Date:** May 2026  

---

## What is B-grid Solo?

The B-grid Solo model is a **simplified 3D atmospheric model** based on
GFDL's B-grid dynamical core. It is the next step up in complexity from
Lorenz 96 — bridging the gap between toy models and full weather models.

| Feature | Lorenz 96 | B-grid Solo |
|---------|-----------|-------------|
| Dimensions | 1D circle | 3D atmosphere |
| Variables | 40 (one type) | 28,200 (four types) |
| Variable types | state only | Ps, T, u, v |
| Obs types | RAW_STATE | Radiosonde |
| Run time | Seconds | Minutes |

---

## Model Grid

```
Grid structure: 60 lon × 30 lat × 5 lev
                ↑         ↑        ↑
         longitude     latitude   vertical

Total variables: ~60 × 30 × 5 × 4 = 36,000
Actual model size: 28,200 (winds on VelJ=29 grid)
```

### Why VelJ = 29 (not 30)?

The B-grid uses **Arakawa B-grid staggering** — temperature and pressure
(T, Ps) sit at grid cell centers, while winds (u, v) sit at cell corners.
Corners of a 30-point latitude grid give 29 interior boundaries.

---

## State Variables

| Symbol | Name | Units | Dimensions | Notes |
|--------|------|-------|------------|-------|
| Ps | Surface Pressure | Pa | 2D (lat × lon) | Ground level only — no vertical dim |
| T | Temperature | K | 3D (lev × lat × lon) | Different at each level |
| u | Zonal Wind | m/s | 3D (lev × VelJ × lon) | East-west component |
| v | Meridional Wind | m/s | 3D (lev × VelJ × lon) | North-south component |

### Why is Ps 2D?

Surface pressure is the **total weight of the atmosphere** pressing down
on one point at the ground. It is measured at the surface only — there is
no such thing as "surface pressure at level 3." Hence no vertical dimension.

---

## Held-Suarez Forcing

The model uses **Held-Suarez forcing** — a standard benchmark for
atmospheric model testing (Held & Suarez, 1994).

- Maintains temperature gradient between tropics and poles
- Relaxes atmosphere toward an equilibrium temperature profile
- Creates realistic jet streams and weather-like patterns
- No moisture, clouds, or ocean — simplified but physically meaningful

Key parameters in `input.nml`:

```fortran
&hs_forcing_nml
   delh   = 60.    ! temperature gradient pole-to-equator
   t_zero = 315.   ! reference temperature (K)
   ka     = -40.   ! atmospheric relaxation timescale
/
```

---

## Key Parameters (input.nml)

| Parameter | Value | Namelist | Description |
|-----------|-------|----------|-------------|
| `nlon` | 60 | `&bgrid_cold_start_nml` | Longitude points |
| `nlat` | 30 | `&bgrid_cold_start_nml` | Latitude points |
| `nlev` | 5 | `&bgrid_cold_start_nml` | Vertical levels |
| `dt_atmos` | 3600 | `&model_nml` | 1 hour time step |
| `days` | 10 | `&model_nml` | Intended run length |
| `ens_size` | 20 | `&filter_nml` | Ensemble members |
| `inf_flavor` | 0 | `&filter_nml` | No inflation (baseline) |
| `cutoff` | 0.20 | `&assim_tools_nml` | ~1145 km localization radius |
| `perturb_from_single_instance` | .true. | `&filter_nml` | Create ensemble by perturbing |
| `perturbation_amplitude` | 0.2 | `&filter_nml` | Ensemble perturbation size |

---

## Observations

The pre-built `obs_seq.in` file (1.54 MB) contains a real
**radiosonde observation network**.

| Observation Type | Variable | Description |
|-----------------|----------|-------------|
| `RADIOSONDE_TEMPERATURE` | T | Temperature from balloon soundings (K) |
| `RADIOSONDE_U_WIND_COMPONENT` | u | Zonal wind from balloons (m/s) |
| `RADIOSONDE_V_WIND_COMPONENT` | v | Meridional wind from balloons (m/s) |
| `RADIOSONDE_SURFACE_PRESSURE` | Ps | Surface pressure (Pa) |

**Statistics:**

- Total observations: 9,600
- Assimilation cycles completed: 2
- Observations per cycle: 4,800
- QC threshold: 3.0

**Key difference from Lorenz 96:** B-grid uses real radiosonde observation
types at 3D locations (lat, lon, pressure level) vs synthetic identity
observations at 1D locations in Lorenz 96.

---

## Workflow

### Step 1 — Compile

```bash
module load intel/21
module load openmpi/4.1.6
cd /gpfs/home/ea25e/space/DART/models/bgrid_solo/work/
./quickbuild.sh
```

Compiles 15 DART executables (more than Lorenz 96's 11).

### Step 2 — Create NetCDF Input File

```bash
ncgen -o perfect_input.nc perfect_input.cdl
```

Converts the CDL text description to NetCDF binary format.
`perfect_input.cdl` ships with DART and contains a valid B-grid
atmospheric state to use as the starting point.

### Step 3 — Generate Truth and Observations

```bash
./perfect_model_obs
```

**Output:**

```
Model size = 28,200
Total obs in sequence = 9,600
2 assimilation windows completed
```

**Produces:**
- `true_state.nc` — truth (shape: 2 × 1 × 30 × 60 for Ps)
- `obs_seq.out` — synthetic observations with noise

### Step 4 — Create Ensemble Starting Point ⚠️ B-grid specific

```bash
cp perfect_input.nc filter_input.nc
```

**This step is unique to B-grid — not needed in Lorenz 96.**

The filter needs `filter_input.nc` to create its initial ensemble.
Because `perturb_from_single_instance = .true.`, it reads this single
state and creates 20 members by adding random noise (amplitude = 0.2).

### Step 5 — Run Data Assimilation

```bash
./filter
```

Only 2 cycles completed — `obs_seq.in` contained observations for
2 assimilation windows only. The `days = 10` setting in the namelist
was overridden by the available observations.

**Produces:**
- `preassim.nc` — prior ensemble (shape: 2 × 20 × 30 × 60 for Ps)
- `analysis.nc` — posterior ensemble
- `obs_seq.final` — observations with DA diagnostics
- `filter_output.nc` — final ensemble state

---

## Why Only 2 Cycles?

The 2 cycles and 5 levels are **completely independent**:

- **2 cycles** = how many times the forecast → observe → correct loop ran in **time**
- **5 levels** = vertical structure of the model at **each moment in time**

Every cycle, all 5 levels were assimilated simultaneously. The 2 just
means it happened twice. To run more cycles, a new `obs_seq.in` file
with more observation times is needed.

---

## Experiment 1 — Default Run Results

**Settings:** `inf_flavor=0` | `cutoff=0.20` | `ens_size=20` | 2 cycles

| Variable | Max Error | Units | Assessment |
|----------|-----------|-------|------------|
| Surface Pressure (Ps) | 1.298 | Pa | ~0.001% of total — excellent |
| Temperature (T) Level 3 | 0.007 | K | 7 thousandths of a degree — excellent |
| U Wind Level 3 | 0.003 | m/s | ~0.01% of typical winds — excellent |
| V Wind Level 3 | 0.003 | m/s | ~0.01% of typical winds — excellent |

> **Note:** Errors appear very small because only 2 cycles ran.
> The ensemble started close to truth (copied from `perfect_input.nc`)
> and had minimal time to diverge. Running 100+ cycles would produce
> more realistic error statistics.

### Error Map Interpretation

- **Red patches** = posterior overestimates truth
- **Blue patches** = posterior underestimates truth
- **White** = near zero error (ideal)
- **Scattered pattern** = random residual errors, no systematic geographic bias
- **Vertical striping in U wind** = B-grid grid structure signature

---

## Output File Reference

| File | Variable shapes | Contents |
|------|----------------|----------|
| `true_state.nc` | ps:(2,1,30,60) t:(2,1,5,30,60) | Truth trajectory |
| `preassim.nc` | ps:(2,20,30,60) t:(2,20,5,30,60) | Prior ensemble + mean + sd |
| `analysis.nc` | ps:(2,20,30,60) t:(2,20,5,30,60) | Posterior ensemble + mean + sd |
| `filter_output.nc` | ps:(1,20,30,60) | Final ensemble state |
| `perfect_output.nc` | ps:(1,1,30,60) | Final truth state |

---

## Key Findings

1. **B-grid successfully compiled** on FSU RCC with Intel/21 + OpenMPI/4.1.6
2. **Extra step required:** `cp perfect_input.nc filter_input.nc` — unique to B-grid
3. **Only 2 cycles ran** — limited by observations in `obs_seq.in`
4. **All 5 vertical levels assimilated** in both cycles — levels and cycles are independent
5. **Surface pressure is 2D** — exists only at ground level, no vertical dimension
6. **Errors very small** at 2 cycles — ensemble had not diverged from truth yet
7. **RMSE near zero at time step 0** — ensemble identical to truth at start (both from same file)

---

## Next Steps

- [ ] Run more cycles — generate new `obs_seq.in` with longer time coverage
- [ ] Experiment 2: Add inflation (`inf_flavor=2`, `inf_initial=1.0`)
- [ ] Experiment 3: Observation impact — remove wind obs or surface pressure
- [ ] Experiment 4: Localization sensitivity (`cutoff=0.10` vs `0.20` vs `0.40`)
- [ ] Experiment 5: Ensemble size (`ens_size=10, 20, 40, 80`)

---

## References

- Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison
  of the dynamical cores of atmospheric general circulation models.
  *Bulletin of the American Meteorological Society*, 75(10), 1825–1830.

- Anderson, J. L., and Coauthors, 2009: The Data Assimilation Research
  Testbed: A community facility. *Bulletin of the American Meteorological
  Society*, 90(9), 1283–1296.